Precalculus

Cynthia Young

Chapter 5

Trigonometric Functions of Real Numbers - all with Video Answers

Educators

Section 1

Trigonometric Functions: The Unit Circle Approach

Problem 1

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\sin \left(\frac{5 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 2

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\cos \left(\frac{5 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 3

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\cos \left(\frac{7 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 4

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\sin \left(\frac{7 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 5

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\sin \left(\frac{3 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 6

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\cos \left(\frac{3 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 7

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\tan \left(\frac{7 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 8

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\cot \left(\frac{7 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 9

Find the exact values of the indicated trigonometric functions using the unit circle.
$$(5 \pi)$$

Abhishek Kumar

Numerade Educator

Problem 10

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\csc \left(\frac{5 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 11

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\tan \left(\frac{4 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 12

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\cot \left(\frac{11 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 13

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\csc \left(\frac{5 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 14

Find the exact values of the indicated trigonometric functions using the unit circle.
$$\cot \left(\frac{2 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 15

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin \left(-\frac{2 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 16

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin \left(-\frac{5 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 17

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin \left(-\frac{\pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 18

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\operatorname{in}\left(-\frac{7 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 19

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{3 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 20

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{5 \pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 21

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{5 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 22

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{7 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 23

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin \left(-\frac{5 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 24

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin (-\pi)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 25

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin \left(-\frac{3 \pi}{2}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 26

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sin \left(-\frac{\pi}{3}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 27

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{\pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 28

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{3 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 29

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{\pi}{2}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 30

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cos \left(-\frac{7 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 31

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\csc \left(-\frac{5 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 32

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\sec \left(-\frac{7 \pi}{4}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 33

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\tan \left(-\frac{11 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 34

Use the unit circle and the fact that sine is an odd function and cosine is an even function to find the exact values of the indicated functions.
$$\cot \left(-\frac{11 \pi}{6}\right)$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 35

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cos \theta=\frac{\sqrt{3}}{2}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 36

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cos \theta=-\frac{\sqrt{3}}{2}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 37

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sin \theta=-\frac{\sqrt{3}}{2}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 38

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sin \theta=\frac{\sqrt{3}}{2}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 39

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sin \theta=0,0 \leq \theta \leq 4 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 40

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sin \theta=-1,0 \leq \theta \leq 4 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 41

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cos \theta=-1,0 \leq \theta \leq 4 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 42

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cos \theta=0,0 \leq \theta \leq 4 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 43

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\tan \theta=-1,0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 44

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cot \theta=1,0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 45

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sec \theta=-\sqrt{2}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 46

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\csc \theta=\sqrt{2}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 47

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\csc \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 48

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sec \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 49

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\tan \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 50

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cot \theta \text { is undefined, } 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 51

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\csc \theta=-2,0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 52

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\cot \theta=-\sqrt{3}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 53

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\sec \theta=\frac{2 \sqrt{3}}{3}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 54

Use the unit circle to find all of the exact values of $\theta$ that make the equation true in the indicated interval.
$$\tan \theta=\frac{\sqrt{3}}{3}, 0 \leq \theta \leq 2 \pi$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 55

Refer to the following:
The average daily temperature in Peoria, Illinois, can be predicted by the formula $T=50-28 \cos \left[\frac{2 \pi(x-31)}{365}\right],$ where $x$ is the number of the day in the year (January $1=1$, February $1=32$, etc.) and $T$ is in degrees Fahrenheit.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 56

Refer to the following:
The average daily temperature in Peoria, Illinois, can be predicted by the formula $T=50-28 \cos \left[\frac{2 \pi(x-31)}{365}\right],$ where $x$ is the number of the day in the year (January $1=1$, February $1=32$, etc.) and $T$ is in degrees Fahrenheit.
Atmospheric Temperature. What is the expected temperature on August $15 ?$ (Assume it is not a leap year.)

Cullen Miller

Numerade Educator

Problem 57

Refer to the following:
The human body temperature normally fluctuates during the day. A person's body temperature can be predicted by the formula $T=99.1-0.5 \sin \left(x+\frac{\pi}{12}\right),$ where $x$ is the number of hours since midnight and $T$ is in degrees Fahrenheit.
What is the person's temperature at $6.00 \mathrm{A} \cdot \mathrm{M} . ?$

Abhishek Kumar

Numerade Educator

Problem 58

Refer to the following:
The human body temperature normally fluctuates during the day. A person's body temperature can be predicted by the formula $T=99.1-0.5 \sin \left(x+\frac{\pi}{12}\right),$ where $x$ is the number of hours since midnight and $T$ is in degrees Fahrenheit.
What is the person's temperature at 9: 00 RM.?

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 59

Refer to the following:
The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula $h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]$ where $x$ is the number of hours since midnight and $h$ is the height of the tide in feet.
What is the height of the tide at 3.00 P.M.?
(IMAGE CANNOT COPY)

Abhishek Kumar

Numerade Educator

Problem 60

Refer to the following:
The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula $h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]$ where $x$ is the number of hours since midnight and $h$ is the height of the tide in feet.
What is the height of the tide at 5.00 A.M.?

