Question 1

Decide whether the expression is or is not an identity.
-$$\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1$$

Accepted Answer

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

Question 2

Rewrite the following as a product of trigonometric functions.
-$$\cos \frac { \pi } { 13 } - \cos \frac { \pi } { 4 }$$

Accepted Answer

A)  $- 2 \sin \frac { 17 \pi } { 104 } \sin \left( \frac { - 9 \pi } { 104 } \right)$ 
B)  $- 2 \cos \frac { 17 \pi } { 104 } \sin \left( \frac { - 9 \pi } { 104 } \right)$ 
C)  $- 2 \cos \frac { 17 \pi } { 104 } \cos \left( \frac { - 9 \pi } { 104 } \right)$ 
D)  $- 2 \sin \frac { 17 \pi } { 104 } \cos \left( \frac { - 9 \pi } { 104 } \right)$ 
A)  $- 2 \sin \frac { 17 \pi } { 104 } \sin \left( \frac { - 9 \pi } { 104 } \right)$ 
B)  $- 2 \cos \frac { 17 \pi } { 104 } \sin \left( \frac { - 9 \pi } { 104 } \right)$ 
C)  $- 2 \cos \frac { 17 \pi } { 104 } \cos \left( \frac { - 9 \pi } { 104 } \right)$ 
D)  $- 2 \sin \frac { 17 \pi } { 104 } \cos \left( \frac { - 9 \pi } { 104 } \right)$

Question 3

Find the exact value by using a half-angle identity.
-$\sin 75 ^ { \circ }$

Accepted Answer

A)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
A)  $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
B)  $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$ 
C)  $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$ 
D)  $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Question 4

Use the fundamental identities to simplify the expression.
-$\cos \theta - \cos \theta \sin ^ { 2 } \theta$

Accepted Answer

A)  $\sin \theta$ 
B)  $\cos ^ { 3 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$ 
A)  $\sin \theta$ 
B)  $\cos ^ { 3 } \theta$ 
C)  $\sec ^ { 2 } \theta$ 
D)  $\tan ^ { 2 } \theta$

Question 5

Rewrite the following as a product of trigonometric functions.
-$\cos 8 ^ { \circ } - \cos 24 ^ { \circ }$

Accepted Answer

A)  $- 2 \sin 16 ^ { \circ } \sin \left( - 8 ^ { \circ } \right)$ 
B)  $- 2 \sin 16 ^ { \circ } \cos \left( - 8 ^ { \circ } \right)$ 
C)  $- 2 \cos 16 ^ { \circ } \sin \left( - 8 ^ { \circ } \right)$ 
D)  $- 2 \cos 16 ^ { \circ } \cos \left( - 8 ^ { \circ } \right)$ 
A)  $- 2 \sin 16 ^ { \circ } \sin \left( - 8 ^ { \circ } \right)$ 
B)  $- 2 \sin 16 ^ { \circ } \cos \left( - 8 ^ { \circ } \right)$ 
C)  $- 2 \cos 16 ^ { \circ } \sin \left( - 8 ^ { \circ } \right)$ 
D)  $- 2 \cos 16 ^ { \circ } \cos \left( - 8 ^ { \circ } \right)$

Question 6

Use identities to write the expression as a single function of x or $$\theta$$
-$\cos \left( \theta + 90 ^ { \circ } \right)$

Accepted Answer

A)  $- \sin \theta$ 
B)  $\cos \theta$ 
C)  $\sin \theta$ 
D)  $- \cos \theta$ 
A)  $- \sin \theta$ 
B)  $\cos \theta$ 
C)  $\sin \theta$ 
D)  $- \cos \theta$

Question 7

Write the product as a sum or difference of trigonometric functions.
-$- 8 \cos ( - \theta ) \sin ( - 6 \theta )$

Accepted Answer

A)  $4 ( \sin 7 \theta - \sin 5 \theta )$ 
B)  $- 4 ( \cos 7 \theta - \cos 5 \theta )$ 
C)  $- 8 ( \cos 7 \theta + \cos 5 \theta )$ 
D)  $4 ( \sin 7 \theta + \sin 5 \theta )$ 
A)  $4 ( \sin 7 \theta - \sin 5 \theta )$ 
B)  $- 4 ( \cos 7 \theta - \cos 5 \theta )$ 
C)  $- 8 ( \cos 7 \theta + \cos 5 \theta )$ 
D)  $4 ( \sin 7 \theta + \sin 5 \theta )$

Question 8

Express the function as a trigonometric function of x.
-$\sin 4 x$

Accepted Answer

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

Question 9

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { \sec \theta \sin \theta } { \tan \theta } - 1$$

