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Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry

Michael Sullivan

Chapter 7

Applications of Trigonometric Functions - all with Video Answers

Educators

Chapter Questions

01:06

Problem 1

The area $K$ of a triangle whose base is $b$ and whose height is $h$ is ___________. (p. A15)

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:24

Problem 1

The amplitude $A$ and period $T$ of $f(x)=5 \sin (4 x)$ are __ and __. (pp. 451-453)

Lucas Finney
Lucas Finney
Numerade Educator
00:34

Problem 1

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
The difference formula for the sine function is $\sin (A-B)=$ (p. 532)

Foster Wisusik
Foster Wisusik
Numerade Educator
00:44

Problem 1

In a right triangle, if the length of the hypotenuse is 5 and the length of one of the other sides is 3 , what is the length of the third side? (pp.A14-A.15)

Erika Bustos
Erika Bustos
Numerade Educator
00:50

Problem 1

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
Write the formula for the distance $d$ from $P_1=\left(x_1, y_1\right)$ to $P_2=\left(x_2, y_2\right) \cdot$ (p. 3)

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:26

Problem 2

If two sides $a$ and $b$ and the included angle $C$ are known in a triangle, then the area $K$ is found using the formula $K=$ ____________.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:04

Problem 2

The motion of an object obeys the equation $d=4 \cos (6 t) . \quad$ no frictional force to retard the motion, and the motion is Such motion is described as _________ _________. The number 4 called if there is such friction. is called the ________.

Lucas Finney
Lucas Finney
Numerade Educator
03:01

Problem 2

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
If $\theta$ is an acute angle, solve the equation $\cos \theta=\frac{\sqrt{3}}{2}$. (pp. 511-516)

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:53

Problem 2

True or False $\sin 52^{\circ}=\cos 48^{\circ}$. (pp. 407-408)

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:01

Problem 2

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
If $\theta$ is an acute angle, solve the equation $\cos \theta=\frac{\sqrt{2}}{2}$. (pp. 511-516)

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:37

Problem 3

The area $K$ of a triangle with sides $a, b$, and $c$ is $K=$ _____________ , where $s=$ ______________.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:08

Problem 3

When a mass hanging from a spring is pulled down and then released, the motion is called _______ _______if there is no frictional force to retard the motion, and the motion is called _________ if there is such friction.

Lucas Finney
Lucas Finney
Numerade Educator
00:46

Problem 3

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.
The two triangles shown are similar. Find the missing length. (pp.A16-A19)

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:17

Problem 3

If $\theta$ is an acute angle, solve the equation $\tan \theta=\frac{1}{2}$. Express your answer in degrees, rounded to one decimal place. (pp. 511-516)

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
00:30

Problem 3

If three sides of a triangle are given, the Law of______ is used to solve the triangle.

AK
Avinash Koya
Numerade Educator
01:16

Problem 4

True or False The area of a triangle equals one-half the product of the lengths of two of its sides times the sine of their included angle.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:13

Problem 4

True or False If the distance $d$ of an object from its rest position at time $t$ is given by a sinusoidal graph, the motion of the object is simple harmonic motion.

Lucas Finney
Lucas Finney
Numerade Educator
00:38

Problem 4

If none of the angles of a triangle is a right angle, the triangle is called_______,
(a) oblique
(b) obtuse
(c) acute
(d) scalene

Arin Asawa
Arin Asawa
Numerade Educator
00:47

Problem 4

If $\theta$ is an acute angle, solve the equation $\sin \theta=\frac{1}{2}$.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:05

Problem 4

If one side and two angles of a triangle are given, which law can be used to solve the triangle?
(a) Law of Sines
(b) Law of Cosines
(c) Either $\mathrm{a}$ or $\mathrm{b}$
(d) The triangle cannot be solved.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
00:55

Problem 5

Given two sides of a triangle, $b$ and $c$, and the included angle $A$, the altitude $h$ from angle $B$ to side $b$ is given by _____________.
(a) $\frac{1}{2} a b \sin A$
(b) $b \sin A$
(c) $c \sin A$
(d) $\frac{1}{2} b c \sin A$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:22

Problem 5

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period $T$, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up.
$a=5 ; \quad T=2$ seconds

Lucas Finney
Lucas Finney
Numerade Educator
00:59

Problem 5

For a triangle with sides $a, b, c$ and opposite angles $A, B, C$, the Law of Sines states that_____,

Arin Asawa
Arin Asawa
Numerade Educator
00:44

Problem 5

The sum of the measures of the two acute angles in a right triangle is
(a) $45^{\circ}$
(b) $90^{\circ}$
(c) $180^{\circ}$
(d) $360^{\circ}$

Melissa Salvador
Melissa Salvador
Numerade Educator
00:46

Problem 5

If two sides and the included angle of a triangle are given, is insufficient information to solve the triangle.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
00:25

Problem 6

Heron's Formula is used to find the area of ____________ triangles.
(a) ASA
(b) SAS
(c) SSS
(d) AAS

Jonathon Brumley
Jonathon Brumley
Numerade Educator
03:24

Problem 6

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period $T$, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up.
$a=10 ; \quad T=3$ seconds

Evan Leonard
Evan Leonard
Numerade Educator
01:42

Problem 6

True or False An oblique triangle in which two sides and an angle are given always results in at least one triangle.

Raushan Kumar
Raushan Kumar
Numerade Educator
00:50

Problem 6

In navigation or surveying, the or from
a point $O$ to a point $P$ equals the acute angle $\theta$ between ray $O P$ and the vertical line through $O$, the north-south line.

Melissa Salvador
Melissa Salvador
Numerade Educator
00:36

Problem 6

True or False Given only the three sides of a triangle, there

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
00:50

Problem 7

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
02:45

Problem 7

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period $T$, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up.
$a=6 ; \quad T=\pi$ seconds

Evan Leonard
Evan Leonard
Numerade Educator
01:13

Problem 7

True or False The Law of Sines can be used to solve triangles where three sides are known.

Arin Asawa
Arin Asawa
Numerade Educator
00:45

Problem 7

True or False In a right triangle, if two sides are known, we can solve the triangle.

Melissa Salvador
Melissa Salvador
Numerade Educator
00:43

Problem 7

True or False The Law of Cosines states that the square of one side of a triangle equals the sum of the squares of the other two sides, minus twice their product.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
00:50

Problem 8

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
03:20

Problem 8

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period $T$, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up.
$a=4 ; T=\frac{\pi}{2}$ seconds

Evan Leonard
Evan Leonard
Numerade Educator
00:53

Problem 8

Triangles for which two sides and the angle opposite one of them are known (SSA) are referred to as the

Mitchell Cutler
Mitchell Cutler
Numerade Educator
00:53

Problem 8

True or False In a right triangle, if we know the two acute angles, we can solve the triangle.

Melissa Salvador
Melissa Salvador
Numerade Educator
01:00

Problem 8

True or False A special case of the Law of Cosines is the which law can be used to solve the triangle? Pythagorean Theorem.
(a) Law of Sines
(b) Law of Cosines
(c) Either a or b
(d) The triangle cannot be solved.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
00:50

Problem 9

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:19

Problem 9

Rework Problem 5 under the same conditions, except that

Lucas Finney
Lucas Finney
Numerade Educator
03:03

Problem 9

In Problems 9-16, solve each triangle.
(Figure Can't Copy)

Mitchell Cutler
Mitchell Cutler
Numerade Educator

Problem 9

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$b=5, B=20^{\circ}$;
find $a, c$, and $A$

Check back soon!

Problem 9

In Problems 9-16, solve each triangle.
(Graph Can't Copy)

Check back soon!
00:50

Problem 10

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:14

Problem 10

Rework Problem 6 under the same conditions, except that at time $t=0$, the object is at its resting position and moving down. at time $t=0$, the object is at its resting position and moving down.

Lucas Finney
Lucas Finney
Numerade Educator
02:24

Problem 10

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 10

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$b=4, B=10^{\circ} ;$ find $a, c$, and $A$

Check back soon!

Problem 10

Solve each triangle.
(Graph Can't Copy)

Check back soon!
00:50

Problem 11

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:14

Problem 11

Rework Problem 7 under the same conditions, except that

Lucas Finney
Lucas Finney
Numerade Educator
02:24

Problem 11

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 11

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
. $a=6, B=40^{\circ}$; find $b, c$, and $A$

Check back soon!

Problem 11

Solve each triangle.
(Graph Can't Copy)

Check back soon!
00:50

Problem 12

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:14

Problem 12

Rework Problem 8 under the same conditions, except that at time $t=0$, the object is at its resting position and moving down. at time $t=0$, the object is at its resting position and moving down.

Lucas Finney
Lucas Finney
Numerade Educator
02:24

Problem 12

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 12

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$a=7, B=50^{\circ}$;
find $b, c$, and $A$

Check back soon!

Problem 12

Solve each triangle.
(Graph Can't Copy)

Check back soon!
00:50

Problem 13

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:37

Problem 13

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=5 \sin (3 t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:24

Problem 13

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 13

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$b=4, A=10^{\circ}$; find $a, c$, and $B$

Check back soon!

Problem 13

Solve each triangle.
(Graph Can't Copy)

Check back soon!
00:50

Problem 14

In Problems 7-14, find the area of each triangle. Round answers to two decimal places.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
02:58

Problem 14

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=4 \sin (2 t)$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
02:24

Problem 14

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 14

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$b=6, A=20^{\circ}$;
find $a, c$, and $B$

Check back soon!

