Question 1

Use identities to write the expression as a single function of x or θ.
-$\cos ( \theta - \pi )$

Accepted Answer

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

Question 2

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \cos ^ { - 1 } ( - 0.3907 )$$

Accepted Answer

A)  $- 1.1698423$ 
B)  $2.0079383$ 
C)  $1.9206367$ 
D)  $1.9721882$ 
A)  $- 1.1698423$ 
B)  $2.0079383$ 
C)  $1.9206367$ 
D)  $1.9721882$

Question 3

Use the cofunction identities to find an angle θ that makes the statement true.
-$\cot 5 \theta = \tan 4 \theta$

Accepted Answer

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

Question 4

Write the product as a sum or difference of trigonometric functions.
-$\cos 36 ^ { \circ } \sin 21 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 } \left( \cos 57 ^ { \circ } + \cos 15 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \sin 57 ^ { \circ } - \sin 15 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \cos 57 ^ { \circ } - \cos 15 ^ { \circ } \right)$ 
D)  $\frac { 1 } { 2 } \left( \sin 57 ^ { \circ } + \sin 15 ^ { \circ } \right)$ 
A)  $\frac { 1 } { 2 } \left( \cos 57 ^ { \circ } + \cos 15 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \sin 57 ^ { \circ } - \sin 15 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \cos 57 ^ { \circ } - \cos 15 ^ { \circ } \right)$ 
D)  $\frac { 1 } { 2 } \left( \sin 57 ^ { \circ } + \sin 15 ^ { \circ } \right)$

Question 5

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\sin s = \frac { 1 } { 4 }$, with $s$ in quadrant II, and $\sin t = - \frac { 1 } { 2 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
B)  $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
A)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
B)  $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$

Question 6

Use Identities to find the exact value.
-$\cos \left( \frac { 19 \pi } { 12 } \right)$

Accepted Answer

A)  $\sqrt { 2 } - \sqrt { 6 }$ 
B)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
C)  $- \sqrt { 6 } - \sqrt { 2 }$ 
D)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
A)  $\sqrt { 2 } - \sqrt { 6 }$ 
B)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$ 
C)  $- \sqrt { 6 } - \sqrt { 2 }$ 
D)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$

Question 7

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \theta$$

Accepted Answer

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

Question 8

Verify that the equation is an identity.
-$$\frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta } = \cot \beta - \cot \alpha$$

Accepted Answer

\[\begin{array} { l } 
\frac { \sin ( \a

Question 9

Factor the trigonometric expression and simplify.
-$\sec ^ { 4 } x - 2 \sec ^ { 2 } x \tan ^ { 2 } x + \tan ^ { 4 } x$

Accepted Answer

A)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
B)  1 
C)  $\sec ^ { 2 } x \left( 1 + \tan ^ { 2 } x \right)$ 
D)  2 
A)  $\sec ^ { 2 } x + \tan ^ { 2 } x$ 
B)  1 
C)  $\sec ^ { 2 } x \left( 1 + \tan ^ { 2 } x \right)$ 
D)  2

Question 10

Determine all solutions of the equation in radians.
-Find $\cos \frac { x } { 2 }$, given that $\cos x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 10 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$

Question 11

Determine all solutions of the equation in radians.
-Find $\cos \frac { \theta } { 2 }$, given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$.

Accepted Answer

A)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 6 } } { 4 }$ 
A)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 10 } } { 4 }$ 
C)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 6 } } { 4 }$

Question 12

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 26 } } { 5 }$ and $\theta$ is in quadrant $\mathrm { I }$.

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  5 
C)  $\frac { \sqrt { 26 } } { 26 }$ 
D)  $\sqrt { 26 }$ 
A)  $\frac { 1 } { 5 }$ 
B)  5 
C)  $\frac { \sqrt { 26 } } { 26 }$ 
D)  $\sqrt { 26 }$

Question 13

Perform the transformation.
-Write $\tan x$ in terms of $\cos x$.

