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Precalculus

David Cohen, Theodore B. Lee, David Sklar

Chapter 7

Graphs of the Trigonometric Functions - all with Video Answers

Educators

Section 1

Trigonometric Functions of Real Numbers

02:15

Problem 1

Specify the period and amplitude for each function.
(GRAPH CAN'T COPY)

Amy Jiang
Amy Jiang
Numerade Educator
02:34

Problem 1

Evaluate each expression (as in Example 1 ).
(a) $\cos (11 \pi / 6)$
(b) $\cos (-11 \pi / 6)$
(c) $\sin (11 \pi / 6)$
(d) $\sin (-11 \pi / 6)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:22

Problem 2

In Exercises 1– 8, evaluate each expression
(a) $\cos (2 \pi / 3)$
(b) $\cos (-2 \pi / 3)$
(c) $\sin (2 \pi / 3)$
(d) $\sin (-2 \pi / 3)$

Amy Jiang
Amy Jiang
Numerade Educator
02:36

Problem 2

Evaluate each expression (as in Example 1 ).
(a) $\cos (2 \pi / 3)$
(b) $\cos (-2 \pi / 3)$
(c) $\sin (2 \pi / 3)$
(d) $\sin (-2 \pi / 3)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:45

Problem 3

Evaluate each expression
(a) $\cos (\pi / 6)$
(b) $\cos (-\pi / 6)$

Amy Jiang
Amy Jiang
Numerade Educator
02:07

Problem 3

Evaluate each expression (as in Example 1 ).
(a) $\cos (\pi / 6)$
(b) $\cos (-\pi / 6)$
(c) $\sin (\pi / 6)$
(d) $\sin (-\pi / 6)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
03:16

Problem 4

Specify the period and amplitude for each function.
(GRAPH CAN'T COPY)

Amy Jiang
Amy Jiang
Numerade Educator
03:26

Problem 4

Evaluate each expression (as in Example 1 ).
(a) $\cos (13 \pi / 4)$
(b) $\cos (-13 \pi / 4)$
(c) $\sin (13 \pi / 4)$
(d) $\sin (-13 \pi / 4)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:36

Problem 5

Specify the period and amplitude for each function.
(GRAPH CAN'T COPY)

Amy Jiang
Amy Jiang
Numerade Educator
01:49

Problem 5

Evaluate each expression (as in Example 1 ).
(a) $\cos (5 \pi / 4)$
(c) $\sin (5 \pi / 4)$
(b) $\cos (-5 \pi / 4)$
(d) $\sin (-5 \pi / 4)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:13

Problem 6

Specify the period and amplitude for each function.
(GRAPH CAN'T COPY)

Amy Jiang
Amy Jiang
Numerade Educator
03:38

Problem 6

Evaluate each expression (as in Example 1 ).
(a) $\cos (9 \pi / 4)$
(b) $\cos (-9 \pi / 4)$
(c) $\sin (9 \pi / 4)$
(d) $\sin (-9 \pi / 4)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
03:24

Problem 7

Specify the period and amplitude for each function.
(GRAPH CAN'T COPY)

Amy Jiang
Amy Jiang
Numerade Educator
03:19

Problem 7

Evaluate each expression (as in Example 1 ).
(a) $\sec (5 \pi / 3)$
(b) $\csc (-5 \pi / 3)$
(c) $\tan (5 \pi / 3)$
(d) $\cot (-5 \pi / 3)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:38

Problem 8

Specify the period and amplitude for each function.
(GRAPH CAN'T COPY)

Amy Jiang
Amy Jiang
Numerade Educator
02:23

Problem 8

Evaluate each expression (as in Example 1 ).
(a) $\sec (7 \pi / 4)$
(b) $\csc (-7 \pi / 4)$
(c) $\tan (7 \pi / 4)$
(d) $\cot (-7 \pi / 4)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:09

Problem 9

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$C$$

Amy Jiang
Amy Jiang
Numerade Educator
01:32

Problem 9

(a) List four positive real-number values of $t$ for which $\cos t=0$.
(b) List four negative real-number values of $t$ for which $\cos t=0$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
04:18

