Trigonometry

David I. Schneider; Callie J. Daniels; John Hornsby

Chapter 2

Acute Angles and Right Triangles - all with Video Answers

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Section 1

Trigonometric Functions of Acute Angles

Problem 1

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
II
A. $\sqrt{3}$
B. 1
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$
$\mathbf{I}$
$\sin 30^{\circ}$

Elizabeth Waters

Elizabeth Waters

Numerade Educator

Problem 1

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
$\mathbf{I}$
1. $\sin 30^{\circ}$
2. $\cos 45^{\circ}$
3. $\tan 45^{\circ}$
4. $\sec 60^{\circ}$
5. $\csc 60^{\circ}$
6. $\cot 30^{\circ}$
II
A. $\sqrt{3}$
B.
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 2

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
II
A. $\sqrt{3}$
B. 1
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$
$\mathbf{I}$
$\sin 30^{\circ}$

Elizabeth Waters

Elizabeth Waters

Numerade Educator

Problem 3

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
II
A. $\sqrt{3}$
B. 1
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$
$\mathbf{I}$
$\tan 45^{\circ}$

Nicole Hoffman

Numerade Educator

Problem 4

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
II
A. $\sqrt{3}$
B. 1
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$
$\mathbf{I}$
$\sec 60^{\circ}$

Nicole Hoffman

Numerade Educator

Problem 5

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
II
A. $\sqrt{3}$
B. 1
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$
$\mathbf{I}$
$\csc 60^{\circ}$

Nicole Hoffman

Numerade Educator

Problem 6

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.
II
A. $\sqrt{3}$
B. 1
C. $\frac{1}{2}$
D. $\frac{\sqrt{3}}{2}$
E. $\frac{2 \sqrt{3}}{3}$
F. $\frac{\sqrt{3}}{3}$
G. 2
H. $\frac{\sqrt{2}}{2}$
I. $\sqrt{2}$
$\mathbf{I}$
$\cot 30^{\circ}$

Nicole Hoffman

Numerade Educator

Problem 7

Find exact values or expressions for $\sin A, \cos A,$ and $\tan A$.

Supratim Roy

Numerade Educator

Problem 8

Find exact values or expressions for $\sin A, \cos A,$ and $\tan A$.

Supratim Roy

Numerade Educator

Problem 9

Find exact values or expressions for $\sin A, \cos A,$ and $\tan A$.

Supratim Roy

Numerade Educator

Problem 10

Find exact values or expressions for $\sin A, \cos A,$ and $\tan A$.

Supratim Roy

Numerade Educator

Problem 11

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$a=5, b=12$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 12

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$a=3, b=4$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 13

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$a=6, c=7$

Erika Bustos

Numerade Educator

Problem 14

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$b=7, c=12$

Erika Bustos

Numerade Educator

Problem 15

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$a=3, c=10$

Erika Bustos

Numerade Educator

Problem 16

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$b=8, c=11$

Erika Bustos

Numerade Educator

Problem 17

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$a=1, c=2$

Erika Bustos

Numerade Educator

Problem 18

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$a=\sqrt{2}, c=2$

Erika Bustos

Numerade Educator

Problem 19

Suppose ABC is a right triangle with sides of lengths a, $b$, and $c$ and right angle at $C$. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable.
$b=2, c=5$

Leslie Deeb

Numerade Educator

Problem 20

Give the six cofunction identities.

Cheryl Glor

Numerade Educator

Problem 21

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\cos 30^{\circ}$

Vysakh M

Numerade Educator

Problem 22

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\sin 45^{\circ}$

Vysakh M

Numerade Educator

Problem 23

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\csc 60^{\circ}$

Vysakh M

Numerade Educator

Problem 24

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\cot 73^{\circ}$

Vysakh M

Numerade Educator

Problem 25

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\sec 39^{\circ}$

Vysakh M

Numerade Educator

Problem 26

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\tan 25.4^{\circ}$

Vysakh M

Numerade Educator

Problem 27

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\sin 38.7^{\circ}$

Vysakh M

Numerade Educator

Problem 28

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\cos \left(\theta+20^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 29

Write each function in terms of its cofunction. Assume all angles involved are acute angles.
$\sec \left(\theta+15^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 30

With a calculator, evaluate $\sin \left(90^{\circ}-\theta\right)$ and $\cos \theta$ for various values of $\theta$. (Check values greater than $90^{\circ}$ and less than $0^{\circ}$.) Comment on the results.