Problem 14

Solve each triangle.
(Graph Can't Copy)

Check back soon!
01:06

Problem 15

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=3, b=4, C=40^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
01:37

Problem 15

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=6 \cos (\pi t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:24

Problem 15

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 15

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$a=5, A=25^{\circ} ;$ find $b, c$, and $B$

Check back soon!

Problem 15

Solve each triangle.
(Graph Can't Copy)

Check back soon!
01:35

Problem 16

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=2, c=1, \quad B=10^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
02:21

Problem 16

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=5 \cos \left(\frac{\pi}{2} t\right)$

Evan Leonard
Evan Leonard
Numerade Educator
02:24

Problem 16

Solve each triangle.
(Figure Can't Copy)

Olga Gnedash
Olga Gnedash
Mira Loma High School

Problem 16

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$a=6, A=40^{\circ} ;$ find $b, c$, and $B$

Check back soon!

Problem 16

Solve each triangle.
(Graph Can't Copy)

Check back soon!
01:24

Problem 17

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$b=1, c=3, A=80^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
02:12

Problem 17

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=-3 \sin \left(\frac{1}{2} t\right)$

Evan Leonard
Evan Leonard
Numerade Educator
View

Problem 17

In Problems 17-24, solve each triangle.
$A=40^{\circ}, \quad B=20^{\circ}, \quad a=2$

Teresa Fuston
Teresa Fuston
Numerade Educator

Problem 17

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$c=9, \quad B=20^{\circ}$; find $b, a$, and $A$

Check back soon!
02:27

Problem 17

In Problems 17-32, solve each triangle.
$a=3, b=4, C=40^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
01:23

Problem 18

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=6, b=4, C=60^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
01:46

Problem 18

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=-2 \cos (2 t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:53

Problem 18

Solve each triangle.
$A=50^{\circ}, \quad C=20^{\circ}, \quad a=3$

Mitchell Cutler
Mitchell Cutler
Numerade Educator

Problem 18

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
. $c=10, \quad A=40^{\circ} ;$ find $b, a$, and $B$

Check back soon!
01:15

Problem 18

Solve each triangle.
18.

Sreeraj P
Sreeraj P
Numerade Educator
01:24

Problem 19

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=3, c=2, \quad B=110^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
02:46

Problem 19

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=6+2 \cos (2 \pi t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:31

Problem 19

Solve each triangle.
$B=70^{\circ}, C=10^{\circ}, \quad b=5$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
03:24

Problem 19

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$a=5, b=3 ;$ find $c, A$, and $B$

Amy Jiang
Amy Jiang
Numerade Educator
03:41

Problem 19

Solve each triangle.
$b=1, c=3, A=80^{\circ}$

AK
Avinash Koya
Numerade Educator

Problem 20

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$b=4, c=1, \quad A=120^{\circ}$

Check back soon!
02:17

Problem 20

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given.
(a) Describe the motion of the object.
(b) What is the maximum displacement from its resting position?
(c) What is the time required for one oscillation?
(d) What is the frequency?
$d=4+3 \sin (\pi t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:41

Problem 20

Solve each triangle.
$A=70^{\circ}, \quad B=60^{\circ}, c=4$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
03:57

Problem 20

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$a=2, b=8 ; \quad$ find $c, A$, and $B$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
04:58

Problem 20

Solve each triangle.
$a=6, b=4, C=60^{\circ}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator

Problem 21

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=12, b=13, c=5$

Check back soon!
06:40

Problem 21

In Problems 21-24, graph each damped vibration curve for $0 \leq t \leq 2 \pi$.
$d(t)=e^{-t / x} \cos (2 t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:11

Problem 21

Solve each triangle.$A=110^{\circ}, \quad C=30^{\circ}, \quad c=3$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
02:22

Problem 21

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$a=2, c=5 ; \quad$ find $b, A$, and $B$

Teresa Fuston
Teresa Fuston
Numerade Educator
03:18

Problem 21

Solve each triangle.
$a=3, c=2, \quad B=110^{\circ}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator

Problem 22

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=4, b=5, c=3$

Check back soon!
08:06

Problem 22

In Problems 21-24, graph each damped vibration curve for $0 \leq t \leq 2 \pi$.
$d(t)=e^{-t / 2 \pi} \cos (2 t)$

Evan Leonard
Evan Leonard
Numerade Educator
02:35

Problem 22

Solve each triangle.
$B=10^{\circ}, \quad C=100^{\circ}, \quad b=2$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
03:38

Problem 22

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle.
$b=4, c=6 ; \quad$ find $a, A$, and $B$

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
03:12

Problem 22

Solve each triangle.
$b=4, c=1, \quad A=120^{\circ}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator

Problem 23

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=2, b=2, c=2$

Check back soon!
07:21

Problem 23

In Problems 21-24, graph each damped vibration curve for $0 \leq t \leq 2 \pi$.
$d(t)=e^{-t / 2 \pi} \cos t$

Evan Leonard
Evan Leonard
Numerade Educator
02:04

Problem 23

Solve each triangle.
$A=40^{\circ}, \quad B=40^{\circ}, \quad c=2$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:23

Problem 23

Geometry The hypotenuse of a right triangle is 5 inches. If one leg is 2 inches, find the degree measure of each angle.

Melissa Salvador
Melissa Salvador
Numerade Educator
02:21

Problem 23

Solve each triangle.
$a=2, b=2, \quad C=50^{\circ}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator

Problem 24

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=3, b=3, c=2$

Check back soon!
08:46

Problem 24

In Problems 21-24, graph each damped vibration curve for $0 \leq t \leq 2 \pi$.
$d(t)=e^{-t / 4 x} \cos t$

Evan Leonard
Evan Leonard
Numerade Educator
02:45

Problem 24

Solve each triangle.
$B=20^{\circ}, \quad C=70^{\circ}, \quad a=1$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:11

Problem 24

Geometry The hypotenuse of a right triangle is 3 feet. If one $\mathrm{leg}$ is 1 foot, find the degree measure of each angle.

Melissa Salvador
Melissa Salvador
Numerade Educator
03:38

Problem 24

Solve each triangle.
$a=3, c=2, \quad B=90^{\circ}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
01:43

Problem 25

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=5, b=8, c=9$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
06:23

Problem 25

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$f(x)=x+\cos x$

Evan Leonard
Evan Leonard
Numerade Educator
View

Problem 25

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$a=3, b=2, \quad A=50^{\circ}$

Teresa Fuston
Teresa Fuston
Numerade Educator
01:23

Problem 25

Geometry A right triangle has a hypotenuse of length 8 inches. If one angle is $35^{\circ}$, find the length of each leg.

Melissa Salvador
Melissa Salvador
Numerade Educator
06:05

Problem 25

Solve each triangle.
$a=12, b=13, c=5$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator

Problem 26

In Problems 15-26, find the area of each triangle. Round answers to two decimal places.
$a=4, b=3, c=6$

Check back soon!
08:13

Problem 26

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$f(x)=x+\cos (2 x)$

Evan Leonard
Evan Leonard
Numerade Educator
01:38

Problem 26

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$b=4, c=3, \quad B=40^{\circ}$

Sreeraj P
Sreeraj P
Numerade Educator
01:12

Problem 26

Geometry A right triangle has a hypotenuse of length 10 centimeters. If one angle is $40^{\circ}$, find the length of each leg-

Jeff Harris
Jeff Harris
Numerade Educator
03:03

Problem 26

Solve each triangle.
$a=4, b=5, c=3$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:20

Problem 27

Area of an ASA Triangle If two angles and the included side are given, the third angle is easy to find. Use the Law of Sines to show that the area $K$ of a triangle with side $a$ and angles $A, B$, and $C$ is
$$
K=\frac{a^2 \sin B \sin C}{2 \sin A}
$$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
07:59

Problem 27

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$f(x)=x-\sin x$

Evan Leonard
Evan Leonard
Numerade Educator
03:56

Problem 27

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$b=5, c=3, \quad B=100^{\circ}$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
00:55

Problem 27

Finding the Angle of Elevation of the Sun At 10 Am on April 26, 2017, a building 300 feet high cast a shadow 50 feet long. What was the angle of elevation of the Sun?

Melissa Salvador
Melissa Salvador
Numerade Educator
02:18

Problem 27

Solve each triangle.
$a=2, b=2, c=2$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
03:54

Problem 28

Area of a Triangle Prove the two other forms of the formula given in Problem 27
$$
K=\frac{b^2 \sin A \sin C}{2 \sin B} \text { and } K=\frac{c^2 \sin A \sin B}{2 \sin C}
$$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
08:19

Problem 28

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$f(x)=x-\cos x$

Evan Leonard
Evan Leonard
Numerade Educator
01:28

Problem 28

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$a=2, \quad c=1, \quad A=120^{\circ}$

Sreeraj P
Sreeraj P
Numerade Educator
00:58

Problem 28

Directing a Laser Beam A laser beam is to be directed through a small hole in the center of a circle of radius $10 \mathrm{feet}$. The origin of the beam is 35 feet from the circle (see the figure). At what angle of elevation should the beam be aimed to ensure that it goes through the hole?

