Question 1

Use a sum or difference identity to find the exact value.
-$$\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }$$

Accepted Answer

A)  $\sec \theta \csc \theta$ 
B)  $- 2 \tan ^ { 2 } \theta$ 
C)  $1 + \cot \theta$ 
D)  $\sin \theta \tan \theta$ 
A)  $\sec \theta \csc \theta$ 
B)  $- 2 \tan ^ { 2 } \theta$ 
C)  $1 + \cot \theta$ 
D)  $\sin \theta \tan \theta$

Question 2

Solve the equation for exact solutions.
-$$\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }$$

Accepted Answer

The answer of Solve the equation for exact solutions.
-\[\frac {...

Question 3

Find the exact value of the expression using the provided information.
-$$\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0$$

Accepted Answer

A)  Not an identity 
B)  Identity 
A)  Not an identity 
B)  Identity

Question 4

Determine all solutions of the equation in radians.
-Find $\cos \frac { \theta } { 2 }$, given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$.

Accepted Answer

A)  $\frac { \sqrt { 5 } } { 5 }$ 
B)  $- \frac { \sqrt { 5 } } { 5 }$ 
C)  $\frac { \sqrt { 30 } } { 10 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$ 
A)  $\frac { \sqrt { 5 } } { 5 }$ 
B)  $- \frac { \sqrt { 5 } } { 5 }$ 
C)  $\frac { \sqrt { 30 } } { 10 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$

Question 5

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-cos 22° = cos 49°cos 27° - sin 49°sin 27°

Accepted Answer

A) True 
 B)False

Question 6

Verify that each equation is an identity.
-$$\frac { 1 + \cos t } { 1 - \cos t } - \frac { 1 - \cos t } { 1 + \cos t } = 4 \cot t \csc t$$

Accepted Answer

\[\begin{array} { l } 
\frac { 1 + \cos

Question 7

Use identities to fill in the blank with the appropriate trigonometric function name.
-$$\cos \frac { \pi } { 11 } - \cos \frac { \pi } { 2 }$$

Accepted Answer

A)  $- 2 \cos \frac { 13 \pi } { 44 } \cos \left( \frac { - 9 \pi } { 44 } \right)$ 
B)  $- 2 \sin \frac { 13 \pi } { 44 } \sin \left( \frac { - 9 \pi } { 44 } \right)$ 
C)  $- 2 \sin \frac { 13 \pi } { 44 } \cos \left( \frac { - 9 \pi } { 44 } \right)$ 
D)  $- 2 \cos \frac { 13 \pi } { 44 } \sin \left( \frac { - 9 \pi } { 44 } \right)$ 
A)  $- 2 \cos \frac { 13 \pi } { 44 } \cos \left( \frac { - 9 \pi } { 44 } \right)$ 
B)  $- 2 \sin \frac { 13 \pi } { 44 } \sin \left( \frac { - 9 \pi } { 44 } \right)$ 
C)  $- 2 \sin \frac { 13 \pi } { 44 } \cos \left( \frac { - 9 \pi } { 44 } \right)$ 
D)  $- 2 \cos \frac { 13 \pi } { 44 } \sin \left( \frac { - 9 \pi } { 44 } \right)$

Question 8

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\tan \left( 30 ^ { \circ } + x \right)$

Accepted Answer

A)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
B)  $- \tan x$ 
C)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
D)  $\tan x$ 
A)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
B)  $- \tan x$ 
C)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
D)  $\tan x$

Question 9

Rewrite the following as a product of trigonometric functions.
-sin 2x = 2 sin x

Accepted Answer

A) Identity 
B) Not an identity 
A) Identity 
B) Not an identity

Question 10

Verify that each equation is an identity.
-$$\cos ( 3 x ) = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$$

Accepted Answer

\[\begin{array} { l } 
\cos ( 3 x ) = \c

Question 11

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { 4 + 7 \tan x } { \sin x } = 4 \csc x + 7 \sec x$$

Accepted Answer

A)  Identity 
B)  Not an Identity 
A)  Identity 
B)  Not an Identity

Question 12

Verify that each equation is an identity.
-$$\cos ^ { 4 } x = \frac { 1 } { 8 } ( 3 + 4 \cos ( 2 x ) + \cos ( 4 x ) )$$

Accepted Answer

\[\begin{array} { l } 
\cos ^ { 4 } x =

Question 13

Use Identities to find the exact value.
-Find $\cos ( s - t )$ given that $\cos s = - \frac { 1 } { 2 }$, with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $t$ in quadrant III.