Problem 10

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$F$$

Amy Jiang
Amy Jiang
Numerade Educator
View

Problem 10

(a) List four positive real numbers $t$ such that $\sin t=1 / 2$.
(b) List four positive real numbers $t$ such that $\sin t=-1 / 2$
(c) List four negative real number $t$ such that $\sin t=1 / 2$.
(d) List four negative real numbers $t$ such that $\sin t=-1 / 2$

Danielle Fairburn
Danielle Fairburn
Numerade Educator
02:24

Problem 11

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$G$$

Amy Jiang
Amy Jiang
Numerade Educator
02:04

Problem 11

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) 2.06
(b) -2.06

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:03

Problem 12

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) 0.55
(b) -0.55

Amy Jiang
Amy Jiang
Numerade Educator
01:43

Problem 12

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places. )
(a) 0.55
(b) -0.55

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:58

Problem 13

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places.)
(a) $\pi / 6$
(b) $\pi / 6+2 \pi$

Amy Jiang
Amy Jiang
Numerade Educator
02:01

Problem 13

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places. )
(a) $\pi / 6$
(b) $\pi / 6+2 \pi$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:21

Problem 14

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$J$$

Amy Jiang
Amy Jiang
Numerade Educator
02:00

Problem 14

Use a calculator to evaluate the six trigonometric functions using the given real-number input. (Round the results to two decimal places. )
(a) 1000
(b) $1000-2 \pi$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
View

Problem 15

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$D$$

Tanvi Garg
Tanvi Garg
Numerade Educator
04:23

Problem 15

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\sin ^{2} t+\cos ^{2} t=1$
(a) $t=\pi / 3$
(b) $t=5 \pi / 4$
(c) $t=-53$

Bobby Barnes
Bobby Barnes
University of North Texas
View

Problem 16

In Exercises 9–20, evaluate each expression without using a calculator or tables
$\sec (3 \pi / 4)$

Tanvi Garg
Tanvi Garg
Numerade Educator
02:25

Problem 16

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\tan ^{2} t+1=\sec ^{2} t$
(a) $t=3 \pi / 4$
(b) $t=-2 \pi / 3$
(c) $t=\sqrt{5}$

Bobby Barnes
Bobby Barnes
University of North Texas
View

Problem 17

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$E$$

Tanvi Garg
Tanvi Garg
Numerade Educator
05:42

Problem 17

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\cot ^{2} t+1=\csc ^{2} t$
(a) $t=-\pi / 6$
(b) $t=7 \pi / 4$
(c) $t=0.12$

Bobby Barnes
Bobby Barnes
University of North Texas
View

Problem 18

Refer to the graph of $y=\sin x$ in the following figure. Specify the coordinates of the indicated points. Give the $x$ -coordinates both in terms of $\pi$ and as calculator approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$I$$

Tanvi Garg
Tanvi Garg
Numerade Educator
04:23

Problem 18

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\tan ^{2} t+1=\sec ^{2} t$
(a) $t=3 \pi / 4$
(b) $t=-2 \pi / 3$
(c) $t=\sqrt{5}$

Bobby Barnes
Bobby Barnes
University of North Texas
03:40

Problem 19

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\sin (-t)=-\sin t$
(a) $t=3 \pi / 2$
(b) $t=-5 \pi / 6$
(c) $t=13.24$

Bobby Barnes
Bobby Barnes
University of North Texas
03:40

Problem 19

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\sin (-t)=-\sin t$
(a) $t=3 \pi / 2$
(b) $t=-5 \pi / 6$
(c) $t=13.24$

Bobby Barnes
Bobby Barnes
University of North Texas
01:07

Problem 20

State whether the function $y=\sin x$ is increasing or decreasing on the given interval. (The terms increasing and decreasing are explained on page $159 .)$
$$-\pi / 2<x<\pi / 2$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
06:06

Problem 20

Check that both sides of the identity are in- deed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\tan (-t)=-\tan t$
(a) $t=-4 \pi / 3$
(b) $t=\pi / 4$
(c) $t=1000$