Melissa Salvador
Melissa Salvador
Numerade Educator
02:40

Problem 28

Solve each triangle.
$a=3, b=3, c=2$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:41

Problem 29

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=40^{\circ}, B=20^{\circ}, \quad a=2$

Teresa Fuston
Teresa Fuston
Numerade Educator
01:41

Problem 29

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=40^{\circ}, B=20^{\circ}, \quad a=2$

Teresa Fuston
Teresa Fuston
Numerade Educator
08:10

Problem 29

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$f(x)=\sin x+\cos x$

Evan Leonard
Evan Leonard
Numerade Educator
01:51

Problem 29

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$a=4, b=5, A=60^{\circ}$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
05:44

Problem 29

Finding the Speed of a Truck A state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle $\theta$ between the highway and the line of observation from the patrol car to the truck is measured. See the illustration.
(a) If the angle measures $15^{\circ}$, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.
(b) If the angle measures $20^{\circ}$, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.
(c) If the speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour or more over the limit, for what angles should the trooper issue a ticket?

Nick Johnson
Nick Johnson
Numerade Educator
02:49

Problem 29

Solve each triangle.
$a=5, b=8, c=9$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:22

Problem 30

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=50^{\circ}, C=20^{\circ}, \quad a=3$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:22

Problem 30

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=50^{\circ}, C=20^{\circ}, a=3$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
13:50

Problem 30

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$f(x)=\sin (2 x)+\cos x$

Evan Leonard
Evan Leonard
Numerade Educator
02:46

Problem 30

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$b=2, c=3, \quad B=40^{\circ}$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:28

Problem 30

Security A security camera in a neighborhood bank is mounted on a wall 9 feet above the floor. What angle of depression should be used if the camera is to be directed to a spot 6 feet above the floor and 12 feet from the wall? 14.0

Melissa Salvador
Melissa Salvador
Numerade Educator
05:01

Problem 30

Solve each triangle.
$a=4, b=3, c=6$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
01:19

Problem 31

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$B=70^{\circ}, C=10^{\circ}, \quad b=5$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:19

Problem 31

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$B=70^{\circ}, C=10^{\circ}, \quad b=5$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
11:37

Problem 31

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$g(x)=\sin x+\sin (2 x)$

Evan Leonard
Evan Leonard
Numerade Educator
03:02

Problem 31

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$b=4, c=6, \quad B=20^{\circ}$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
02:22

Problem 31

Parallax One method of measuring the distance from Earth to a star is the parallax method. The idea behind computing this distance is to measure the angle formed between Earth and the star at two different points in time. Typically, the measurements are taken so that the side opposite the angle is as large as possible. Therefore, the optimal approach is to measure the angle when Earth is on opposite sides of the Sun, as shown in the figure.
(a) Proxima Centauri is 4.22 light-years from Earth. If 1 light-year is about 5.9 trillion miles, how many miles is Proxima Centauri from Earth? $2.4898 \times 10^{13}$ miles
(b) The mean distance from Earth to the Sun is $93,000,000$ miles What is the parallax of Proxima Centauri?

Melissa Salvador
Melissa Salvador
Numerade Educator
02:41

Problem 31

Solve each triangle.
$a=10, b=8, c=5$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:09

Problem 32

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=70^{\circ}, \quad B=60^{\circ}, \quad c=4$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:09

Problem 32

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=70^{\circ}, \quad B=60^{\circ}, \quad c=4$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
07:10

Problem 32

In Problems 25-32, use the method of adding $y$-coordinates to graph each function.
$g(x)=\cos (2 x)+\cos x$

Evan Leonard
Evan Leonard
Numerade Educator
00:42

Problem 32

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$a=3, b=7, \quad A=70^{\circ}$

Sreeraj P
Sreeraj P
Numerade Educator
02:55

Problem 32

Parallax See Problem 31. The star 61 Cygni, sometimes called Bessel's Star (after Friedrich Bessel, who measured the distance from Earth to the star in 1838), is a star in the constellation Cygnus
(a) If 61 Cygni is 11.14 light-years from Earth and 1 lightyear is about 5.9 trillion miles, how many miles is 61 Cygni from Earth? $6.5726 \times 10^{13} \mathrm{mi}$
(b) The mean distance from Earth to the Sun is $93,000,000$ miles. What is the parallax of 61 Cygni? 0.000081

Melissa Salvador
Melissa Salvador
Numerade Educator
02:55

Problem 32

Solve each triangle.
$a=9, b=7, c=10$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:07

Problem 33

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=110^{\circ}, C=30^{\circ}, c=3$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:07

Problem 33

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$A=110^{\circ}, C=30^{\circ}, \quad c=3$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:10

Problem 33

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y$-coordinates to graph each function on the interval $[0,2 \pi]$.
$f(x)=\sin (2 x) \sin x$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:51

Problem 33

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$a=2, c=1, \quad C=100^{\circ}$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
05:09

Problem 33

Finding the Bearing of an Aircraft A DC-9 aircraft leaves Midway Airport from runway 4 RIGHT, whose bearing is $\mathrm{N} 40^{\circ} \mathrm{E}$. After flying for $\frac{1}{2}$ mile, the pilot requests permission to turn $90^{\circ}$ and head toward the southeast. The permission is granted.After the airplane goes 1 mile in this direction, what bearing should the control tower use to locate the aircraft?

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
02:29

Problem 33

In Problems 33-42, solve each triangle using either the Law of Sines or the Law of Cosines.
$B=20^{\circ}, C=75^{\circ}, b=5$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:07

Problem 34

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$B=10^{\circ}, C=100^{\circ}, \quad b=2$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:07

Problem 34

In Problems 29-34, use the results of Problem 27 or 28 to find the area of each triangle. Round answers to two decimal places.
$B=10^{\circ}, C=100^{\circ}, \quad b=2$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:08

Problem 34

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y$-coordinates to graph each function on the interval $[0,2 \pi]$.
$F(x)=\sin (3 x) \sin x$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:12

Problem 34

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$b=4, c=5, \quad B=95^{\circ}$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
02:40

Problem 34

Finding the Bearing of a Ship A ship leaves the port of Miami with a bearing of $\mathrm{S} 80^{\circ} \mathrm{E}$ and a speed of 15 knots. After 1 hour, the ship turns $90^{\circ}$ toward the south. After 2 hours, maintaining the same speed, what is the bearing to the ship from the port? $\$ 16.6^{\prime} \mathrm{E}$

Melissa Salvador
Melissa Salvador
Numerade Educator
03:45

Problem 34

Solve each triangle using either the Law of Sines or the Law of Cosines.
$A=50^{\circ}, B=55^{\circ}, c=9$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:56

Problem 35

Area of a Segment Find the area of the segment (shaded in blue in the figure) of a circle whose radius is 8 feet, formed by a central angle of $70^{\circ}$.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:07

Problem 35

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y$-coordinates to graph each function on the interval $[0,2 \pi]$.
$G(x)=\cos (4 x) \cos (2 x)$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:55

Problem 35

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$a=2, c=1, \quad C=25^{\circ}$

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:38

Problem 35

Niagara Falls Incline Railway Situated between Portage Road and the Niagara Parkway directly across from the Canadian Horseshoe Falls, the Falls Incline Railway is a funicular that carries passengers up an embankment to Table Rock Observation Point. If the length of the track is 51.8 meters and the angle of inclination is $36^{\circ} 2^{\prime}$, determine the height of the embankment. $30.5 \mathrm{~m}$
Source: wwwniagaraparkscom
36. Willis Tower The Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurements:
- The angle of elevation from her position to the top of the building is $34^{\circ}$.
- The distance from her position to the top of the building is 2593 feet.

Melissa Salvador
Melissa Salvador
Numerade Educator
04:28

Problem 35

Solve each triangle using either the Law of Sines or the Law of Cosines.
$a=6, b=8, c=9$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:15

Problem 36

Area of a Segment Find the area of the segment of a circle whose radius is 5 inches, formed by a central angle of $40^{\circ}$.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:04

Problem 36

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y$-coordinates to graph each function on the interval $[0,2 \pi]$.
$h(x)=\cos (2 x) \cos (x)$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:46

Problem 36

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s).
$b=4, c=5, \quad B=40^{\circ}$

Eleanor Johnson
Eleanor Johnson
Numerade Educator
05:00

Problem 36

Willis Tower The Willis Tower in Chicago is the second tallest building in the United States and is topped by a high antenna. A surveyor on the ground makes the following measurements:
- The angle of elevation from her position to the top of the building is $34^{\circ}$.
- The distance from her position to the top of the building is 2593 feet.

Melissa Salvador
Melissa Salvador
Numerade Educator
02:19

Problem 36

Solve each triangle using either the Law of Sines or the Law of Cosines.
$a=14, b=7, A=85^{\circ}$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
03:29

Problem 37

Cost of a Triangular Lot The dimensions of a triangular lot are 100 feet by 50 feet by 75 feet. If the price of such land is $$\$ 3$$ per square foot, how much does the lot cost?