Accepted Answer

A)  $\frac { \pm \cos x \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$ 
B)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { \cos x }$ 
C)  $\pm \sqrt { 1 - \cos ^ { 2 } x }$ 
D)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$ 
A)  $\frac { \pm \cos x \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$ 
B)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { \cos x }$ 
C)  $\pm \sqrt { 1 - \cos ^ { 2 } x }$ 
D)  $\frac { \pm \sqrt { 1 - \cos ^ { 2 } x } } { 1 - \cos ^ { 2 } x }$

Question 14

Use a sum or difference identity to find the exact value.
-$$\frac { \tan 65 ^ { \circ } + \tan 85 ^ { \circ } } { 1 - \tan 65 ^ { \circ } \tan 85 ^ { \circ } }$$

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Use a calculator to give the value to the nearest degree.
-$$\theta = \cos ^ { - 1 } ( 0.8910 )$$

Accepted Answer

A)  $27 ^ { \circ }$ 
B)  $25 ^ { \circ }$ 
C)  $63 ^ { \circ }$ 
D)  $29 ^ { \circ }$ 
A)  $27 ^ { \circ }$ 
B)  $25 ^ { \circ }$ 
C)  $63 ^ { \circ }$ 
D)  $29 ^ { \circ }$

Question 17

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth.
-$$\cos ^ { 2 } x - 1 = 0$$

Accepted Answer

A)  $\{ 2 n \pi \}$ 
B)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \right\}$ 
C)  $\{ n \pi \}$ 
D)  $\left\{ \frac { \pi } { 3 } + n \pi \right\}$ 
A)  $\{ 2 n \pi \}$ 
B)  $\left\{ \frac { \pi } { 2 } + 2 n \pi \right\}$ 
C)  $\{ n \pi \}$ 
D)  $\left\{ \frac { \pi } { 3 } + n \pi \right\}$

Question 18

Solve the equation for x, where x is restricted to the given interval.
-$y = 7 \sin x$, for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Accepted Answer

A)  $x = \arcsin \frac { y } { 7 }$ 
B)  $x = \arcsin 7 y$ 
C)  $x = \frac { 1 } { 7 } \arcsin y$ 
D)  $x = 7 \arcsin y$ 
A)  $x = \arcsin \frac { y } { 7 }$ 
B)  $x = \arcsin 7 y$ 
C)  $x = \frac { 1 } { 7 } \arcsin y$ 
D)  $x = 7 \arcsin y$

Question 19

Use a sum or difference identity to find the exact value.
-$$\frac { \tan 70 ^ { \circ } - \tan ( - 50 ) ^ { \circ } } { 1 + \tan 70 ^ { \circ } \tan ( - 50 ) ^ { \circ } }$$

Accepted Answer

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

Question 20

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { \sin x + \csc x } { \sin x - \csc x } = - 2 \tan ^ { 2 } x$$

Accepted Answer

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

Use identities to write the expression as a single function of x or θ. - $\cos ( \theta - \pi )$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \cos ^ { - 1 } ( - 0.3907 )$

Use the cofunction identities to find an angle θ that makes the statement true. - $\cot 5 \theta = \tan 4 \theta$

Write the product as a sum or difference of trigonometric functions. - $\cos 36 ^ { \circ } \sin 21 ^ { \circ }$

Find the exact value of the expression using the provided information. -Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\sin s = \frac { 1 } { 4 }$ , with $s$ in quadrant II, and $\sin t = - \frac { 1 } { 2 }$ , with $t$ in quadrant IV.

Use Identities to find the exact value. - $\cos \left( \frac { 19 \pi } { 12 } \right)$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \theta$

Verify that the equation is an identity. - $\frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta } = \cot \beta - \cot \alpha$

Factor the trigonometric expression and simplify. - $\sec ^ { 4 } x - 2 \sec ^ { 2 } x \tan ^ { 2 } x + \tan ^ { 4 } x$

Determine all solutions of the equation in radians. -Find $\cos \frac { x } { 2 }$ , given that $\cos x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Determine all solutions of the equation in radians. -Find $\cos \frac { \theta } { 2 }$ , given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$ .

Use the fundamental identities to find the value of the trigonometric function. -Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 26 } } { 5 }$ and $\theta$ is in quadrant $\mathrm { I }$ .

Perform the transformation. -Write $\tan x$ in terms of $\cos x$ .