Accepted Answer

A)  0 
B)  $- \sec ^ { 2 } \theta$ 
C)  1 
D)  $1 - \sin \theta$ 
A)  0 
B)  $- \sec ^ { 2 } \theta$ 
C)  1 
D)  $1 - \sin \theta$

Question 10

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sin \theta$ if $\sec \theta = - \frac { 8 } { 5 }$ and $\tan \theta < 0$

Accepted Answer

A)  $\frac { 39 } { 64 }$ 
B)  $\frac { \sqrt { 39 } } { 8 }$ 
C)  $- \frac { \sqrt { 39 } } { 8 }$ 
D)  $- \frac { 5 } { 8 }$ 
A)  $\frac { 39 } { 64 }$ 
B)  $\frac { \sqrt { 39 } } { 8 }$ 
C)  $- \frac { \sqrt { 39 } } { 8 }$ 
D)  $- \frac { 5 } { 8 }$

Question 11

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { 4 + 7 \tan x } { \sin x } = 4 \csc x + 7 \sec x$$

Accepted Answer

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

Question 12

Find the exact value of the expression using the provided information.
-Find $\cos ( s - t )$ given that $\cos s = \frac { \sqrt { 5 } } { 5 }$, with $s$ in quadrant $I$, and $\sin t = - \frac { \sqrt { 3 } } { 2 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $- \frac { \sqrt { 5 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } + 2 \sqrt { 15 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } - \sqrt { 15 } } { 9 }$ 
D)  $\frac { \sqrt { 5 } - 2 \sqrt { 15 } } { 10 }$ 
A)  $- \frac { \sqrt { 5 } } { 10 }$ 
B)  $\frac { \sqrt { 5 } + 2 \sqrt { 15 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } - \sqrt { 15 } } { 9 }$ 
D)  $\frac { \sqrt { 5 } - 2 \sqrt { 15 } } { 10 }$

Question 13

Use identities to write the expression as a single function of x or $$\theta$$
-$$\tan \left( x - \frac { \pi } { 3 } \right)$$

Accepted Answer

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

Question 14

Verify that each equation is an identity.
-$$\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\cot...

Question 15

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }$$

Accepted Answer

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

Question 16

Solve the equation for solutions in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Round to the nearest degree.
-$\sin 2 \theta = \cos \theta$

Accepted Answer

A)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
A)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$

Question 17

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \sin ^ { - 1 } ( - 0.4848 )$$

Accepted Answer

A)  $- 0.5412698$ 
B)  $2.6365103$ 
C)  $- 0.4888889$ 
D)  $- 0.5061345$ 
A)  $- 0.5412698$ 
B)  $2.6365103$ 
C)  $- 0.4888889$ 
D)  $- 0.5061345$

Question 18

Find the exact value of the real number y.
-$$y = \arcsin \left( - \frac { \sqrt { 3 } } { 2 } \right)$$

Accepted Answer

A)  $\pi$ 
B)  $- \frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$ 
A)  $\pi$ 
B)  $- \frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 7 \pi } { 6 }$

Question 19

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 20

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cot \theta$ if $\sec \theta = - \frac { \sqrt { 53 } } { 7 }$ and $\theta$ is in quadrant II.

Accepted Answer

A)  $\frac { 2 } { 7 }$ 
B)  $\frac { \sqrt { 53 } } { 2 }$ 
C)  $- \frac { 7 } { 2 }$ 
D)  $- \frac { \sqrt { 53 } } { 2 }$ 
A)  $\frac { 2 } { 7 }$ 
B)  $\frac { \sqrt { 53 } } { 2 }$ 
C)  $- \frac { 7 } { 2 }$ 
D)  $- \frac { \sqrt { 53 } } { 2 }$

Decide whether the expression is or is not an identity. - $\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1$

Rewrite the following as a product of trigonometric functions. - $\cos \frac { \pi } { 13 } - \cos \frac { \pi } { 4 }$

Find the exact value by using a half-angle identity. - $\sin 75 ^ { \circ }$

Use the fundamental identities to simplify the expression. - $\cos \theta - \cos \theta \sin ^ { 2 } \theta$

Rewrite the following as a product of trigonometric functions. - $\cos 8 ^ { \circ } - \cos 24 ^ { \circ }$

Use identities to write the expression as a single function of x or $\theta$ - $\cos \left( \theta + 90 ^ { \circ } \right)$

Write the product as a sum or difference of trigonometric functions. - $- 8 \cos ( - \theta ) \sin ( - 6 \theta )$

Express the function as a trigonometric function of x. - $\sin 4 x$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { \sec \theta \sin \theta } { \tan \theta } - 1$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\sec \theta = - \frac { 8 } { 5 }$ and $\tan \theta < 0$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { 4 + 7 \tan x } { \sin x } = 4 \csc x + 7 \sec x$

Find the exact value of the expression using the provided information. -Find $\cos ( s - t )$ given that $\cos s = \frac { \sqrt { 5 } } { 5 }$ , with $s$ in quadrant $I$ , and $\sin t = - \frac { \sqrt { 3 } } { 2 }$ , with $t$ in quadrant IV.