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:17

Problem 37

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y$-coordinates to graph each function on the interval $[0,2 \pi]$.
$H(x)=2 \sin (3 x) \cos (x)$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
03:01

Problem 37

Finding the Length of a Ski Lift Consult the figure. To find the length of the span of a proposed ski lift from $P$ to $Q$, a surveyor measures $\angle D P Q$ to be $25^{\circ}$ and then walks back a distance of 1000 feet to $R$ and measures $\angle P R Q$ to be $15^{\circ}$. What is the distance from $P$ to $Q$ ? (Graph can't Copy)

Mitchell Cutler
Mitchell Cutler
Numerade Educator
00:59

Problem 37

Chicago Skyscrapers The angle of inclination from the base of the John Hancock Center to the top of the main structure of the Willis Tower is approximately $10.3^{\circ}$. If the main structure of the Willis Tower is 1450 feet tall, how far apart are the two skyscrapers? Assume the bases of the two buildings are at the same elevation. $7979 \mathrm{ft}$
Source: www. willistower.com

Melissa Salvador
Melissa Salvador
Numerade Educator
03:24

Problem 37

Solve each triangle using either the Law of Sines or the Law of Cosines.
$B=35^{\circ}, C=65^{\circ}, a=15$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:43

Problem 38

Amount of Material to Make a Tent A cone-shaped tent is made from a circular piece of canvas 24 feet in diameter by removing a sector with central angle $100^{\circ}$ and connecting the ends. What is the surface area of the tent?

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:36

Problem 38

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y$-coordinates to graph each function on the interval $[0,2 \pi]$.
$g(x)=2 \sin x \cos (3 x)$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
View

Problem 38

Finding the Height of a Mountain Use the illustration in Problem 37 to find the height $Q D$ of the mountain.

Teresa Fuston
Teresa Fuston
Numerade Educator
04:01

Problem 38

Estimating the Width of the Mississippi River A tourist at the top of the Gateway Arch (height $630 \mathrm{feet}$ ) in St. Louis, Missouri, observes a boat moored on the Illinois side of the Mississippi River 2070 feet directly across from the Arch. She also observes a boat moored on the Missouri side directly across from the first boat (see the figure). Given that $B=\cot ^{-1} \frac{67}{55}$, estimate the width of the Mississippi River at the St. Louis riverfront. $1650 \mathrm{ft}$
Source: U.S Army Corps of Engineers

Melissa Salvador
Melissa Salvador
Numerade Educator
05:23

Problem 38

Solve each triangle using either the Law of Sines or the Law of Cosines.
$a=4, c=5, B=55^{\circ}$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
01:35

Problem 39

Property Area A lot for sale in a subdivision has the shape of a quadrilateral as shown in the figure. Find the area of the lot to the nearest square foot.
(GRAPH CANT COPY)

Eleanor Johnson
Eleanor Johnson
Numerade Educator
06:15

Problem 39

In Problems 39-44, an object of mass $m$ (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is $T$ (in seconds) under simple harmonic motion.
(a) Write an equation that relates the displacement $d$ of the object from its rest position after $t$ seconds.
(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.
$m=25, a=10, b=0.7, \quad T=5$

Evan Leonard
Evan Leonard
Numerade Educator
05:47

Problem 39

Finding the Height of an Airplane An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. How high is the airplane? (Graph can't Copy)

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
01:22

Problem 39

Finding the Pitch of a Roof A carpenter is preparing to put a roof on a garage that is 20 feet by 40 feet by 20 feet. A steel support beam 46 feet in length is positioned in the center of the garage. To support the roof, another beam will be attached to the top of the center beam (see the figure). At what angle of elevation is the new beam? In other words, what is the pitch of the roof? $69.0^{\circ}$

Melissa Salvador
Melissa Salvador
Numerade Educator
02:24

Problem 39

Solve each triangle using either the Law of Sines or the Law of Cosines.
$A=10^{\circ}, a=3, b=10$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
04:36

Problem 40

Dimensions of Home Plate The dimensions of home plate at any major league baseball stadium are shown. Find the area of home plate.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
04:47

Problem 40

In Problems 39-44, an object of mass $m$ (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is $T$ (in seconds) under simple harmonic motion.
(a) Write an equation that relates the displacement $d$ of the object from its rest position after $t$ seconds.
(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.
$m=20, \quad a=15, \quad b=0.75, \quad T=6$

Evan Leonard
Evan Leonard
Numerade Educator
03:12

Problem 40

Finding the Height of the Bridge over the Royal Gorge The highest bridge in the world is the bridge over the Royal Gorge of the Arkansas River in Colorado. Sightings to the same point at water level directly under the bridge are taken from each side of the 880 -foot-long bridge, as indicated in the figure. How high is the bridge?
Source: Guinness Book of World Records

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:53

Problem 40

Shooting Free Throws in Basketball The eyes of a basketball player are 6 feet above the floor. The player is at the free-throw line, which is 15 feet from the center of the basket rim (see the figure). What is the angle of elevation from the player's eyes to the center of the rim?

Melissa Salvador
Melissa Salvador
Numerade Educator
02:31

Problem 40

Solve each triangle using either the Law of Sines or the Law of Cosines.
$A=65^{\circ}, B=72^{\circ}, b=7$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
02:33

Problem 41

Computing Areas See the figure. Find the area of the shaded region enclosed in a semicircle of diameter 10 inches. The length of the chord $P Q$ is 8 inches.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
05:06

Problem 41

In Problems 39-44, an object of mass $m$ (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is $T$ (in seconds) under simple harmonic motion.
(a) Write an equation that relates the displacement $d$ of the object from its rest position after $t$ seconds.
(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.
$m=30, a=18, \quad b=0.6, \quad T=4$

Evan Leonard
Evan Leonard
Numerade Educator
02:31

Problem 41

Land Dimensions A triangular plot of land has one side along a straight road measuring 200 feet. A second side makes a $50^{\circ}$ angle with the road, and the third side makes a $43^{\circ}$ angle with the road. How long are the other two sides?

Narayan Hari
Narayan Hari
Numerade Educator
02:45

Problem 41

Geometry Find the value of the angle $\theta$ (see the figure) in degrees rounded to the nearest tenth of a degree.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
03:06

Problem 41

Solve each triangle using either the Law of Sines or the Law of Cosines.
$b=5, c=12, A=60^{\circ}$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
02:46

Problem 42

Geometry See the figure, which shows a circle of radius $r$ with center at $O$. Find the area $K$ of the shaded region as a function of the central angle $\theta$.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
05:11

Problem 42

In Problems 39-44, an object of mass $m$ (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is $T$ (in seconds) under simple harmonic motion.
(a) Write an equation that relates the displacement $d$ of the object from its rest position after $t$ seconds.
(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.
$m=15, a=16, \quad b=0.65, \quad T=5$

Evan Leonard
Evan Leonard
Numerade Educator
03:33

Problem 42

Distance between Runners Two runners in a marathon determine that the angles of elevation of a news helicopter covering the race are $38^{\circ}$ and $45^{\circ}$. If the helicopter is 1700 feet directly above the finish line, how far apart are the runners?

Nick Johnson
Nick Johnson
Numerade Educator
07:42

Problem 42

Surveillance Satellites A surveillance satellite circles Earth at a height of $h$ miles above the surface. Suppose that $d$ is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration.
(a) Find an equation that relates the central angle $\theta$ (in radians) to the height $h$.
(b) Find an equation that relates the observable distance $d$ and $\theta$.
(c) Find an equation that relates $d$ and $h$.
(d) If $d$ is to be 2500 miles, how high must the satellite orbit aboye Earth?

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
02:17

Problem 42

Solve each triangle using either the Law of Sines or the Law of Cosines.
$a=10, b=10, c=15$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
12:16

Problem 43

Approximating the Area of a Lake To approximate the area of a lake, a surveyor walks around the perimeter of the lake, taking the measurements shown in the illustration. Using this technique, what is the approximate area of the lake?
(GRAPH CANT COPY)

Mutahar Mehkri
Mutahar Mehkri
Numerade Educator
05:06

Problem 43

In Problems 39-44, an object of mass $m$ (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is $T$ (in seconds) under simple harmonic motion.
(a) Write an equation that relates the displacement $d$ of the object from its rest position after $t$ seconds.
(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.
$m=10, a=5, \quad b=0.8, \quad T=3$

Evan Leonard
Evan Leonard
Numerade Educator
03:27

Problem 43

Landscaping Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is $30^{\circ}$, and from a second position 40 feet farther along this path it is $20^{\circ}$. What is the height of the tree?

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
09:17

Problem 43

Drive Wheel of an Engine The drive wheel of an engine is 13 inches in diameter, and the pulley on the rotary pump is 5 inches in diameter. If the shafts of the drive wheel and the pulley are 2 feet apart, what length of belt is required to join them as shown in the figure?

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
02:16

Problem 43

Distance to the Green A golfer hits an errant tee shot that lands in the rough. A marker in the center of the fairway is 150 yards from the center of the green. While standing on the marker and facing the green, the golfer turns $110^{\circ}$ toward his ball. He then paces off 35 yards to his ball. See the figure. How far is the ball from the center of the green?

Narayan Hari
Narayan Hari
Numerade Educator
02:02

Problem 44

Bermuda Triangle The Bermuda Triangle is roughly defined by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. The distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton are approximately 1028, 1046, and 965 miles, respectively. Ignoring the curvature of Earth, approximate the area of the Bermuda Triangle.
(GRAPH CANT COPY)

Jonathon Brumley
Jonathon Brumley
Numerade Educator
05:18

Problem 44

In Problems 39-44, an object of mass $m$ (in grams) attached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is $T$ (in seconds) under simple harmonic motion.
(a) Write an equation that relates the displacement $d$ of the object from its rest position after $t$ seconds.
(b) Graph the equation found in part (a) for 5 oscillations using a graphing utility.
$m=10, a=5, \quad b=0.7, \quad T=3$

Evan Leonard
Evan Leonard
Numerade Educator
02:14

Problem 44

Construction A loading ramp 10 feet long that makes an angle of $18^{\circ}$ with the horizontal is to be replaced by one that makes an angle of $12^{\circ}$ with the horizontal. How long is the new ramp?