Abhishek Kumar

Numerade Educator

Problem 61

Refer to the following:
The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula $h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]$ where $x$ is the number of hours since midnight and $h$ is the height of the tide in feet.
A woman has been yo-yo dieting for years. Her weight changes throughout the year as she gains and loses weight. Her weight in a particular month can be determined by the formula $w(x)=145+10 \cos \left(\frac{\pi}{6} x\right),$ where $x$ is the month and $w$ is in pounds. If $x=1$ corresponds to January, how much does she weigh in June?

Abhishek Kumar

Numerade Educator

Problem 62

Refer to the following:
The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula $h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]$ where $x$ is the number of hours since midnight and $h$ is the height of the tide in feet.
How much does the woman in Exercise 61 weigh in December?

Abhishek Kumar

Numerade Educator

Problem 63

Refer to the following:
The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula $h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]$ where $x$ is the number of hours since midnight and $h$ is the height of the tide in feet.
The average number of guests visiting the Magic Kingdom at Walt Disney World per day is given by $n(x)=30,000+20,000 \sin \left[\frac{\pi}{2}(x+1)\right],$ where $n$ is the number of guests and $x$ is the month. If January corresponds to $x=1$ how many people on average are visiting the Magic Kingdom per day in February?

Abhishek Kumar

Numerade Educator

Problem 64

Refer to the following:
The height of the water in a harbor changes with the tides. The height of the water at a particular hour during the day can be determined by the formula $h(x)=5+4.8 \sin \left[\frac{\pi}{6}(x+4)\right]$ where $x$ is the number of hours since midnight and $h$ is the height of the tide in feet.
How many guests are visiting the Magic Kingdom in Exercise 63 in December?

Abhishek Kumar

Numerade Educator

Problem 65

Explain the mistake that is made.
Use the unit circle to evaluate $\tan \left(\frac{5 \pi}{6}\right)$ exactly.
$\begin{array}{l}\text { Tangent is the } \\ \text { ratio of sine } \\ \text { to cosine. }\end{array} \tan \left(\frac{5 \pi}{6}\right)=\frac{\sin \left(\frac{5 \pi}{6}\right)}{\cos \left(\frac{5 \pi}{6}\right)}$
Use the unit circle to $\begin{array}{l}\text { identify sine } \\ \text { and cosine. }\end{array} \quad \sin \left(\frac{5 \pi}{6}\right)=-\frac{\sqrt{3}}{2}$ and $\cos \left(\frac{5 \pi}{6}\right)=\frac{1}{2}$
Substitute values for $\begin{array}{l}\text { sine and } \\ \text { cosine. }\end{array} \quad \tan \left(\frac{5 \pi}{6}\right)=\frac{-(\sqrt{3} / 2)}{1 / 2}$
Simplify. $\tan \left(\frac{5 \pi}{6}\right)=-\sqrt{3}$
This is incorrect. What mistake was made?

Abhishek Kumar

Numerade Educator

Problem 66

Explain the mistake that is made.
Use the unit circle to evaluate $\sec \left(\frac{11 \pi}{6}\right)$ exactly.
Solution:
$\begin{array}{l}\text { Secant is the reciprocal } \\ \text { of cosine. }\end{array} \quad \sec \left(\frac{11 \pi}{6}\right)=\frac{1}{\cos \left(\frac{11 \pi}{6}\right)}$
$\begin{array}{l}\text { Use the unit circle } \\ \text { to evaluate cosine. }\end{array} \quad \cos \left(\frac{11 \pi}{6}\right)=-\frac{1}{2}$
$\begin{array}{l}\text { Substitute the value } \\ \text { for cosine. }\end{array} \quad \sec \left(\frac{11 \pi}{6}\right)=\frac{1}{-\frac{1}{2}}$
Simplify. $\quad \sec \left(\frac{11 \pi}{6}\right)=-2$
This is incorrect. What mistake was made?

Abhishek Kumar

Numerade Educator

Problem 67

Determine whether each statement is true or false.
$\sin (2 n \pi+\theta)=\sin \theta, n$ an integer.

Robert Leedy

Numerade Educator

Problem 68

Determine whether each statement is true or false.
$\cos (2 n \pi+\theta)=\cos \theta, n$ an integer.

Abhishek Kumar

Numerade Educator

Problem 69

Determine whether each statement is true or false.
$\sin \theta=1$ when $\theta=\frac{(2 n+1) \pi}{2}, n$ an integer.

Abhishek Kumar

Numerade Educator

Problem 70

Determine whether each statement is true or false.
$\cos \theta=1$ when $\theta=n \pi, n$ an integer.

Abhishek Kumar

Numerade Educator

Problem 71

Determine whether each statement is true or false.
$\tan (\theta+2 n \pi)=\tan \theta, n$ an integer.

Abhishek Kumar

Numerade Educator

Problem 72

Determine whether each statement is true or false.
$\tan \theta=0$ if and only if $\theta=\frac{(2 n+1) \pi}{2}, n$ an integer.