Use identities to write the expression as a single function of x or $\theta$ - $\tan \left( x - \frac { \pi } { 3 } \right)$

Verify that each equation is an identity. - $\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\sin 2 \theta = \cos \theta$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \sin ^ { - 1 } ( - 0.4848 )$

Find the exact value of the real number y. - $y = \arcsin \left( - \frac { \sqrt { 3 } } { 2 } \right)$

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 the fundamental identities to find the value of the trigonometric function. -Find $\cot \theta$ if $\sec \theta = - \frac { \sqrt { 53 } } { 7 }$ and $\theta$ is in quadrant II.

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 8: Trigonometric Identities and Equations

Decide whether the expression is or is not an identity. - sin⁡2x+cos⁡2x=1\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1sin2x+cos2x​=1

Rewrite the following as a product of trigonometric functions. - cos⁡π13−cos⁡π4\cos \frac { \pi } { 13 } - \cos \frac { \pi } { 4 }cos13π​−cos4π​

Find the exact value by using a half-angle identity. - sin⁡75∘\sin 75 ^ { \circ }sin75∘

Use the fundamental identities to simplify the expression. - cos⁡θ−cos⁡θsin⁡2θ\cos \theta - \cos \theta \sin ^ { 2 } \thetacosθ−cosθsin2θ

Rewrite the following as a product of trigonometric functions. - cos⁡8∘−cos⁡24∘\cos 8 ^ { \circ } - \cos 24 ^ { \circ }cos8∘−cos24∘

Use identities to write the expression as a single function of x or θ\thetaθ - cos⁡(θ+90∘)\cos \left( \theta + 90 ^ { \circ } \right)cos(θ+90∘)

Write the product as a sum or difference of trigonometric functions. - −8cos⁡(−θ)sin⁡(−6θ)- 8 \cos ( - \theta ) \sin ( - 6 \theta )−8cos(−θ)sin(−6θ)

Express the function as a trigonometric function of x. - sin⁡4x\sin 4 xsin4x

Perform the indicated operations and simplify the result so there are no quotients. - sec⁡θsin⁡θtan⁡θ−1\frac { \sec \theta \sin \theta } { \tan \theta } - 1tanθsecθsinθ​−1

Use the fundamental identities to find the value of the trigonometric function. -Find sin⁡θ\sin \thetasinθ if sec⁡θ=−85\sec \theta = - \frac { 8 } { 5 }secθ=−58​ and tan⁡θ<0\tan \theta < 0tanθ<0

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - 4+7tan⁡xsin⁡x=4csc⁡x+7sec⁡x\frac { 4 + 7 \tan x } { \sin x } = 4 \csc x + 7 \sec xsinx4+7tanx​=4cscx+7secx

Use identities to write the expression as a single function of x or θ\thetaθ - tan⁡(x−π3)\tan \left( x - \frac { \pi } { 3 } \right)tan(x−3π​)

Verify that each equation is an identity. - cot⁡2x=(csc⁡x−1)(csc⁡x+1)\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )cot2x=(cscx−1)(cscx+1)

Perform the indicated operations and simplify the result so there are no quotients. - sin⁡θcos⁡θ+cos⁡θsin⁡θ\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta }cosθsinθ​+sinθcosθ​

Solve the equation for solutions in the interval [0 [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Round to the nearest degree. - sin⁡2θ=cos⁡θ\sin 2 \theta = \cos \thetasin2θ=cosθ

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=sin⁡−1(−0.4848)y = \sin ^ { - 1 } ( - 0.4848 )y=sin−1(−0.4848)

Find the exact value of the real number y. - y=arcsin⁡(−32)y = \arcsin \left( - \frac { \sqrt { 3 } } { 2 } \right)y=arcsin(−23​​)

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 the fundamental identities to find the value of the trigonometric function. -Find cot⁡θ\cot \thetacotθ if sec⁡θ=−537\sec \theta = - \frac { \sqrt { 53 } } { 7 }secθ=−753​​ and θ\thetaθ is in quadrant II.