Gregory Higby
Gregory Higby
Numerade Educator
13:16

Problem 44

Drive Wheel of an Engine Rework Problem 43 if the belt is crossed, as shown in the figure.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
05:39

Problem 44

Navigation An airplane flies due north from Ft. Myers to Sarasota, a distance of 150 miles, and then turns through an angle of $50^{\circ}$ and flies to Orlando, a distance of 100 miles. See the figure.
(a) How far is it directly from Ft. Myers to Orlando?
(b) What bearing should the pilot use to fly directly from Ft. Myers to Orlando?
84

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:12

Problem 45

The Flatiron Building Completed in 1902 in New York City, the Flatiron Building is triangular shaped and bounded by 22 nd Street, Broadway, and 5 th Avenue. The building measures approximately 87 feet on the 22 nd Street side, 190 feet on the Broadway side, and 173 feet on the 5 th Avenue side. Approximate the ground area covered by the building.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
05:16

Problem 45

In Problems 45-50, the distance d (in meters) of the bob of a pendulum of mass $m$ (in kilograms) from its rest position at time $t$ (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
(a) Describe the motion of the object. Be sure to give the mass and damping factor.
(b) What is the initial displacement of the bob? That is, what is the displacement at $t=0$ ?
(c) Graph the motion using a graphing utility.
(d) What is the displacement of the bob at the start of the second oscillation?
(e) What happens to the displacement of the bob as time increases without bound?
$d=-20 e^{-0.7 t / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^2-\frac{0.49}{1600} t}\right)$

Evan Leonard
Evan Leonard
Numerade Educator
02:13

Problem 45

Commercial Navigation Adam must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles. A second flight option flies first to Kansas City and then connects to St. Louis. The bearing from Oklahoma City to Kansas City is $\mathrm{N} 29.6^{\circ} \mathrm{E}$, and the bearing from Oklahoma City to St. Louis is $\mathrm{N} 577^{\circ} \mathrm{E}$. The bearing from St. Louis to Oklahoma City is $\mathrm{S} 577^{\circ} \mathrm{W}$, and the bearing from St. Louis to Kansas City is N79.4 W. How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct flight?
Source: www.landings.com

Eleanor Johnson
Eleanor Johnson
Numerade Educator
16:03

Problem 45

The Gibb's Hill Lighthouse, Southampton, Bermuda In operation since 1846, the Gibb's Hill Lighthouse stands 117 feet high on a hill 245 feet high, so its beam of light is 362 feet above sea level. A brochure states that ships
40 miles away can see the light and planes flying at 10,000 feet can see it 120 miles away. Verify the accuracy of these statements. What assumption did the brochure make about the height of the ship?

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator
08:28

Problem 45

Avoiding a Tropical Storm A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West Indies, a distance of 600 nautical miles. To avoid a tropical storm, the captain heads out of San Juan in a direction of $20^{\circ}$ off a direct heading to Barbados. The captain maintains the 15 -knot speed for 10 hours, after which time the path to Barbados becomes clear of storms.
(a) Through what angle should the captain turn to head directly to Barbados?
(b) Once the turn is made, how long will it be before the ship reaches Barbados if the same 15-knot speed is maintained?

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
01:30

Problem 46

Area of a Quadrilateral Bretschneider's Formula is a Heron-type formula that can be used to find the area of a general quadrilateral.
$$
K=\sqrt{(s-a)(s-b)(s-c)(s-d)-a b c d \cos ^2 \theta}
$$
where $a, b, c$, and $d$ are the side lengths, $\theta$ is half the sum of two opposite angles, and $s$ is half the perimeter.
(a) Show that if a triangle is considered a quadrilateral with one side equal to 0, Bretschneider's Formula reduces to Heron's Formula.
(b) Use Bretschneiders's Formula to find the area of the lot for sale in Problem 39, rounded to the nearest square foot. Do the results agree?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
05:16

Problem 46

In Problems 45-50, the distance d (in meters) of the bob of a pendulum of mass $m$ (in kilograms) from its rest position at time $t$ (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
(a) Describe the motion of the object. Be sure to give the mass and damping factor.
(b) What is the initial displacement of the bob? That is, what is the displacement at $t=0$ ?
(c) Graph the motion using a graphing utility.
(d) What is the displacement of the bob at the start of the second oscillation?
(e) What happens to the displacement of the bob as time increases without bound?
$d=-20 e^{-0.8 / 40} \cos \left(\sqrt{\left(\frac{2 \pi}{5}\right)^2-\frac{0.64}{1600} t}\right)$

Evan Leonard
Evan Leonard
Numerade Educator
03:47

Problem 46

Time Lost to a Navigation Error In attempting to fly from city $P$ to city $Q$, an aircraft followed a course that was $10^{\circ}$ in error, as indicated in the figure. After flying a distance of 50 miles, the pilot corrected the course by turning at point $R$ and flying 300 miles farther. If the constant speed of the aircraft was 250 miles per hour, how much time was lost due to the error?

Joseph Lentino
Joseph Lentino
Numerade Educator
01:18

Problem 46

Problems $46-49$ are based on material leamed earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Determine whether $x-3$ is a factor of $x^4+2 x^3-21 x^2+19 x-3$

Melissa Salvador
Melissa Salvador
Numerade Educator
07:27

Problem 46

Revising a Flight Plan In attempting to fly from Chicago to Louisville, a distance of 330 miles, a pilot inadvertently took a course that was $10^{\circ}$ in error, as indicated in the figure.
(a) If the aircraft maintains an average speed of 220 miles per hour, and if the error in direction is discovered after 15 minutes, through what angle should the pilot turn to head toward Louisville?
(b) What new average speed should the pilot maintain so that the total time of the trip is 90 minutes?

Linda Hand
Linda Hand
Numerade Educator
12:58

Problem 47

Geometry Refer to the figure. If $|O A|=1$, show that:
(a) Area $\triangle O A C=\frac{1}{2} \sin \alpha \cos \alpha$
(b) Area $\triangle O C B=\frac{1}{2}|O B|^2 \sin \beta \cos \beta$
(c) Area $\triangle O A B=\frac{1}{2}|O B| \sin (\alpha+\beta)$
(d) $|O B|=\frac{\cos \alpha}{\cos \beta}$
(e) $\sin (\alpha+\beta)=\sin \alpha \cos \beta+\cos \alpha \sin \beta$

Ziya Ogron
Ziya Ogron
Numerade Educator
04:22

Problem 47

In Problems 45-50, the distance d (in meters) of the bob of a pendulum of mass $m$ (in kilograms) from its rest position at time $t$ (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
(a) Describe the motion of the object. Be sure to give the mass and damping factor.
(b) What is the initial displacement of the bob? That is, what is the displacement at $t=0$ ?
(c) Graph the motion using a graphing utility.
(d) What is the displacement of the bob at the start of the second oscillation?
(e) What happens to the displacement of the bob as time increases without bound?
$d=-30 e^{-0.60 / 80} \cos \left(\sqrt{\left(\frac{2 \pi}{7}\right)^2-\frac{0.36}{6400} t}\right)$

Evan Leonard
Evan Leonard
Numerade Educator
03:47

Problem 47

Rescue at Sea Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates that the ship is located $\mathrm{N} 55^{\circ} \mathrm{E}$; the call to Station Baker indicates that the ship is located $\mathrm{S} 60^{\circ} \mathrm{E}$.
(a) How far is each station from the ship?
(b) If a helicopter capable of flying 200 miles per hour is dispatched from the station nearest the ship, how long will it take to reach the ship?

Pawan Yadav
Pawan Yadav
Numerade Educator
01:36

Problem 47

Problems $46-49$ are based on material leamed earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the exact value of $\sin 15^{\circ}$.

Melissa Salvador
Melissa Salvador
Numerade Educator
06:09

Problem 47

Major League Baseball Field A major league baseball diamond is actually a square 90 feet on a side. The pitching rubber is located 60.5 feet from home plate on a line joining home plate and second base.
(a) How far is it from the pitching rubber to first base?
(b) How far is it from the pitching rubber to second base?
(c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
01:12

Problem 48

Geometry Refer to the figure, in which a unit circle is drawn. The line segment $D B$ is tangent to the circle, and $\theta$ is acute.
(a) Express the area of $\triangle O B C$ in terms of $\sin \theta$ and $\cos \theta$.
(b) Express the area of $\triangle O B D$ in terms of $\sin \theta$ and $\cos \theta$.
(c) The area of the sector $\widehat{O B C}$ of the circle is $\frac{1}{2} \theta$, where $\theta$ is measured in radians. Use the results of parts (a) and (b) and the fact that
Area $\triangle O B C<$ Area $\overparen{O B C}<$ Area $\triangle O B D$
to show that
$$
1<\frac{\theta}{\sin \theta}<\frac{1}{\cos \theta}
$$
(GRAPH CANT COPY)

R M
R M
Numerade Educator
04:44

Problem 48

In Problems 45-50, the distance d (in meters) of the bob of a pendulum of mass $m$ (in kilograms) from its rest position at time $t$ (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
(a) Describe the motion of the object. Be sure to give the mass and damping factor.
(b) What is the initial displacement of the bob? That is, what is the displacement at $t=0$ ?
(c) Graph the motion using a graphing utility.
(d) What is the displacement of the bob at the start of the second oscillation?
(e) What happens to the displacement of the bob as time increases without bound?
$d=-30 e^{-0.5 t / 70} \cos \left(\sqrt{\left(\frac{\pi}{2}\right)^2-\frac{0.25}{4900} t}\right)$

Evan Leonard
Evan Leonard
Numerade Educator
01:55

Problem 48

Distance to the Moon At exactly the same time, Tom and Alice measured the angle of elevation to the moon while standing exactly $300 \mathrm{~km}$ apart. The angle of elevation to the moon for Tom was $49.8974^{\circ}$, and the angle of elevation to the moon for Alice was $49.9312^{\circ}$. See the figure. To the nearest $1000 \mathrm{~km}$, how far was the moon from Earth when the measurement was obtained?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:51

Problem 48

Problems $46-49$ are based on material leamed earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Evaluate $\frac{f(x)-f(4)}{x-4}$, where $f(x)=\sqrt{x}$ for $x=5,4.5$, and 4.1. Round results to three decimal places.