Vysakh M

Numerade Educator

Problem 31

Find one solution for each equation. Assume all angles involved are acute angles.
$\tan \alpha=\cot \left(\alpha+10^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 32

Find one solution for each equation. Assume all angles involved are acute angles.
$\cos \theta=\sin \left(2 \theta-30^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 33

Find one solution for each equation. Assume all angles involved are acute angles.
$\sin \left(2 \theta+10^{\circ}\right)=\cos \left(3 \theta-20^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 34

Find one solution for each equation. Assume all angles involved are acute angles.
$\sec \left(\beta+10^{\circ}\right)=\csc \left(2 \beta+20^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 35

Find one solution for each equation. Assume all angles involved are acute angles.
$\tan \left(3 B+4^{\circ}\right)=\cot \left(5 B-10^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 36

Find one solution for each equation. Assume all angles involved are acute angles.
$\cot \left(5 \theta+2^{\circ}\right)=\tan \left(2 \theta+4^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 37

Find one solution for each equation. Assume all angles involved are acute angles.
$\sin \left(\theta-20^{\circ}\right)=\cos \left(2 \theta+5^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 38

Find one solution for each equation. Assume all angles involved are acute angles.
$\cos \left(2 \theta+50^{\circ}\right)=\sin \left(2 \theta-20^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 39

Find one solution for each equation. Assume all angles involved are acute angles.
$\sec \left(3 \beta+10^{\circ}\right)=\csc \left(\beta+8^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 40

Find one solution for each equation. Assume all angles involved are acute angles.
$\csc \left(\beta+40^{\circ}\right)=\sec \left(\beta-20^{\circ}\right)$

Khushbu Rani

Numerade Educator

Problem 41

Determine whether each statement is true or false.
$\sin 50^{\circ}>\sin 40^{\circ}$

Vysakh M

Numerade Educator

Problem 42

Determine whether each statement is true or false.
$\tan 28^{\circ} \leq \tan 40^{\circ}$

Vysakh M

Numerade Educator

Problem 43

Determine whether each statement is true or false.
$\sin 46^{\circ}<\cos 46^{\circ}$
$\left(\right.$ Hint $\left.: \cos 46^{\circ}=\sin 44^{\circ}\right)$

Vysakh M

Numerade Educator

Problem 44

Determine whether each statement is true or false.
$\cos 28^{\circ}<\sin 28^{\circ}$
$\left(\right.$ Hint $\left.: \sin 28^{\circ}=\cos 62^{\circ}\right)$

Vysakh M

Numerade Educator

Problem 45

Determine whether each statement is true or false.
$\tan 41^{\circ}<\cot 41^{\circ}$

Vysakh M

Numerade Educator

Problem 46

Determine whether each statement is true or false.
$\cot 30^{\circ}<\tan 40^{\circ}$

Vysakh M

Numerade Educator

Problem 47

Determine whether each statement is true or false.
$\sec 60^{\circ}>\sec 30^{\circ}$

Vysakh M

Numerade Educator

Problem 48

Determine whether each statement is true or false.
$\csc 20^{\circ}<\csc 30^{\circ}$

Vysakh M

Numerade Educator

Problem 49

Give the exact value of each expression.
$\tan 30^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 50

Give the exact value of each expression.
$\cot 30^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 51

Give the exact value of each expression.
$\sin 30^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 52

Give the exact value of each expression.
$\cos 30^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 53

Give the exact value of each expression.
$\sec 30^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 54

Give the exact value of each expression.
$\csc 30^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 55

Give the exact value of each expression.
$\csc 45^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 56

Give the exact value of each expression.
$\sec 45^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 57

Give the exact value of each expression.
$\cos 45^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 58

Give the exact value of each expression.
$\cot 45^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 59

Give the exact value of each expression.
$\tan 45^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 60

Give the exact value of each expression.
$\sin 45^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 61

Give the exact value of each expression.
$\sin 60^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 62

Give the exact value of each expression.
$\cos 60^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 63

Give the exact value of each expression.
$\tan 60^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 64

Give the exact value of each expression.
$\csc 60^{\circ}$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 65

Work each problem.
What value of A between $0^{\circ}$ and $90^{\circ}$ will produce the output shown on the graphing calculator screen?