Abhishek Kumar

Numerade Educator

Problem 73

Determine whether each statement is true or false.
Is cosecant an even or an odd function? Justify your answer.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 74

Determine whether each statement is true or false.
Is tangent an even or an odd function? Justify your answer.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 75

Find all the values of $\theta, 0 \leq \theta \leq 2 \pi,$ for which the equation $\sin \theta=\cos \theta$ is true.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 76

Find all the values of $\theta(\theta$ is any real number) for which the equation $\sin \theta=\cos \theta$ is true.

Abhishek Kumar

Numerade Educator

Problem 77

Find all the values of $\theta, 0 \leq \theta \leq 2 \pi,$ for which the equation $2 \sin \theta=\csc \theta$ is true.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 78

Find all the values of $\theta, 0 \leq \theta \leq 2 \pi,$ for which the
equation $\cos \theta=\frac{1}{4} \sec \theta$ is true.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 79

Find all the values of $\theta(\theta$ is any real number) for which the equation $3 \csc \theta=4 \sin \theta$ is true.

Abhishek Kumar

Numerade Educator

Problem 80

Find all the values of $\theta(\theta$ is any real number) for which the equation $4 \cos \theta=3 \sec \theta$ is true.

Abhishek Kumar

Numerade Educator

Problem 81

Does there exist an angle $0 \leq \theta<2 \pi$ such that $\tan \theta=\cot \theta ?$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 82

Does there exist an angle $0 \leq \theta<2 \pi$ such that $\sec \theta=\csc (-\theta) ?$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 83

Use a calculator to approximate $\sin 423^{\circ} .$ What do you expect $\sin \left(-423^{\circ}\right)$ to be? Verify your answer with a calculator.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 84

Use a calculator to approximate $\cos 227^{\circ} .$ What do you expect $\cos \left(-227^{\circ}\right)$ to be? Verify your answer with a calculator.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 85

Use a calculator to approximate $\tan 81^{\circ} .$ What do you expect $\tan \left(-81^{\circ}\right)$ to be? Verify your answer with a calculator.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 86

Use a calculator to approximate cse $211^{\circ} .$ What do you expect $\csc \left(-211^{\circ}\right)$ to be? Verify your answer with a calculator.

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 87

For Exercises $87-90$, refer to the following:
Set the calculator in parametric and radian modes and let
$$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$
(TABLE CANNOT COPY)
Set the window so that $0 \leq \mathrm{T} \leq 2 \pi,$ step $=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2$ and $-2 \leq Y \leq 2 .$ To approximate the sine or cosine of a T value, use the $[\text { TRACE }]$ key, type in the T value, and read the corresponding coordinates from the screen.
Approximate $\cos \left(\frac{\pi}{3}\right),$ take 5 steps of $\frac{\pi}{15}$ each, and read the $x$ -coordinate.

Victor Salazar

Numerade Educator

Problem 88

For Exercises $87-90$, refer to the following:
Set the calculator in parametric and radian modes and let
$$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$
(TABLE CANNOT COPY)
Set the window so that $0 \leq \mathrm{T} \leq 2 \pi,$ step $=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2$ and $-2 \leq Y \leq 2 .$ To approximate the sine or cosine of a T value, use the $[\text { TRACE }]$ key, type in the T value, and read the corresponding coordinates from the screen.
Approximate $\sin \left(\frac{\pi}{3}\right),$ take 5 steps of $\frac{\pi}{15}$ each, and read the $y$ -coordinate.

Victor Salazar

Numerade Educator

Problem 89

For Exercises $87-90$, refer to the following:
Set the calculator in parametric and radian modes and let
$$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$
(TABLE CANNOT COPY)
Set the window so that $0 \leq \mathrm{T} \leq 2 \pi,$ step $=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2$ and $-2 \leq Y \leq 2 .$ To approximate the sine or cosine of a T value, use the $[\text { TRACE }]$ key, type in the T value, and read the corresponding coordinates from the screen.
Approximate $\sin \left(\frac{2 \pi}{3}\right)$ to four decimal places.

Victor Salazar

Numerade Educator

Problem 90

For Exercises $87-90$, refer to the following:
Set the calculator in parametric and radian modes and let
$$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$
(TABLE CANNOT COPY)
Set the window so that $0 \leq \mathrm{T} \leq 2 \pi,$ step $=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2$ and $-2 \leq Y \leq 2 .$ To approximate the sine or cosine of a T value, use the $[\text { TRACE }]$ key, type in the T value, and read the corresponding coordinates from the screen.
Approximate $\cos \left(\frac{5 \pi}{4}\right)$ to four decimal places.

Victor Salazar

Numerade Educator

Problem 91

$$\int_{0}^{\pi} \sin x d x$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 92

$$\int_{\pi / 4}^{5 \pi / 6} \cos x d x$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 93

$$\int_{7 \pi / 6}^{5 \pi / 4} \sec ^{2} x d x$$

Kimberly Waterbury

Kimberly Waterbury

Numerade Educator

Problem 94

$$\int_{S \pi / 3}^{11 \pi / 6} \csc x \cot x d x$$

Abhishek Kumar

Numerade Educator