Melissa Salvador
Melissa Salvador
Numerade Educator
07:45

Problem 48

Little League Baseball Field According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base.
(a) How far is it from the pitching rubber to first base?
(b) How far is it from the pitching rubber to second base?
(c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
10:35

Problem 49

The Cow Problem* A cow is tethered to one corner of a square barn, 10 feet by 10 feet, with a rope 100 feet long. What is the maximum grazing area for the cow? See the illustration.
(GRAPH CANT COPY)

Nick Johnson
Nick Johnson
Numerade Educator
04:37

Problem 49

In Problems 45-50, the distance d (in meters) of the bob of a pendulum of mass $m$ (in kilograms) from its rest position at time $t$ (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
(a) Describe the motion of the object. Be sure to give the mass and damping factor.
(b) What is the initial displacement of the bob? That is, what is the displacement at $t=0$ ?
(c) Graph the motion using a graphing utility.
(d) What is the displacement of the bob at the start of the second oscillation?
(e) What happens to the displacement of the bob as time increases without bound?
$d=-15 e^{-0.9 t / 30} \cos \left(\sqrt{\left(\frac{\pi}{3}\right)^2-\frac{0.81}{900} t}\right)$

Evan Leonard
Evan Leonard
Numerade Educator
05:36

Problem 49

Finding the Lean of the Leaning Tower of Pisa The famous Leaning Tower of Pisa was originally 184.5 feet high.* At a distance of 123 feet from the base of the tower, the angle of elevation to the top of the tower is found to be $60^{\circ}$. Find $\angle R P Q$ indicated in the figure. Also, find the perpendicular distance from $R$ to $P Q$.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
03:17

Problem 49

Problems $46-49$ are based on material leamed earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Solve $2 \sin ^2 \theta-\sin \theta-1=0$ for $0 \leq \theta<2 \pi$.

Melissa Salvador
Melissa Salvador
Numerade Educator
02:41

Problem 49

Finding the Length of a Guy Wire The height of a radio tower is 500 feet, and the ground on one side of the tower slopes upward at an angle of $10^{\circ}$ (see the figure).
(a) How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 100 feet from the base of the tower?
(b) How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a position 100 feet from the base on the flat side?

Victor Salazar
Victor Salazar
Numerade Educator
10:35

Problem 50

Another Cow Problem If the barn in Problem 49 is rectangular, 10 feet by 20 feet, what is the maximum grazing area for the cow?

Nick Johnson
Nick Johnson
Numerade Educator
05:16

Problem 50

In Problems 45-50, the distance d (in meters) of the bob of a pendulum of mass $m$ (in kilograms) from its rest position at time $t$ (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction.
(a) Describe the motion of the object. Be sure to give the mass and damping factor.
(b) What is the initial displacement of the bob? That is, what is the displacement at $t=0$ ?
(c) Graph the motion using a graphing utility.
(d) What is the displacement of the bob at the start of the second oscillation?
(e) What happens to the displacement of the bob as time increases without bound?
$d=-10 e^{-0.8 / 50} \cos \left(\sqrt{\left(\frac{2 \pi}{3}\right)^2-\frac{0.64}{2500} t}\right)$

Evan Leonard
Evan Leonard
Numerade Educator
02:08

Problem 50

Crankshafts on Cars On a certain automobile, the crankshaft is 3 inches long and the connecting rod is 9 inches long
(see the figure). At the time when $\angle O P Q$ is $15^{\circ}$, how far is the piston $(P)$ from the center $(O)$ of the crankshaft?

Carson Merrill
Carson Merrill
Numerade Educator
02:57

Problem 50

Finding the Length of a Guy Wire A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of $5^{\circ}$. See the figure. How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower?

Aman Gupta
Aman Gupta
Numerade Educator
02:46

Problem 51

Perfect Triangles A perfect triangle is one having integers for sides for which the area is numerically equal to the perimeter. Show that the triangles with the given side lengths are perfect.
(a) $9,10,17$
(b) $6,25,29$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
02:16

Problem 51

Loudspeaker A loudspeaker diaphragm is oscillating in simple harmonic motion described by the equation $d=a \cos (\omega t)$ with a frequency of 520 hertz (cycles per second) and a maximum displacement of 0.80 millimeter. Find $\omega$ and then determine the equation that describes the movement of the diaphragm.

Evan Leonard
Evan Leonard
Numerade Educator
08:33

Problem 51

Constructing a Highway U.S. 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay?

Ryan Pollard
Ryan Pollard
Numerade Educator
01:33

Problem 51

Identifying Remains The Purkait triangle, located at the proximal end of the femur, has been used to identify the gender of fragmented skeletal remains. See the figure.
(a) Given $\overline{A B}=30.1 \mathrm{~mm}, \overline{A C}=51.4 \mathrm{~mm}$, and $A=89.2^{\circ}$, find the length of $\overline{B C}$.
(b) If the average length of $\overline{B C}$ is $59.4 \mathrm{~mm}$ for males and $53.3 \mathrm{~mm}$ for females, which gender would be identified for the measurements in part (a)?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:33

Problem 52

If $h_1, h_2$, and $h_3$ are the altitudes dropped from $P, Q$, and $R$, respectively, in a triangle (see the figure), show that
$$
\frac{1}{h_1}+\frac{1}{h_2}+\frac{1}{h_3}=\frac{s}{K}
$$
where $K$ is the area of the triangle and $s=\frac{1}{2}(a+b+c)$.

Narayan Hari
Narayan Hari
Numerade Educator
03:42

Problem 52

Colossus Added to Six Flags St. Louis in 1986, the Colossus is a giant Ferris wheel. Its diameter is 165 feet, it rotates at a rate of about 1.6 revolutions per minute, and the bottom of the wheel is 15 feet above the ground. Determine an equation that relates a rider's height above the ground at time $t$. Assume the passenger begins the ride at the bottom of the wheel.

Lucas Finney
Lucas Finney
Numerade Educator
02:24

Problem 52

Calculating Distances at Sea The navigator of a ship at sea spots two lighthouses that she knows to be 3 miles apart along a straight seashore. She determines that the angles formed between two line-of-sight observations of the lighthouses and the line from the ship directly to shore are $15^{\circ}$ and $35^{\circ}$. See the illustration.
(a) How far is the ship from lighthouse $P$ ?
(b) How far is the ship from lighthouse $Q$ ?
(c) How far is the ship from shore?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:45

Problem 52

Identifying Remains Like the Purkait triangle in Problem 51, the metric triangle is located at the proximal end of the femur and has been used to identify the gender of fragmented skeletal remains. See the figure.
(a) If $\overline{A C}=48.8 \mathrm{~mm}, \overline{B C}=62.2 \mathrm{~mm}$, and $C=89^{\circ}$, find the length of $\overline{A B}$.
(b) If $\overline{A B}<80 \mathrm{~mm}$ typically indicates a female and $\overline{A B}>100 \mathrm{~mm}$ typically indicates a male, which gender, if any, would be identified from the measurements in part (a)?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
02:07

Problem 53

Show that a formula for the altitude $h$ from a vertex to the opposite side $a$ of a triangle is
$$
h=\frac{a \sin B \sin C}{\sin A}
$$
$$
\sin A
$$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:13

Problem 53

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the equation $d=a \sin (\omega t)$. If a tuning fork for the note $\mathrm{A}$ above middle $\mathrm{C}$ on an
even-tempered scale $\left(A_4\right.$, the tone by which an orchestra tunes itself) has a frequency of 440 hertz (cycles per second), find $\omega$. If the maximum displacement of the end of the tuning fork is 0.01 millimeter, determine the equation that describes the movement of the tuning fork.

Lucas Finney
Lucas Finney
Numerade Educator
02:16

Problem 53

Designing an Awning An awning that covers a sliding glass door that is 88 inches tall forms an angle of $50^{\circ}$ with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the Sun is more than $65^{\circ}$. See the figure. Find the length $L$ of the awning.

Pawan Yadav
Pawan Yadav
Numerade Educator
05:17

Problem 53

Soccer Angles A soccer goal is 8 yards wide. Suppose a goalie is standing on her line in the center of her goal as a striker from the opposing team moves the ball towards her. The near post angle, $\alpha$, is formed by rays extending from the ball to the near post and the goalie. Similarly, the far post angle, $\beta$, is formed by rays extending from the ball to the far post and the goalie. See the figure.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
04:10

Problem 54

Refer to Problem 109 in Section 6.5.
(a) Show that the area of a regular dodecagon is given by $K=3 \cot \left(\frac{\pi}{12}\right) a^2$ or $K=12 \tan \left(\frac{\pi}{12}\right) r^2$, where $a$ is the length of one of the sides and $r$ is the radius of the inscribed circle.
(b) Given that each interior angle of a regular $n$-sided polygon $(n \geq 3)$ measures $\frac{(n-2) \cdot 180^{\circ}}{n}$, generalize these formulas for any such polygon.