Vysakh M

Numerade Educator

Problem 66

Work each problem.
A student was asked to give the exact value of $\sin 45^{\circ} .$ Using a calculator, he gave the answer $0.7071067812 .$ Explain why the teacher did not give him credit.

Lauren Shelton

Numerade Educator

Problem 67

Work each problem.
Find the equation of the line that passes through the origin and makes a $30^{\circ}$ angle with the $x$ -axis.

Erika Bustos

Numerade Educator

Problem 68

Work each problem.
Find the equation of the line that passes through the origin and makes a $60^{\circ}$ angle with the $x$ -axis.

Erika Bustos

Numerade Educator

Problem 69

Work each problem.
What angle does the line $y=\sqrt{3} x$ make with the positive $x$ -axis?

Khushbu Rani

Numerade Educator

Problem 70

Work each problem.
What angle does the line $y=\frac{\sqrt{3}}{3} x$ make with the positive $x$ -axis?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 71

Consider an equilateral triangle with each side having length $2 k$.
(a) What is the measure of each angle?
(b) Label one angle $A$. Drop a perpendicular from $A$ to the side opposite $A$. Two $30^{\circ}$ angles are formed at $A$, and two right triangles are formed. What is the length of the sides opposite the $30^{\circ}$ angles?
(c) What is the length of the perpendicular in part (b)?
(d) From the results of parts (a)-(c), complete the following statement: In a $30^{\circ}-60^{\circ}$ right triangle, the hypotenuse is always ______ times as long as the shorter leg, and the longer leg has a length that is _______ times as long as that of the shorter leg. Also, the shorter leg is opposite the _________ angle, and the longer leg is opposite the __________ angle.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 72

Consider a square with each side of length $k$.
(a) Draw a diagonal of the square. What is the measure of each angle formed by a side of the square and this diagonal?
(b) What is the length of the diagonal?
(c) From the results of parts (a) and (b), complete the following statement:
In a $45^{\circ}-45^{\circ}$ right triangle, the hypotenuse has a length that is ________ times as long as either leg.

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 73

Find the exact value of the variables in each figure.

Matthew Markham

Matthew Markham

Numerade Educator

Problem 74

Find the exact value of the variables in each figure.

Matthew Markham

Matthew Markham

Numerade Educator

Problem 75

Find the exact value of the variables in each figure.

Matthew Markham

Matthew Markham

Numerade Educator

Problem 76

Find the exact value of the variables in each figure.

Matthew Markham

Matthew Markham

Numerade Educator

Problem 77

Find a formula for the area of each figure in terms of $s$.

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 78

Find a formula for the area of each figure in terms of $s$.

Jennifer Stoner

Jennifer Stoner

Numerade Educator

Problem 79

With a graphing calculator, find the coordinates of the point of intersection of the graphs of $y=x$ and $y=\sqrt{1-x^{2}}$. These coordinates are the cosine and sine of what angle between $0^{\circ}$ and $90^{\circ} ?$

Mohamed Mohamed

Mohamed Mohamed

Numerade Educator

Problem 80

Suppose we know the length of one side and one acute angle of a $30^{\circ}-60^{\circ}$ right triangle. Is it possible to determine the measures of all the sides and angles of the triangle?

Lauren Shelton

Numerade Educator

Problem 81

The figure shows a $45^{\circ}$ central angle in a circle with radius 4 units. To find the coordinates of point $P$ on the circle.
Sketch a line segment from $P$ perpendicular to the $x$ -axis.

Khushbu Rani

Numerade Educator

Problem 82

The figure shows a $45^{\circ}$ central angle in a circle with radius 4 units. To find the coordinates of point $P$ on the circle.
Use the trigonometric ratios for a $45^{\circ}$ angle to label the sides of the right triangle sketched in Exercise 81.

Khushbu Rani

Numerade Educator

Problem 83

The figure shows a $45^{\circ}$ central angle in a circle with radius 4 units. To find the coordinates of point $P$ on the circle.
Which sides of the right triangle give the coordinates of point $P ?$ What are the coordinates of $P ?$

Khushbu Rani

Numerade Educator

Problem 84

The figure shows a $45^{\circ}$ central angle in a circle with radius 4 units. To find the coordinates of point $P$ on the circle.
The figure at the right shows a $60^{\circ}$ central angle in a circle of radius 2 units. Follow the same procedure as in Exercises $81-83$ to find the coordinates of $P$ in the figure.

Jill Tolbert

Numerade Educator