Question 1

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

Accepted Answer

A)  $\frac { 4 \tan x - 3 \tan ^ { 3 } x } { 1 + \tan ^ { 4 } x - 6 \tan ^ { 2 } x }$ 
B)  $\frac { 3 \tan x - \tan ^ { 3 } x } { 1 - 3 \tan ^ { 2 } x }$ 
C)  $\frac { 4 \tan x - 3 \tan ^ { 4 } x } { 1 + \tan ^ { 3 } x - 6 \tan ^ { 2 } x }$ 
D)  $\frac { 4 \tan x - 4 \tan ^ { 3 } x } { 1 + \tan ^ { 4 } x - 6 \tan ^ { 2 } x }$ 
A)  $\frac { 4 \tan x - 3 \tan ^ { 3 } x } { 1 + \tan ^ { 4 } x - 6 \tan ^ { 2 } x }$ 
B)  $\frac { 3 \tan x - \tan ^ { 3 } x } { 1 - 3 \tan ^ { 2 } x }$ 
C)  $\frac { 4 \tan x - 3 \tan ^ { 4 } x } { 1 + \tan ^ { 3 } x - 6 \tan ^ { 2 } x }$ 
D)  $\frac { 4 \tan x - 4 \tan ^ { 3 } x } { 1 + \tan ^ { 4 } x - 6 \tan ^ { 2 } x }$

Question 2

Using a sum or difference identity, write the following as an expression involving functions of x.
-$$\sqrt { \frac { 1 - \cos 66 ^ { \circ } } { 2 } }$$

Accepted Answer

A)  $\tan 33 ^ { \circ }$ 
B)  $\sin 33 ^ { \circ }$ 
C)  $\cot 33 ^ { \circ }$ 
D)  $\cos 33 ^ { \circ }$ 
A)  $\tan 33 ^ { \circ }$ 
B)  $\sin 33 ^ { \circ }$ 
C)  $\cot 33 ^ { \circ }$ 
D)  $\cos 33 ^ { \circ }$

Question 3

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-Find $\sin \theta$ if $\cos \theta = \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  $- \frac { \sqrt { 5 } } { 3 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $- \frac { 3 } { 2 }$ 
D)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
A)  $- \frac { \sqrt { 5 } } { 3 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $- \frac { 3 } { 2 }$ 
D)  $\frac { 3 \sqrt { 7 } } { 7 }$

Question 4

Solve the problem.
-$$\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1$$

Accepted Answer

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

Question 5

Use an identity to write the expression as a single trigonometric function or as a single number.
-Find $\sin \frac { \theta } { 2 }$, given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$.

Accepted Answer

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

Question 6

Determine all solutions of the equation in radians.
-$\cos 2 \theta = \frac { 5 } { 6 }$ and $\theta$ terminates in quadrant $\mathrm { I } \quad$ Find $\sin \theta$.

Accepted Answer

A)  $\sin \theta = \frac { \sqrt { 3 } } { 6 }$ 
B)  $\sin \theta = 0$ 
C)  $\sin \theta = 2 \sqrt { 3 }$ 
D)  $\sin \theta = - \frac { \sqrt { 3 } } { 6 }$ 
A)  $\sin \theta = \frac { \sqrt { 3 } } { 6 }$ 
B)  $\sin \theta = 0$ 
C)  $\sin \theta = 2 \sqrt { 3 }$ 
D)  $\sin \theta = - \frac { \sqrt { 3 } } { 6 }$

Question 7

Use the fundamental identities to simplify the expression.
-$\tan \frac { 7 \pi } { 12 }$

Accepted Answer

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

Question 8

Solve the problem.
-$$\frac { \tan \theta } { \sec \theta }$$

Accepted Answer

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

Question 9

Solve the problem.
-$$\sin \frac { x } { 2 } \cos \frac { x } { 2 } = \frac { \sin x } { 2 }$$

Accepted Answer

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

Question 10

Use a sum or difference identity to find the exact value.
-$$\sin 14 ^ { \circ } = \frac { 1 } { 76 ^ { \circ } }$$

Accepted Answer

A)  $\sin$ 
B)  $\cos$ 
C)  $\csc$ 
D)  $\sec$ 
A)  $\sin$ 
B)  $\cos$ 
C)  $\csc$ 
D)  $\sec$

Question 11

Verify that the equation is an identity.
-$$\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\tan...

