Deck 9: Trigonometric Identities and Equations

Question 1

Given $\sin y=-\frac{2}{5}, \cos x=-\frac{1}{3}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\cos (x-y)$

Answer 1

$\frac{\sqrt{21}-4 \sqrt{2}}{15}$

Question 2

Given $\sin y=-\frac{2}{5}, \cos x=-\frac{1}{3}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\sin (x+y)$

Answer 2

$\frac{2-2 \sqrt{41}}{15}$

Question 3

Given $\sin y=-\frac{2}{5}, \cos x=-\frac{1}{3}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\tan \frac{x}{2}$

Answer 3

$-\sqrt{2}$

Question 4

Given $\sin y=-\frac{2}{5}, \cos x=-\frac{1}{3}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\cos 2 x$

Question 5

Express $(\sec x+\tan x)(\sec x-\tan x)$ in terms of $\sin x$ and $\cos x$ , and simplify.

Question 6

Graph $\cot x-\csc x$ , and use the graph to conjecture an identity. Verify your conjecture analytically.
Use an identity to write each expression as a trigonometric function of $\theta$ alone.

Question 7

Verify that each equation is an identity.

- $\frac{\sin ^{2} 2 x}{1+\cos 2 x}=2 \sin ^{2} x$

Question 8

Verify that each equation is an identity.

- $2 \csc ^{2} \theta=\frac{1}{1-\cos \theta}+\frac{1}{1+\cos \theta}$

Question 9

Use an identity to write each expression as a trigonometric function of $\theta$ alone.
- $\sin \left(\frac{3 \pi}{2}-\theta\right) \quad$

Question 10

Use an identity to write each expression as a trigonometric function of $\theta$ alone.
- $\cos \left(180^{\circ}+\theta\right)$

Question 11

Consider the function defined by $f(x)=-\tan ^{-1} x$ .
(a) Sketch the graph.
(b) Give the domain and range.
(c) Explain why there is no $x$ such that $f(x)=\frac{\pi}{2}$ .

Question 12

Find the exact value of each expression.
- $\arcsin \left(\frac{1}{2}\right)$

Question 13

Find the exact value of each expression.
- $\sin \left(2 \arccos \frac{1}{5}\right)$

Question 14

Find the exact value of each expression.
- $\sec ^{-1}(2)$

Question 15

Find the exact value of each expression.
- $\cos ^{-1}\left(\cos \frac{3 \pi}{2}\right)$

Question 16

Find the exact value of each expression.
- $\sin \left(\cos ^{-1} \frac{\sqrt{2}}{5}\right)$

Question 17

Find the exact value of each expression.
- $\tan ^{-1}\left(\frac{\sqrt{3}}{3}\right)$

Question 18

Write as an algebraic expression in $u, u>0$ .
- $\csc \left(\sin ^{-1} u\right)$

Question 19

Write as an algebraic expression in $u, u>0$ .
- $\cos (\arctan u)$

Question 20

Solve each equation or inequality over the indicated interval.
- $2 \sin x+\sqrt{3} \leq 0, \quad[0,2 \pi)$

Question 21

Solve each equation or inequality over the indicated interval.
- $2 \sin ^{2} x+3 \sin x=-1,[0,2 \pi)$

Question 22

Solve each equation or inequality over the indicated interval.
- $\sin 2 \theta+3 \cos \theta=0, \quad\left[0^{\circ}, 360^{\circ}\right)$

Question 23

Solve each equation or inequality over the indicated interval.
- $3 \tan ^{2} \theta-1=0,\left[0^{\circ}, 360^{\circ}\right)$

Question 24

Solve each equation or inequality over the indicated interval.
- $\csc \frac{x}{3}=\sin \frac{x}{3},[0,2 \pi)$

Question 25

Suppose that the formula $A(t)=\frac{1}{3} \sin \frac{\pi}{4} t+\frac{2}{3}$ describes the motion formed by a rhythmically moving arm during an 8-minute time period where $A(t)$ is the angle (in radians) formed by the arm at time $t$ (in minutes).
(a) Give the domain and range of $A$ .
(b) Graph $A(t)$ over its domain.
(c) Use the graph to determine the maximum and minimum values of $A(t)$ and when they occur.
(d) Find $A(1)$ analytically and check your result graphically. Use symmetry to find $A(3)$ .
(e) When is the angle $\frac{2}{3}$ radians?
(f) Write the equation $A=\frac{1}{3} \sin \frac{\pi}{4} t+\frac{2}{3}$ as an equation involving arcsine by solving for $t$ .