Andrew Bassila
Andrew Bassila
Numerade Educator
01:43

Problem 54

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the equation $d=a \sin (\omega t)$. If a tuning fork for the note $\mathrm{E}$ above middle $\mathrm{C}$ on an even-tempered scale $\left(\mathrm{E}_4\right)$ has a frequency of approximately 329.63 hertz (cycles per second), find $\omega$. If the maximum displacement of the end of the tuning fork is 0.025 millimeter, determine the equation that describes the movement of the tuning fork.

Lucas Finney
Lucas Finney
Numerade Educator
02:12

Problem 54

Finding Distances A forest ranger is walking on a path inclined at $5^{\circ}$ to the horizontal directly toward a 100 -foot-tall fire observation tower. The angle of elevation from the path to the top of the tower is $40^{\circ}$. How far is the ranger from the tower at this time?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:17

Problem 54

Covering the Angles In soccer, a defending goalkeeper wants to take up a position that bisects the angle that needs to be covered. The keeper stands square to the ball - that is, perpendicular to the line of bisection - at a point such that the area covered (shaded) lies completely outside the goal. How far will the goalkeeper be from the center of the goal line if an attacking striker is 24 yards from the near post and 30 yards from the far post?

Prabhu Ramji
Prabhu Ramji
Numerade Educator

Problem 55

Inscribed Circle For Problems 55-58, the lines that bisect each angle of a triangle meet in a single point $O$, and the perpendicular distance $r$ from $O$ to each side of the triangle is the same. The circle with center at $O$ and radius $r$ is called the inscribed circle of the triangle (see the figure).
Apply the formula from Problem 53 to triangle $O P Q$ to show that
$$
r=\frac{c \sin \frac{A}{2} \sin \frac{\mathrm{B}}{2}}{\cos \frac{\mathrm{C}}{2}}
$$

Check back soon!
02:54

Problem 55

Charging a Capacitor See the illustration. If a charged capacitor is connected to a coil by closing a switch, energy is transferred to the coil and then back to the capacitor in an oscillatory motion. The voltage $V$ (in volts) across the capacitor will gradually diminish to 0 with time $t$ (in seconds).
(a) Graph the function relating $V$ and $t$ :
$$
V(t)=e^{-t / 3} \cos (\pi t) \quad 0 \leq t \leq 3
$$
(b) At what times $t$ will the graph of $V$ touch the graph of $y=e^{-t / 3}$ ? When does the graph of $V$ touch the graph of $y=-e^{-t / 3}$ ?
(c) When will the voltage $V$ be between -0.4 and 0.4 volt?
(GRAPH CANT COPY)

Evan Leonard
Evan Leonard
Numerade Educator
01:36

Problem 55

Great Pyramid of Cheops One of the original Seven Wonders of the World, the Great Pyramid of Cheops was built about $2580 \mathrm{BC}$. Its original height was 480 feet 11 inches, but owing to the loss of its topmost stones, it is now shorter. Find the current height of the Great Pyramid using the information given in the illustration.
Source: Guinness Book of World Records

Eleanor Johnson
Eleanor Johnson
Numerade Educator
02:25

Problem 55

Wrigley Field, Home of the Chicago Cubs The distance from home plate to the fence in dead center in Wrigley Field is 400 feet (see the figure). How far is it from the fence in dead center to third base?

Nick Johnson
Nick Johnson
Numerade Educator
02:28

Problem 56

Inscribed Circle For Problems 55-58, the lines that bisect each angle of a triangle meet in a single point $O$, and the perpendicular distance $r$ from $O$ to each side of the triangle is the same. The circle with center at $O$ and radius $r$ is called the inscribed circle of the triangle (see the figure).
Use the result of Problem 55 and the result of Problem 61 in Section 7.3 to show that
$$
\cot \frac{C}{2}=\frac{s-c}{r}
$$
where $s=\frac{1}{2}(a+b+c)$.

Eleanor Johnson
Eleanor Johnson
Numerade Educator
02:36

Problem 56

The Sawtooth Curve An oscilloscope often displays a sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes.
(a) Use a graphing utility to graph the following function, which can be used to approximate the sawtooth curve.
$$
f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x) \quad 0 \leq x \leq 4
$$
(b) A better approximation to the sawtooth curve is given by
$$
f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x)+\frac{1}{8} \sin (8 \pi x)
$$
Use a graphing utility to graph this function for $0 \leq x \leq 4$ and compare the result to the graph obtained in part (a).
(c) A third and even better approximation to the sawtooth curve is given by
$$
f(x)=\frac{1}{2} \sin (2 \pi x)+\frac{1}{4} \sin (4 \pi x)+\frac{1}{8} \sin (8 \pi x)+\frac{1}{16} \sin (16 \pi x)
$$
(IMAGE CANT COPY)
Use a graphing utility to graph this function for $0 \leq x \leq 4$ and compare the result to the graphs obtained in parts (a) and (b).
(d) What do you think the next approximation to the sawtooth curve is?

Lucas Finney
Lucas Finney
Numerade Educator
02:59

Problem 56

Determining the Height of an Aircraft Two sensors are spaced 700 feet apart along the approach to a small airport. When an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is $20^{\circ}$, and from the second sensor to the aircraft it is $15^{\circ}$. Determine how high the aircraft is at this time.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
02:45

Problem 56

Little League Baseball The distance from home plate to the fence in dead center at the Oak Lawn Little League field is 280 feet. How far is it from the fence in dead center to third base?

Allison Knapp
Allison Knapp
Numerade Educator
02:28

Problem 57

Inscribed Circle For Problems 55-58, the lines that bisect each angle of a triangle meet in a single point $O$, and the perpendicular distance $r$ from $O$ to each side of the triangle is the same. The circle with center at $O$ and radius $r$ is called the inscribed circle of the triangle (see the figure).
Show that $\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\frac{s}{r}$.

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:20

Problem 57

Touch-Tone Phones On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by
$$
y=\sin (2 \pi h) \text { and } y=\sin (2 \pi h t)
$$
where $l$ and $h$ are the low and high frequencies (cycles per second) shown in the illustration. For example, if you touch 7 , the low frequency is $l=852$ cycles per second and the high frequency is $h=1209$ cycles per second. The sound emitted by touching 7 is
$$
y=\sin [2 \pi(852) t]+\sin [2 \pi(1209) t]
$$
Use a graphing utility to graph the sound emitted by touching 7 .
Touch-Tone phone
(IMAGE CANT COPY)

Lucas Finney
Lucas Finney
Numerade Educator
07:25

Problem 57

Mercury The distance from the Sun to Earth is approximately $149,600,000$ kilometers $(\mathrm{km})$. The distance from the Sun to Mercury is approximately $57,910,000 \mathrm{~km}$. The elongation angle $\alpha$ is the angle formed between the line of sight from Earth to the Sun and the line of sight
from Earth to Mercury. See the figure. Suppose that the elongation angle for Mercury is $15^{\circ}$. Use this information to find the possible distances between Earth and Mercury.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:00

Problem 57

Building a Swing Set Clint is building a wooden swing set for his children. Each supporting end of the swing set is to be an A-frame constructed with two 10 -foot-long 4 by 4 's joined at a $45^{\circ}$ angle. To prevent the swing set from tipping over, Clint wants to secure the base of each A-frame to concrete footings. How far apart should the footings for each A-frame be?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
03:30

Problem 58

Inscribed Circle For Problems 55-58, the lines that bisect each angle of a triangle meet in a single point $O$, and the perpendicular distance $r$ from $O$ to each side of the triangle is the same. The circle with center at $O$ and radius $r$ is called the inscribed circle of the triangle (see the figure).
Show that the area $K$ of triangle $P Q R$ is $K=r s$, where $s=\frac{1}{2}(a+b+c)$. Then show that
$$
r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}
$$

Anurag Kumar
Anurag Kumar
Numerade Educator
01:58

Problem 58

Use a graphing utility to graph the sound emitted by the * key on a Touch-Tone phone. See Problem 57.

Evan Leonard
Evan Leonard
Numerade Educator
05:10

Problem 58

Venus The distance from the Sun to Earth is approximately $149,600,000 \mathrm{~km}$. The distance from the Sun to Venus is approximately $108,200,000 \mathrm{~km}$. The elongation angle $\alpha$ is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Venus. Suppose that the elongation angle for Venus is $10^{\circ}$. Use this information to find the possible distances between Earth and Venus.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
04:23

Problem 58

Rods and Pistons Rod $O A$ rotates about the fixed point $O$ so that point $A$ travels on a circle of radius $r$. Connected to point $A$ is another rod $A B$ of length $L>2 r$, and point $B$ is connected to a piston. See the figure. Show that the distance $x$ between point $O$ and point $B$ is given by
$$
x=r \cos \theta+\sqrt{r^2 \cos ^2 \theta+L^2-r^2}
$$
where $\theta$ is the angle of rotation of $\operatorname{rod} O A$.