Use a sum or difference identity to find the exact value. - $\frac { \tan 65 ^ { \circ } + \tan 85 ^ { \circ } } { 1 - \tan 65 ^ { \circ } \tan 85 ^ { \circ } }$

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

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( 0.8910 )$

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $\cos ^ { 2 } x - 1 = 0$

Solve the equation for x, where x is restricted to the given interval. - $y = 7 \sin x$ , for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Use a sum or difference identity to find the exact value. - $\frac { \tan 70 ^ { \circ } - \tan ( - 50 ) ^ { \circ } } { 1 + \tan 70 ^ { \circ } \tan ( - 50 ) ^ { \circ } }$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { \sin x + \csc x } { \sin x - \csc x } = - 2 \tan ^ { 2 } x$

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

Review of Basic Concepts

Filters

Exam 7: Trigonometric Identities and Equations

Use identities to write the expression as a single function of x or θ. - cos⁡(θ−π)\cos ( \theta - \pi )cos(θ−π)

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cos⁡−1(−0.3907)\mathrm { y } = \cos ^ { - 1 } ( - 0.3907 )y=cos−1(−0.3907)

Use the cofunction identities to find an angle θ that makes the statement true. - cot⁡5θ=tan⁡4θ\cot 5 \theta = \tan 4 \thetacot5θ=tan4θ

Write the product as a sum or difference of trigonometric functions. - cos⁡36∘sin⁡21∘\cos 36 ^ { \circ } \sin 21 ^ { \circ }cos36∘sin21∘

Use Identities to find the exact value. - cos⁡(19π12)\cos \left( \frac { 19 \pi } { 12 } \right)cos(1219π​)

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - sin⁡θ+1cos⁡θ+cot⁡θ=tan⁡θ\frac { \sin \theta + 1 } { \cos \theta + \cot \theta } = \tan \thetacosθ+cotθsinθ+1​=tanθ

Verify that the equation is an identity. - sin⁡(α−β)sin⁡αsin⁡β=cot⁡β−cot⁡α\frac { \sin ( \alpha - \beta ) } { \sin \alpha \sin \beta } = \cot \beta - \cot \alphasinαsinβsin(α−β)​=cotβ−cotα

Factor the trigonometric expression and simplify. - sec⁡4x−2sec⁡2xtan⁡2x+tan⁡4x\sec ^ { 4 } x - 2 \sec ^ { 2 } x \tan ^ { 2 } x + \tan ^ { 4 } xsec4x−2sec2xtan2x+tan4x

Determine all solutions of the equation in radians. -Find cos⁡x2\cos \frac { x } { 2 }cos2x​ , given that cos⁡x=14\cos x = \frac { 1 } { 4 }cosx=41​ and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ .

Determine all solutions of the equation in radians. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that sin⁡θ=14\sin \theta = \frac { 1 } { 4 }sinθ=41​ and θ\thetaθ terminates in 0<θ<90∘0 < \theta < 90 ^ { \circ }0<θ<90∘ .

Use the fundamental identities to find the value of the trigonometric function. -Find cot⁡θ\cot \thetacotθ if csc⁡θ=265\csc \theta = \frac { \sqrt { 26 } } { 5 }cscθ=526​​ and θ\thetaθ is in quadrant I\mathrm { I }I .

Perform the transformation. -Write tan⁡x\tan xtanx in terms of cos⁡x\cos xcosx .

Use a sum or difference identity to find the exact value. - tan⁡65∘+tan⁡85∘1−tan⁡65∘tan⁡85∘\frac { \tan 65 ^ { \circ } + \tan 85 ^ { \circ } } { 1 - \tan 65 ^ { \circ } \tan 85 ^ { \circ } }1−tan65∘tan85∘tan65∘+tan85∘​

Perform the indicated operations and simplify the result so there are no quotients. - sec⁡θ−1sec⁡θ\sec \theta - \frac { 1 } { \sec \theta }secθ−secθ1​

Use a calculator to give the value to the nearest degree. - θ=cos⁡−1(0.8910)\theta = \cos ^ { - 1 } ( 0.8910 )θ=cos−1(0.8910)

Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - cos⁡2x−1=0\cos ^ { 2 } x - 1 = 0cos2x−1=0

Solve the equation for x, where x is restricted to the given interval. - y=7sin⁡xy = 7 \sin xy=7sinx , for xxx in [−π2,π2]\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right][−2π​,2π​]

Use a sum or difference identity to find the exact value. - tan⁡70∘−tan⁡(−50)∘1+tan⁡70∘tan⁡(−50)∘\frac { \tan 70 ^ { \circ } - \tan ( - 50 ) ^ { \circ } } { 1 + \tan 70 ^ { \circ } \tan ( - 50 ) ^ { \circ } }1+tan70∘tan(−50)∘tan70∘−tan(−50)∘​

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - sin⁡x+csc⁡xsin⁡x−csc⁡x=−2tan⁡2x\frac { \sin x + \csc x } { \sin x - \csc x } = - 2 \tan ^ { 2 } xsinx−cscxsinx+cscx​=−2tan2x