Bobby Barnes
Bobby Barnes
University of North Texas
04:31

Problem 21

Check that both sides of the identity are indeed equal for the given values of the variable $t$. For part $(c)$ of each problem, use your calculator.
$\sin (t+2 \pi)=\sin t$
(a) $t=5 \pi / 3$
(b) $t=-3 \pi / 2$
(c) $t=\sqrt{19}$

Bobby Barnes
Bobby Barnes
University of North Texas
04:31

Problem 21

Check that both sides of the identity are indeed equal for the given values of the variable t. For part (c) of each problem, use your calculator.
$\sin (t+2 \pi)=\sin t$
(a) $t=5 \pi / 3$
(b) $t=-3 \pi / 2$
(c) $t=\sqrt{19}$

Bobby Barnes
Bobby Barnes
University of North Texas
01:57

Problem 22

State whether the function $y=\sin x$ is increasing or decreasing on the given interval. (The terms increasing and decreasing are explained on page $159 .)$
$$-5 \pi / 2<x<-2 \pi$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
05:17

Problem 22

In Exercises $15-22,$ check that both sides of the identity are indeed equal for the given values of the variable $t .$ For part $(c)$ of each problem, use your calculator.
$\cos (t+2 \pi)=\cos t$
(a) $t=-5 \pi / 3$
(b) $t=\pi$
(c) $t=-\sqrt{3}$

Marcella Sippey
Marcella Sippey
Numerade Educator
00:43

Problem 23

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$
\cos 2 t=2 \cos t ; t=\pi / 6
$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
00:53

Problem 23

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\cos 2 t=2 \cos t ; t=\pi / 6$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
00:51

Problem 24

Exercises 21–30 are calculator exercises. (Set your calculator to the radian mode.) In Exercises 21–26, where
numerical answers are required, round your results to
three decimal places
Evaluate $\sin (0.78)$

Eric Tung
Eric Tung
Numerade Educator
00:43

Problem 24

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$\sin 2 t=2 \sin t ; t=\pi / 2$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:25

Problem 25

Refer to the graph of $y=\cos x$ in the following figure. Specify the coordinates of the indicated points.
Give the $x$-coordinates both in terms of $\pi$ and as calculator
approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$A$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:57

Problem 25

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
If $\sin t=-3 / 5$ and $\pi<t<3 \pi / 2,$ compute cos $t$ and $\tan t$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:28

Problem 26

Refer to the graph of $y=\cos x$ in the following figure. Specify the coordinates of the indicated points.
Give the $x$-coordinates both in terms of $\pi$ and as calculator
approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$G$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:58

Problem 26

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
If $\cos t=5 / 13$ and $3 \pi / 2<t<2 \pi,$ compute $\sin t$ and $\cot t$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
00:42

Problem 27

Refer to the graph of $y=\cos x$ in the following figure. Specify the coordinates of the indicated points.
Give the $x$-coordinates both in terms of $\pi$ and as calculator
approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$E$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
00:58

Problem 27

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
If $\sin t=\sqrt{3} / 4$ and $\pi / 2<t<\pi,$ compute $\tan t$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
00:58

Problem 28

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
If $\sec s=-\sqrt{13} / 2$ and $\sin s>0,$ compute tan $s$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:34

Problem 28

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
If $\sec s=-\sqrt{13} / 2$ and $\sin s>0,$ compute tan $s$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:18

Problem 29

Refer to the graph of $y=\cos x$ in the following figure. Specify the coordinates of the indicated points.
Give the $x$-coordinates both in terms of $\pi$ and as calculator
approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$I$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
06:38

Problem 29

If $\tan \alpha=12 / 5$ and $\cos \alpha>0,$ compute $\sec \alpha, \cos \alpha,$ and $\sin \alpha$

Bobby Barnes
Bobby Barnes
University of North Texas
01:09

Problem 30

Refer to the graph of $y=\cos x$ in the following figure. Specify the coordinates of the indicated points.
Give the $x$-coordinates both in terms of $\pi$ and as calculator
approximations rounded to three decimal places.
(GRAPH CAN'T COPY)
$$F$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
04:00