Carson Merrill
Carson Merrill
Numerade Educator
03:28

Problem 59

A triangle has vertices $A(0,0), B(1,0)$, and $C$, where $C$ is the point on the unit circle corresponding to an angle of $105^{\circ}$ when it is drawn in standard position. Find the area of the triangle. State the answer in complete simplified form with a rationalized denominator.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:27

Problem 59

CBL Experiment Pendulum motion is analyzed to estimate simple harmonic motion. A plot is generated with the position of the pendulum over time. The graph is used to find a sinusoidal curve of the form $y=A \cos [B(x-C)]+D$. Determine the amplitude, period, and frequency. (Activity 16, Real-World Math with the CBL System.)

Evan Leonard
Evan Leonard
Numerade Educator
03:27

Problem 59

The Original Ferris Wheel George Washington Gale Ferris, Jr., designed the original Ferris wheel for the 1893 World's Columbian Exposition in Chicago, Illinois. The wheel had 36 equally spaced cars each the size of a school bus. The distance between adjacent cars was approximately 22 feet. Determine the diameter of the wheel to the nearest foot.
Source: Carnegie Library of Pittsburgh, www.clpgh.org

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:03

Problem 59

Geometry Show that the length $d$ of a chord of a circle of radius $r$ is given by the formula
$$
d=2 r \sin \frac{\theta}{2}
$$
where $\theta$ is the central angle formed by the radii to the ends of the chord. See the figure. Use this result to derive the fact that $\sin \theta<\theta$, where $\theta>0$ is measured in radians.

Yuou Sun
Yuou Sun
Numerade Educator
00:48

Problem 60

What do you do first if you are asked to find the area of a triangle and are given two sides and the included angle?

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:28

Problem 60

CBL Experiment The sound from a tuning fork is collected over time. A model of the form $y=A \cos [B(x-C)]$ is fitted to the data. Determine the amplitude, frequency, and period of the graph.
(Activity 23, Real-World Math with the CBL System.)

Evan Leonard
Evan Leonard
Numerade Educator
02:33

Problem 60

Mollweide's Formula For any triangle, Mollweide's Formula (named after Karl Mollweide, 1774-1825) states that
$$
\frac{a+b}{c}=\frac{\cos \left[\frac{1}{2}(A-B)\right]}{\sin \left(\frac{1}{2} C\right)}
$$
Derive it.

Eleanor Johnson
Eleanor Johnson
Numerade Educator
06:44

Problem 60

For any triangle, show that
$$
\cos \frac{C}{2}=\sqrt{\frac{s(s-c)}{a b}}
$$
where $s=\frac{1}{2}(a+b+c)$.

Linda Hand
Linda Hand
Numerade Educator
00:42

Problem 61

What do you do first if you are asked to find the area of a triangle and are given three sides?

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:01

Problem 61

Use a graphing utility to graph the function $f(x)=\frac{\sin x}{x}, x>0$. Based on the graph, what do you conjecture about the value of $\frac{\sin x}{x}$ for $x$ close to 0 ?

Lucas Finney
Lucas Finney
Numerade Educator
05:25

Problem 61

Mollweide's Formula Another form of Mollweide's Formula is
$$
\frac{a-b}{c}=\frac{\sin \left[\frac{1}{2}(A-B)\right]}{\cos \left(\frac{1}{2} C\right)}
$$
Derive it.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
02:20

Problem 61

For any triangle, show that
$$
\sin \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{a b}}
$$
where $s=\frac{1}{2}(a+b+c)$.

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:14

Problem 62

State the area of an SAS triangle in words.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:18

Problem 62

Use a graphing utility to graph $y=x \sin x, y=x^2 \sin x$, and and $y=\frac{1}{x^3} \sin x$ for $x>0$. What patterns do you observe?

Lucas Finney
Lucas Finney
Numerade Educator
02:00

Problem 62

For any triangle, derive the formula
$$
a=b \cos C+c \cos B
$$

Yujie Wang
Yujie Wang
College of San Mateo
02:43

Problem 62

Use the Law of Cosines to prove the identity
$$
\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{a^2+b^2+c^2}{2 a b c}
$$

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
01:40

Problem 63

Problems 63-66 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Without graphing, determine whether the quadratic function $f(x)=-3 x^2+12 x+5$ has a maximum value or a minimum value, and then find the value.

Edward Downes
Edward Downes
Numerade Educator
01:18

Problem 63

. Use a graphing utility to graph $y=\frac{1}{x} \sin x, y=\frac{1}{x^2} \sin x$, and $y=\frac{1}{x^3} \sin x$ for $x>0$. What patterns do you observe?

Lucas Finney
Lucas Finney
Numerade Educator
07:05

Problem 63

Law of Tangents For any triangle, derive the Law of Tangents:
$$
\frac{a-b}{a+b}=\frac{\tan \left[\frac{1}{2}(A-B)\right]}{\tan \left[\frac{1}{2}(A+B)\right]}
$$

Ryan Pollard
Ryan Pollard
Numerade Educator
01:15

Problem 63

What do you do first if you are asked to solve a triangle and are given two sides and the included angle?

Eleanor Johnson
Eleanor Johnson
Numerade Educator
03:35

Problem 64

Problems 63-66 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Solve the inequality: $\frac{x+1}{x^2-9} \leq 0$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:58

Problem 64

How would you explain to a friend what simple harmonic motion is? How would you explain damped motion?

Evan Leonard
Evan Leonard
Numerade Educator
06:07

Problem 64

Circumscribing a Triangle Show that
$$
\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2 r}
$$
where $r$ is the radius of the circle circumscribing the triangle $P Q R$ whose sides are $a, b$, and $c$, as shown in the figure.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:08

Problem 64

What do you do first if you are asked to solve a triangle and are given three sides?

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
03:55

Problem 65

Problems 63-66 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
$P=\left(-\frac{\sqrt{7}}{3}, \frac{\sqrt{2}}{3}\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

Allison Knapp
Allison Knapp
Numerade Educator
03:03

Problem 65

Problems $65-68$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
The function $f(x)=\frac{x-3}{x-4}, x \neq 4$, is one-to-one. Find its inverse function.

Evan Leonard
Evan Leonard
Numerade Educator
04:14

Problem 65

Make up three problems involving oblique triangles. One should result in one triangle, the second in two triangles, and the third in no triangle.

Mitchell Cutler
Mitchell Cutler
Numerade Educator
02:02

Problem 65

Make up an applied problem that requires using the Law of Cosines.

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:30

Problem 66

Problems 63-66 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Establish the identity: $\csc \theta-\sin \theta=\cos \theta \cot \theta$

Jonathon Brumley
Jonathon Brumley
Numerade Educator
02:27

Problem 66

Problems $65-68$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Write as a single logarithm: $\log _7 x+3 \log _7 y-\log _7(x+y)$

Evan Leonard
Evan Leonard
Numerade Educator
01:14

Problem 66

What do you do first if you are asked to solve a triangle and are given one side and two angles?

Mitchell Cutler
Mitchell Cutler
Numerade Educator
02:11

Problem 66

Write down your strategy for solving an oblique triangle.

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
04:28

Problem 67

Problems $65-68$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Solve: $\log (x+1)+\log (x-2)=1$

Evan Leonard
Evan Leonard
Numerade Educator
01:27

Problem 67

What do you do first if you are asked to solve a triangle and are given two sides and the angle opposite one of them?

Mitchell Cutler
Mitchell Cutler
Numerade Educator
01:56

Problem 67

State the Law of Cosines in words.

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
07:44

Problem 68

Problems $65-68$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Given $\cos \alpha=\frac{4}{5}, 0<\alpha<\frac{\pi}{2}$, find the exact value of:
(a) $\cos \frac{\alpha}{2}$
(b) $\sin \frac{\alpha}{2}$
(c) $\tan \frac{\alpha}{2}$

Evan Leonard
Evan Leonard
Numerade Educator
00:34

Problem 68

Solve Example 6 using right-triangle geometry. Comment on which solution, using the Law of Sines or using right triangles, you prefer. Give reasons.

Mrinal Rana
Mrinal Rana
Numerade Educator
02:05

Problem 68

Problems 68-71 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Graph: $R(x)=\frac{2 x+1}{x-3}$

Yujie Wang
Yujie Wang
College of San Mateo
01:25

Problem 69

Problems $69-72$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Solve: $3 x^3+4 x^2-27 x-36=0$

Erika Bustos
Erika Bustos
Numerade Educator
04:08

Problem 69

Problems 68-71 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Solve $4^x=3^{x+1}$. If the solution is irrational, express it both in exact form and as a decimal rounded to three places.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:27

Problem 70

Problems $69-72$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the exact distance between $P_1=(-1,-7)$ and

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:42

Problem 70

Problems 68-71 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Given $\tan \theta=-\frac{2 \sqrt{6}}{5}$ and $\cos \theta=-\frac{5}{7}$, find the exact value of each of the four remaining trigonometric functions.

Sriram Soundarrajan
Sriram Soundarrajan
Numerade Educator
02:04

Problem 71

Problems $69-72$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the exact value of $\tan \left[\cos ^{-1}\left(-\frac{7}{8}\right)\right]$. $P_2=(2,-1)$. Then approximate the distance to two decimal places.

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:45

Problem 71

Problems 68-71 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find an equation for the graph.

Eleanor Johnson
Eleanor Johnson
Numerade Educator
01:47

Problem 72

Problems $69-72$ are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Graph $y=4 \sin \left(\frac{1}{2} x\right)$. Show at least two periods.

Eleanor Johnson
Eleanor Johnson
Numerade Educator