Accepted Answer

A)  $\frac { - 3 + 4 \sqrt { 3 } } { 10 }$ 
B)  $\frac { 3 + 4 \sqrt { 3 } } { 10 }$ 
C)  $\frac { 3 - 4 \sqrt { 5 } } { 10 }$ 
D)  $\frac { 3 - 4 \sqrt { 3 } } { 10 }$ 
A)  $\frac { - 3 + 4 \sqrt { 3 } } { 10 }$ 
B)  $\frac { 3 + 4 \sqrt { 3 } } { 10 }$ 
C)  $\frac { 3 - 4 \sqrt { 5 } } { 10 }$ 
D)  $\frac { 3 - 4 \sqrt { 3 } } { 10 }$

Question 14

Use an identity to write the expression as a single trigonometric function or as a single number.
-$2 \cos ^ { 2 } 4 x - 1$

Accepted Answer

A)  $\frac { 1 } { 2 } \sin 16 x$ 
B)  $\cos 8 x$ 
C)  $\cos 4 x$ 
D)  $2 \sin 4 x$ 
A)  $\frac { 1 } { 2 } \sin 16 x$ 
B)  $\cos 8 x$ 
C)  $\cos 4 x$ 
D)  $2 \sin 4 x$

Question 15

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { 1 + \tan ^ { 2 } x } { \sec x }$$

Accepted Answer

A)  1 
B)  $- \sec x$ 
C)  $\sec x$ 
D)  $\csc x$ 
A)  1 
B)  $- \sec x$ 
C)  $\sec x$ 
D)  $\csc x$

Question 16

Decide whether the expression is or is not an identity.
-tan 80°

Accepted Answer

A) cot 10° 
B) tan 10° 
C) cot 170° 
D) cot 100° 
A) cot 10° 
B) tan 10° 
C) cot 170° 
D) cot 100°

Question 17

Use the fundamental identities to simplify the expression.
-$\cot \left( 2 \theta - 20 ^ { \circ } \right) = \tan \left( \theta - 10 ^ { \circ } \right)$

Accepted Answer

A)  $\theta = 10 ^ { \circ }$ 
B)  $\theta = 40 ^ { \circ }$ 
C)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
D)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
A)  $\theta = 10 ^ { \circ }$ 
B)  $\theta = 40 ^ { \circ }$ 
C)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
D)  $\theta = \frac { 83 ^ { \circ } } { 7 }$

Question 18

Using a sum or difference identity, write the following as an expression involving functions of x.
-$$1 - \cos ^ { 2 } \theta$$

Accepted Answer

A)  $\sin \theta$ 
B)  $- \sin ^ { 2 } \theta$ 
C)  $\sin ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$ 
A)  $\sin \theta$ 
B)  $- \sin ^ { 2 } \theta$ 
C)  $\sin ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$

Question 19

Use identities to fill in the blank with the appropriate trigonometric function name.
-Find $\cos \theta$ if $\tan \theta = 4$ and $\sin \theta < 0$.

Accepted Answer

A)  $- 17$ 
B)  $\frac { 1 } { 4 }$ 
C)  $- \frac { \sqrt { 17 } } { 17 }$ 
D)  $- \frac { \sqrt { 17 } } { 4 }$ 
A)  $- 17$ 
B)  $\frac { 1 } { 4 }$ 
C)  $- \frac { \sqrt { 17 } } { 17 }$ 
D)  $- \frac { \sqrt { 17 } } { 4 }$

Question 20

Find the exact value of the expression using the provided information.
-$\sin 3 x$

Accepted Answer

A)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
B)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
C)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$ 
A)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
B)  $1 - 8 \sin ^ { 2 } x \cos ^ { 2 } x$ 
C)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
D)  $3 \sin x - 4 \sin ^ { 3 } x$

Use a sum or difference identity to find the exact value. - $\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }$

Solve the equation for exact solutions. - $\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }$

Find the exact value of the expression using the provided information. - $\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0$

Determine all solutions of the equation in radians. -Find $\cos \frac { \theta } { 2 }$ , given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$ .