Problem 30

Show that the equation is not an identity by evaluating both sides using the given value of t and noting that the results are unequal.
$$
\text { If } \cot \theta=-1 / \sqrt{3} \text { and } \cos \theta<0, \text { compute } \csc \theta \text { and } \sin \theta
$$

Bobby Barnes
Bobby Barnes
University of North Texas
02:03

Problem 31

In the expression $\sqrt{9}-x^{2},$ make the substitution $x=3 \sin \theta\left(0<\theta<\frac{\pi}{2}\right),$ and show that the result is $3 \cos \theta$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:03

Problem 31

In the expression $\sqrt{9-x^{2}},$ make the substitution $x=3 \sin \theta\left(0<\theta<\frac{\pi}{2}\right),$ and show that the result is $3 \cos \theta$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
00:49

Problem 32

Make the substitution $u=2 \cos \theta$ in the expression $1 / \sqrt{4-u^{2}},$ and simplify the result. (Assume that $0<\theta<\pi .)$

James Kiss
James Kiss
Numerade Educator
01:48

Problem 32

Make the substitution $u=2 \cos \theta$ in the expression $1 / \sqrt{4-u^{2}},$ and simplify the result. (Assume that $0<\theta<\pi .)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:17

Problem 33

Exercises 21–30 are calculator exercises. (Set your calculator to the radian mode.) In Exercises 21–26, where
numerical answers are required, round your results to
three decimal places In the expression $\left(x^{2}-100\right)^{1 / 2},$ make the substitution $x=10 \sec \theta\left(0<\theta<\frac{\pi}{2}\right)$ and simplify the result.

Heather Zimmers
Heather Zimmers
Numerade Educator
03:13

Problem 33

In the expression $1 /\left(u^{2}-25\right)^{3 / 2},$ make the substitution $u=5 \sec \theta\left(0<\theta<\frac{\pi}{2}\right),$ and show that the result is
$\left(\cot ^{3} \theta\right) / 125$.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:52

Problem 34

State whether the function $y=\cos x$ is increasing or decreasing on the given interval.
$$6 \pi<x<7 \pi$$

Lourence Gonhovi
Lourence Gonhovi
Numerade Educator
02:41

Problem 34

In the expression $1 /\left(x^{2}+5\right)^{2},$ replace $x$ by $\sqrt{5} \tan \theta$ and show that the result is $\left(\cos ^{4} \theta\right) / 25$.

Bobby Barnes
Bobby Barnes
University of North Texas
02:37

Problem 35

In the expression $1 / \sqrt{u^{2}+7},$ let $u=\sqrt{7} \tan \theta,$ where $0<\theta<\pi / 2,$ and simplify the result.

Vysakh M
Vysakh M
Numerade Educator
02:50

Problem 36

In the expression $\sqrt{x^{2}-a^{2}} / x(a>0),$ let $x=a \sec \theta\left(0<\theta<\frac{\pi}{2}\right),$ and simplify the result.

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:24

Problem 37

(a) If $\sin t=2 / 3,$ find $\sin (-t)$
(b) If $\sin \phi=-1 / 4,$ find $\sin (-\phi)$
(c) If $\cos \alpha=1 / 5,$ find $\cos (-\alpha)$
(d) If $\cos s=-1 / 5,$ find $\cos (-s)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:11

Problem 38

(a) If $\sin t=0.35,$ find $\sin (-t)$
(b) If $\sin \phi=-0.47,$ find $\sin (-\phi)$
(c) If $\cos \alpha=0.21,$ find $\cos (-\alpha)$
(d) If $\cos s=-0.56,$ find $\cos (-s)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
03:34

Problem 39

If $\cos t=-1 / 3\left(\frac{\pi}{2}<t<\pi\right),$ compute the following:
(a) $\sin (-t)+\cos (-t)$
(b) $\sin ^{2}(-t)+\cos ^{2}(-t)$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
05:51

Problem 40

If $\sin (-s)=3 / 5\left(\pi<s<\frac{3 \pi}{2}\right),$ compute:
(a) $\sin s$
(b) $\cos (-s)$
(c) $\cos s$
(d) $\tan s+\tan (-s)$