Question 26

Given $\sin y=\frac{3}{5}, \cos x=-\frac{1}{4}$ with $\frac{\pi}{2}<x<\pi$ and $\frac{\pi}{2}<\mathrm{y}<\pi$ , find the exact values for the following:
- $\cos (x+y)$

Question 27

Given $\sin y=\frac{3}{5}, \cos x=-\frac{1}{4}$ with $\frac{\pi}{2}<x<\pi$ and $\frac{\pi}{2}<\mathrm{y}<\pi$ , find the exact values for the following:
- $\sin (x+y)$

Question 28

Given $\sin y=\frac{3}{5}, \cos x=-\frac{1}{4}$ with $\frac{\pi}{2}<x<\pi$ and $\frac{\pi}{2}<\mathrm{y}<\pi$ , find the exact values for the following:
- $\cos \frac{x}{2}$

Question 29

Given $\sin y=\frac{3}{5}, \cos x=-\frac{1}{4}$ with $\frac{\pi}{2}<x<\pi$ and $\frac{\pi}{2}<\mathrm{y}<\pi$ , find the exact values for the following:
- $\tan 2 x$

Question 30

Express $\cot ^{2} x-\csc ^{2} x$ in terms of $\sin x$ and $\cos x$ , and simplify.

Question 31

Graph $\tan x+\sec x$ , and use the graph to conjecture an identity. Verify your conjecture analytically.

Question 32

Verify that each equation is an identity.

- $(\sin x+\cos x)^{2}=\sin 2 x+1$

Question 33

Verify that each equation is an identity.

- $\csc ^{2} x+\cot ^{2} x=\frac{1+\cos ^{2} x}{1-\cos ^{2} x}$

Question 34

Use an identity to write each expression as a trigonometric function of $\theta$ alone.
- $\sin (3 \pi+\theta)$

Question 35

Use an identity to write each expression as a trigonometric function of $\theta$ alone.
- $\cos \left(270^{\circ}+\theta\right)$

Question 36

Consider the function defined by $f(x)=-\frac{1}{3} \sin ^{-1} x$ .
(a) Sketch the graph.
(b) Give the domain and range.
(c) Explain why $f(-2)$ is not defined.

Question 37

Find the exact value of each expression.
- $\arctan \left(\frac{\sqrt{3}}{3}\right)$

Question 38

Find the exact value of each expression.
- $\sin \left(2 \arccos \frac{3}{7}\right)$

Question 39

Find the exact value of each expression.
- $\sec ^{-1}\left(-\frac{2 \sqrt{3}}{3}\right)$

Question 40

Find the exact value of each expression.
- $\sin ^{-1}\left(\sin \frac{5 \pi}{4}\right)$

Question 41

Find the exact value of each expression.
- $\sin \left(\cos ^{-1} \frac{2}{3}\right)$

Question 42

Find the exact value of each expression.
- $\tan ^{-1}(1)$

Question 43

Write as an algebraic expression in $u, u>0$ .
- $\csc \left(\tan ^{-1} u\right)$

Question 44

Write as an algebraic expression in $u, u>0$ .
- $\cot (\arcsin u)$

Question 45

Solve each equation or inequality over the indicated interval.
- $2 \cos x-1 \leq 0, \quad[-\pi, \pi]$

Question 46

Solve each equation or inequality over the indicated interval.
- $\sin ^{2} x+\sin x=0, \quad[0,2 \pi)$

Question 47

Solve each equation or inequality over the indicated interval.
- $\sin 2 \theta+2 \sin \theta=0, \quad\left[0^{\circ}, 360^{\circ}\right)$

Question 48

Solve each equation or inequality over the indicated interval.
- $\cos ^{2} \theta-2 \cos \theta-3=0, \quad\left[0^{\circ}, 360^{\circ}\right)$

Question 49

Solve each equation or inequality over the indicated interval.
- $\sec \frac{x}{4}=\cos \frac{x}{4}, \quad[0,4 \pi)$

Question 50

Suppose that the formula $A(t)=-\frac{1}{4} \cos \frac{\pi}{8} t+\frac{1}{4}$ describes the motion formed by a rhythmically moving arm during a 16 minute time period where $A(t)$ is the angle (in radians) formed by the arm at time $t$ (in minutes).
(a) Give the domain and range of $A$ .
(b) Graph $A(t)$ over its domain.
(c) Use the graph to determine the maximum and minimum values of $A(t)$ and when they occur.
(d) Find $A(6)$ analytically and check your result graphically. Use symmetry to find $A(10)$ .
(e) When is the angle $\frac{1}{4}$ radians?
(f) Write the equation $A=-\frac{1}{4} \cos \frac{\pi}{8} t+\frac{1}{4}$ as an equation involving arcsine by solving for $t$ .

Question 51

Given $\sin y=-\frac{1}{3}, \cos x=-\frac{2}{5}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\cos (x-y)$

Question 52

Given $\sin y=-\frac{1}{3}, \cos x=-\frac{2}{5}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\sin (x+y)$

Question 53

Given $\sin y=-\frac{1}{3}, \cos x=-\frac{2}{5}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\tan \frac{y}{2}$

Question 54

Given $\sin y=-\frac{1}{3}, \cos x=-\frac{2}{5}$ with $\frac{\pi}{2}<x<\pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\sin 2 x$

Question 55

Express $\left(\tan ^{2} x+\csc ^{2} x\right)-\left(\sec ^{2} x+\cot ^{2} x\right)$ in terms of $\sin x$ and $\cos x$ , and simplify.