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -cos 22° = cos 49°cos 27° - sin 49°sin 27°

Verify that each equation is an identity. - $\frac { 1 + \cos t } { 1 - \cos t } - \frac { 1 - \cos t } { 1 + \cos t } = 4 \cot t \csc t$

Use identities to fill in the blank with the appropriate trigonometric function name. - $\cos \frac { \pi } { 11 } - \cos \frac { \pi } { 2 }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\tan \left( 30 ^ { \circ } + x \right)$

Rewrite the following as a product of trigonometric functions. -sin 2x = 2 sin x

Verify that each equation is an identity. - $\cos ( 3 x ) = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { 4 + 7 \tan x } { \sin x } = 4 \csc x + 7 \sec x$

Verify that each equation is an identity. - $\cos ^ { 4 } x = \frac { 1 } { 8 } ( 3 + 4 \cos ( 2 x ) + \cos ( 4 x ) )$

Use Identities to find the exact value. -Find $\cos ( s - t )$ given that $\cos s = - \frac { 1 } { 2 }$ , with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $t$ in quadrant III.

Use an identity to write the expression as a single trigonometric function or as a single number. - $2 \cos ^ { 2 } 4 x - 1$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { 1 + \tan ^ { 2 } x } { \sec x }$

Decide whether the expression is or is not an identity. -tan 80°

Use the fundamental identities to simplify the expression. - $\cot \left( 2 \theta - 20 ^ { \circ } \right) = \tan \left( \theta - 10 ^ { \circ } \right)$

Using a sum or difference identity, write the following as an expression involving functions of x. - $1 - \cos ^ { 2 } \theta$

Use identities to fill in the blank with the appropriate trigonometric function name. -Find $\cos \theta$ if $\tan \theta = 4$ and $\sin \theta < 0$ .

Find the exact value of the expression using the provided information. - $\sin 3 x$

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 5: Trigonometric Identities

Use a sum or difference identity to find the exact value. - sin⁡θ1+sin⁡θ−sin⁡θ1−sin⁡θ\frac { \sin \theta } { 1 + \sin \theta } - \frac { \sin \theta } { 1 - \sin \theta }1+sinθsinθ​−1−sinθsinθ​

Solve the equation for exact solutions. - tan⁡x+cot⁡xtan⁡x−cot⁡x=1sin⁡2x−cos⁡2x\frac { \tan x + \cot x } { \tan x - \cot x } = \frac { 1 } { \sin ^ { 2 } x - \cos ^ { 2 } x }tanx−cotxtanx+cotx​=sin2x−cos2x1​

Find the exact value of the expression using the provided information. - sec⁡2x−tan⁡2x=0\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0sec2x−tan2x​=0

Determine all solutions of the equation in radians. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that cos⁡θ=−35\cos \theta = - \frac { 3 } { 5 }cosθ=−53​ and θ\thetaθ terminates in 90∘<θ<180∘90 ^ { \circ } < \theta < 180 ^ { \circ }90∘<θ<180∘ .

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -cos 22° = cos 49°cos 27° - sin 49°sin 27°

Verify that each equation is an identity. - 1+cos⁡t1−cos⁡t−1−cos⁡t1+cos⁡t=4cot⁡tcsc⁡t\frac { 1 + \cos t } { 1 - \cos t } - \frac { 1 - \cos t } { 1 + \cos t } = 4 \cot t \csc t1−cost1+cost​−1+cost1−cost​=4cottcsct

Use identities to fill in the blank with the appropriate trigonometric function name. - cos⁡π11−cos⁡π2\cos \frac { \pi } { 11 } - \cos \frac { \pi } { 2 }cos11π​−cos2π​

Use an identity to write the expression as a single trigonometric function or as a single number. - tan⁡(30∘+x)\tan \left( 30 ^ { \circ } + x \right)tan(30∘+x)