Bobby Barnes
Bobby Barnes
University of North Texas
02:02

Problem 41

Use one of the identities $\cos (t+2 \pi k)=\cos t \operatorname{or} \sin (t+2 \pi k)=\sin t$ to evaluate each expression.
(a) $\cos \left(\frac{\pi}{4}+2 \pi\right)$
(b) $\sin \left(\frac{\pi}{3}+2 \pi\right)$
(c) $\sin \left(\frac{\pi}{2}-6 \pi\right)$

Bobby Barnes
Bobby Barnes
University of North Texas
07:55

Problem 42

Use one of the identities $\cos (t+2 \pi k)=\cos t \operatorname{or} \sin (t+2 \pi k)=\sin t$ to evaluate each expression.
(a) $\sin (17 \pi / 4)$
(b) $\sin (-17 \pi / 4)$
(c) $\cos 11 \pi$
(d) $\cos (53 \pi / 4)$
(e) $\tan (-7 \pi / 4)$
(f) $\cos (7 \pi / 4)$
(g) $\sec \left(\frac{11 \pi}{6}+2 \pi\right)$
(h) $\csc \left(2 \pi-\frac{\pi}{3}\right)$

Bobby Barnes
Bobby Barnes
University of North Texas
01:01

Problem 43

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\sin ^{2} t+\cos ^{2} t}{\tan ^{2} t+1}$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
00:44

Problem 44

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\sec ^{2} t-1}{\tan ^{2} t}$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:15

Problem 45

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\sec ^{2} \theta-\tan ^{2} \theta}{1+\cot ^{2} \theta}$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:26

Problem 46

Use the Pythagorean identities to simplify the given expressions.
$$\frac{\csc ^{4} \theta-\cot ^{4} \theta}{\csc ^{2} \theta+\cot ^{2} \theta}$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
04:24

Problem 47

Prove that the equations are identities.
$$\csc t=\sin t+\cot t \cos t$$

Bobby Barnes
Bobby Barnes
University of North Texas
01:48

Problem 48

Prove that the equations are identities.
$$\sin ^{2} t-\cos ^{2} t=\frac{1-\cot ^{2} t}{1+\cot ^{2} t}$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:35

Problem 49

Prove that the equations are identities.
$$\frac{1}{1+\sec s}+\frac{1}{1-\sec s}=-2 \cot ^{2} s$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:44

Problem 50

Prove that the equations are identities.
$$\frac{1+\tan s}{1-\tan s}=\frac{\sec ^{2} s+2 \tan s}{2-\sec ^{2} s}$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
02:09

Problem 51

Prove that the equations are identities.
$$\frac{\sec s+\cot s \csc s}{\cos s}=\csc ^{2} s \sec ^{2} s$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
01:14

Problem 52

Prove that the equations are identities.
$$(\tan \theta)\left(1-\cot ^{2} \theta\right)+(\cot \theta)\left(1-\tan ^{2} \theta\right)=0$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
04:47

Problem 53

Prove that the equations are identities.
$(\cos \alpha \cos \beta-\sin \alpha \sin \beta)(\cos \alpha \cos \beta+\sin \alpha \sin \beta)$
$$
=\cos ^{2} \alpha-\sin ^{2} \beta
$$

Bobby Barnes
Bobby Barnes
University of North Texas
09:41

Problem 54

Prove that the equations are identities.
$\cot \theta+\tan \theta+1=\frac{\cot \theta}{1-\tan \theta}+\frac{\tan \theta}{1-\cot \theta}$

Bobby Barnes
Bobby Barnes
University of North Texas
02:23

Problem 55

If $\sec t=13 / 5$ and $3 \pi / 2<t<2 \pi,$ evaluate
$$
\frac{2 \sin t-3 \cos t}{4 \sin t-9 \cos t}
$$

Zachary Mitchell
Zachary Mitchell
Numerade Educator
08:46

Problem 56

If sec $t=\left(b^{2}+1\right) / 2 b$ and $\pi<t<3 \pi / 2,$ find $\tan t$ and sin $t .$ (Note: $b$ is negative. Why?) You should assume that $b<-1$.