Question 56

Graph $\csc x+\cot x$ , and use the graph to conjecture an identity. Verify your conjecture analytically.

Question 57

Verify that each equation is an identity.

- $\frac{2 \tan x}{1+\tan ^{2} x}=\sin 2 x$

Question 58

Verify that each equation is an identity.

- $\sec ^{2} x-1=\frac{\tan ^{2} x+\cos ^{2} x-1}{1-\cos ^{2} x}$

Question 59

Use an identity to write each expression as a trigonometric function of $\theta$ alone.
- $\sin \left(\theta+90^{\circ}\right)$

Question 60

Use an identity to write each expression as a trigonometric function of $\theta$ alone.
- $\cos (\theta-3 \pi)$

Question 61

Consider the function defined by $f(x)=3 \cos ^{-1} x$ .
(a) Sketch the graph.
(b) Give the domain and range.
(c) Explain why $f(1.2)$ is not defined.

Question 62

Find the exact value of each expression.
- $\arcsin \frac{\sqrt{3}}{2}$

Question 63

Find the exact value of each expression.
- $\sin \left(2 \arccos \frac{3}{5}\right)$

Question 64

Find the exact value of each expression.
- $\sec ^{-1}(-2)$

Question 65

Find the exact value of each expression.
- $\tan ^{-1}\left(\tan \frac{5 \pi}{6}\right)$

Question 66

Find the exact value of each expression.
- $\cos \left(\sin ^{-1} \frac{2}{7}\right)$

Question 67

Find the exact value of each expression.
- $\tan ^{-1}(\sqrt{3})$

Question 68

Write as an algebraic expression in $u, u>0$ .
- $\sec \left(\tan ^{-1} u\right)$

Question 69

Write as an algebraic expression in $u, u>0$ .
- $\sec (\arccos u)$

Question 70

Solve each equation or inequality over the indicated interval.
- $1+\sqrt{2} \sin x \geq 0, \quad[0,2 \pi)$

Question 71

Solve each equation or inequality over the indicated interval.
- $\sec ^{2} x=2 \tan x, \quad[0,2 \pi)$

Question 72

Solve each equation or inequality over the indicated interval.
- $\sin 2 \theta=2 \cos ^{2} \theta, \quad\left[0^{\circ}, 360^{\circ}\right)$

Question 73

Solve each equation or inequality over the indicated interval.
- $2 \sin ^{2} \theta-5 \sin \theta+2=0, \quad\left[0^{\circ}, 360^{\circ}\right)$

Question 74

Solve each equation or inequality over the indicated interval.
- $\cot \frac{x}{4}=\tan \frac{x}{4}, \quad[0,4 \pi)$

Question 75

Suppose that the formula $A(t)=-\frac{1}{2} \cos \frac{\pi}{8} t+\frac{1}{6}$ describes the motion formed by a rhythmically moving arm during a 16 minute time period where $A(t)$ is the angle (in radians) formed by the arm at time $t$ (in minutes).
(a) Give the domain and range of $A$ .
(b) Graph $A(t)$ over its domain.
(c) Use the graph to determine the maximum and minimum values of $A(t)$ and when they occur.
(d) Find $A(6)$ analytically and check your result graphically. Use symmetry to find $A(10)$ .
(e) When is the angle $\frac{1}{6}$ radians?
(f) Write the equation $A=-\frac{1}{2} \cos \frac{\pi}{8} t+\frac{1}{6}$ as an equation involving arcsine by solving for $t$ .

Question 76

Given $\sin y=-\frac{4}{5}, \cos x=\frac{1}{2}$ with $\frac{3 \pi}{2}<x<2 \pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\cos (x+y)$

Question 77

Given $\sin y=-\frac{4}{5}, \cos x=\frac{1}{2}$ with $\frac{3 \pi}{2}<x<2 \pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\sin (x-y)$

Question 78

Given $\sin y=-\frac{4}{5}, \cos x=\frac{1}{2}$ with $\frac{3 \pi}{2}<x<2 \pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\tan \frac{x}{2}$

Question 79

Given $\sin y=-\frac{4}{5}, \cos x=\frac{1}{2}$ with $\frac{3 \pi}{2}<x<2 \pi$ and $\pi<y<\frac{3 \pi}{2}$ , find the exact values for the following:
- $\sin 2 x$

Question 80

Express $\frac{2}{\sec ^{2} x}+\frac{2}{\csc ^{2} x}$ in terms of $\sin x$ and $\cos x$ , and simplify.