Bobby Barnes
Bobby Barnes
University of North Texas
02:54

Problem 57

Use the accompanying figure to explain why the following four identities are valid. (The identities can be used to provide an algebraic foundation for the reference-angle technique that we've used to evaluate the trigonometric functions.)
(GRAPH CAN'T COPY)
(i) $\sin (t+\pi)=-\sin t \quad$ (iii) $\cos (t+\pi)=-\cos t$
(ii) $\sin (t-\pi)=-\sin t$
(iv) $\cos (t-\pi)=-\cos t$

Bobby Barnes
Bobby Barnes
University of North Texas
01:34

Problem 58

Use two of the results in Exercise 57 to verify the identity $\tan (t+\pi)=\tan t .$ (You'll see the graphical aspect of this identity in Section $7.5 .)$

Bobby Barnes
Bobby Barnes
University of North Texas
09:32

Problem 59

In the equation $x^{4}+6 x^{2} y^{2}+y^{4}=32,$ make the substitutions $x=X \cos \frac{\pi}{4}-Y \sin \frac{\pi}{4} \quad$ and $\quad y=X \sin \frac{\pi}{4}+Y \cos \frac{\pi}{4}$
and show that the result simplifies to $X^{4}+Y^{4}=16$ (Hint: Evaluate the trigonometric functions, simplify the expressions for $x$ and $y,$ take out the common factor, and then substitute.)

Bobby Barnes
Bobby Barnes
University of North Texas
07:45

Problem 60

Suppose that $\tan \theta=2$ and $0<\theta<\pi / 2$
(a) Compute $\sin \theta$ and $\cos \theta$
(b) Using the values obtained in part (a), make the substitutions $x=X \cos \theta-Y \sin \theta \quad$ and $\quad y=X \sin \theta+Y \cos \theta$
in the expression $7 x^{2}-8 x y+y^{2},$ and simplify the result.

Bobby Barnes
Bobby Barnes
University of North Texas
01:35

Problem 61

In this exercise, we are going to find the minimum value of the function
$$
f(t)=\tan ^{2} t+9 \cot ^{2} t \quad 0<t<\frac{\pi}{2}
$$
(a) Set your calculator in the radian mode and complete the table. Round the values you obtain to two decimal places.
(b) Of the seven outputs you calculated in part (a), which is the smallest? What is the corresponding input?
(c) Prove that $\tan ^{2} t+9 \cot ^{2} t=(\tan t-3 \cot t)^{2}+6$
(d) Use the identity in part (c) to explain why $\tan ^{2} t+9 \cot ^{2} t \geq 6$
(e) The inequality in part (d) tells us that $f(t)$ is never less than 6. Furthermore, in view of part (c), $f(t)$ will equal 6 when $\tan t-3 \cot t=0 .$ From this last equation, show that $\tan ^{2} t=3,$ and conclude that $t=\pi / 3 .$ In summary, the minimum value of $f$ is $6,$ and this occurs when $t=\pi / 3 .$ How do these values compare with your answers in part (b)?

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
13:49

Problem 62

Let $f(\theta)=\sin \theta \cos \theta\left(0 \leq \theta \leq \frac{\pi}{2}\right)$
(a) Set your calculator in the radian mode and complete the table. Round the results to two decimal places.$$\begin{array}{ccccccccc}
\theta & 0 & \frac{\pi}{10} & \frac{\pi}{5} & \frac{\pi}{4} & \frac{3 \pi}{10} & \frac{2 \pi}{5} & \frac{\pi}{2} \\
\hline f(\theta) & & & & & & & \\
\hline
\end{array}$$
(b) What is the largest value of $f(\theta)$ in your table in part (a)?
(c) Show that $\sin \theta \cos \theta \leq 1 / 2$ for all real numbers $\theta$ in the interval $0 \leq \theta \leq \pi / 2 .$ Hint: Use the inequality $\sqrt{a b} \leq(a+b) / 2[\text { given in Exercise } 40(b)$ on
page $111]$, with $a=\sin \theta$ and $b=\cos \theta$
(d) Does the inequality $\sin \theta \cos \theta \leq \frac{1}{2}$ hold for all real numbers $\theta ?$

Bobby Barnes
Bobby Barnes
University of North Texas
06:05

Problem 63

Consider the equation
$$
2 \sin ^{2} t-\sin t=2 \sin t \cos t-\cos t
$$
(a) Evaluate each side of the equation when $t=\pi / 6$
(b) Evaluate each side of the equation when $t=\pi / 4$
(c) Is the given equation an identity?

Bobby Barnes
Bobby Barnes
University of North Texas
06:43

Problem 64

Suppose that $f(t)=(\sin t \cos t)(2 \sin t-1)(2 \cos t-1)(\tan t-1)$
(a) Compute each of the following: $f(0), f(\pi / 6), f(\pi / 4)$ $f(\pi / 3),$ and $f(\pi / 2)$
(b) Is the equation $f(t)=0$ an identity?

Bobby Barnes
Bobby Barnes
University of North Texas
03:10

Problem 65

Complete the tables in Exercises 65. Round (or, for exact values, simply report) the answers to six decimal places.
a)
$$
\begin{array}{lll}
t & 1-\frac{1}{2} t^{2} & \cos t \\
\hline 0.02 & & \\
0.05 & & \\
0.1 & & \\
0.2 & & \\
0.3 & & \\
\hline
\end{array}
$$
b) On the same set of axes, graph the functions $1-\frac{1}{2} t^{2}$ and $\cos t .$ Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and -1 to 1 in the $y$ -direction. What do you observe?

Bobby Barnes
Bobby Barnes
University of North Texas
03:23

Problem 66

Complete the tables.$ Round (or, for exact values, simply report) the answers to six decimal places.
(TABLE CAN'T COPY)
(b) On the same set of axes, graph the functions $t-\frac{1}{6} t^{3}$ and sin $t .$ Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and -1 to 1 in the $y$ -direction. What do you observe?

Bobby Barnes
Bobby Barnes
University of North Texas
03:49

Problem 67

Exercises $65-68$ are known as Taylor polynomials, after the English mathematician Brook Taylor $(1685-1731) .$ The theory of Taylor polynomials is developed in calculus.
(a) $$
\begin{array}{lll}
t & 1-\frac{1}{2} t^{2} & \cos t \\
\hline 0.02 & & \\
0.05 & & \\
0.1 & & \\
0.2 & & \\
0.3 & & \\
\hline
\end{array}
$$
(b) On the same set of axes, graph the functions $1-\frac{1}{2} t^{2}$ and $\cos t$. Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and -1 to 1 in the $y$ -direction. What do you observe?

Bobby Barnes
Bobby Barnes
University of North Texas
04:53

Problem 68

Complete the tables. Round (or, for exact values, simply report) the answers to six decimal places.
(TABLE CAN'T COPY)
(b) On the same set of axes, graph the functions $x^{2}+x$ $\frac{1}{3} x^{3}+x^{2}+x,$ and $e^{x} \sin x .$ Use a window extending from $-\pi$ to $\pi$ in the $x$ -direction and -10 to 10 in the y-direction. What do you observe?

Bobby Barnes
Bobby Barnes
University of North Texas
06:45

Problem 69

The figure on the following page shows two $x$ -y coordinate systems. (The same unit of length is used on all four axes.) In the coordinate system on the left, the curve is a portion of the unit circle
$$
x^{2}+y^{2}=1
$$
and $A$ is the point $(1,0) .$ The points $B, C, D, E,$ and $F$ are located on the circle according to the information in the following table.$$\begin{array}{llllll}
\hline \text { arc } & \widehat{A B} & \widehat{A C} & \widehat{A D} & \widehat{A E} & \widehat{A F} \\
\text { length } & \frac{\pi}{12} & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{5 \pi}{12} \\
\hline
\end{array}$$
Determine the $y$ -coordinates of the points $P, Q, R, S,$ and
T. Give an exact expression for each answer and, where appropriate, a calculator approximation rounded to three decimal places.

Julian Wong
Julian Wong
Numerade Educator