Question 12

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\cos \left( \theta + \frac { \pi } { 2 } \right)$

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 13

Find the exact value by using a half-angle identity.
-Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$, with $\mathrm { s }$ in quadrant $\mathrm { I }$, and $\sin t = \frac { 1 } { 4 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 6 }$ 
B)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 6 }$ 
C)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 12 }$ 
D)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 12 }$ 
A)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 6 }$ 
B)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 6 }$ 
C)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 12 }$ 
D)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 12 }$

Question 14

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { \sin 52 ^ { \circ } } { 1 + \cos 52 ^ { \circ } }$$

Accepted Answer

A)  $\tan 26 ^ { \circ }$ 
B)  $\cos 26 ^ { \circ }$ 
C)  $\sin 26 ^ { \circ }$ 
D)  $\cot 26 ^ { \circ }$ 
A)  $\tan 26 ^ { \circ }$ 
B)  $\cos 26 ^ { \circ }$ 
C)  $\sin 26 ^ { \circ }$ 
D)  $\cot 26 ^ { \circ }$

Question 15

Use a sum or difference identity to find the exact value.
-$\sin \frac { 7 \pi } { 12 }$

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

Find the exact value of the expression using the provided information.
-Find $\tan ( s + t )$ 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)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$ 
A)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } + \sqrt { 15 } } { - 11 }$

Question 18

Verify that the equation is an identity.
-$$\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\sin...

Question 19

Find the exact value of the expression using the provided information.
-Find $\tan ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$, with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$, with $t$ in quadrant II.

Accepted Answer

A)  $\frac { 20 } { 3 }$ 
B)  $- \frac { 220 } { 171 }$ 
C)  $\frac { 220 } { 21 }$ 
D)  $- \frac { 220 } { 221 }$ 
A)  $\frac { 20 } { 3 }$ 
B)  $- \frac { 220 } { 171 }$ 
C)  $\frac { 220 } { 21 }$ 
D)  $- \frac { 220 } { 221 }$

Question 20

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$\cos 2 x = 2 \cos ^ { 2 } x - 1$$

Accepted Answer

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

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

Using a sum or difference identity, write the following as an expression involving functions of x. - $\sqrt { \frac { 1 - \cos 66 ^ { \circ } } { 2 } }$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. -Find $\sin \theta$ if $\cos \theta = \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Solve the problem. - $\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1$

Use an identity to write the expression as a single trigonometric function or as a single number. -Find $\sin \frac { \theta } { 2 }$ , given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$ .

Determine all solutions of the equation in radians. - $\cos 2 \theta = \frac { 5 } { 6 }$ and $\theta$ terminates in quadrant $\mathrm { I } \quad$ Find $\sin \theta$ .

Use the fundamental identities to simplify the expression. - $\tan \frac { 7 \pi } { 12 }$

Solve the problem. - $\frac { \tan \theta } { \sec \theta }$

Solve the problem. - $\sin \frac { x } { 2 } \cos \frac { x } { 2 } = \frac { \sin x } { 2 }$

Use a sum or difference identity to find the exact value. - $\sin 14 ^ { \circ } = \frac { 1 } { 76 ^ { \circ } }$

Verify that the equation is an identity. - $\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\cos \left( \theta + \frac { \pi } { 2 } \right)$

Find the exact value by using a half-angle identity. -Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$ , with $\mathrm { s }$ in quadrant $\mathrm { I }$ , and $\sin t = \frac { 1 } { 4 }$ , with $\mathrm { t }$ in quadrant II.

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { \sin 52 ^ { \circ } } { 1 + \cos 52 ^ { \circ } }$

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 12 }$

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

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

Verify that the equation is an identity. - $\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$

Find the exact value of the expression using the provided information. -Find $\tan ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$ , with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$ , with $t$ in quadrant II.

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\cos 2 x = 2 \cos ^ { 2 } x - 1$

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

Express the function as a trigonometric function of x. - tan⁡4x\tan 4 xtan4x

Using a sum or difference identity, write the following as an expression involving functions of x. - 1−cos⁡66∘2\sqrt { \frac { 1 - \cos 66 ^ { \circ } } { 2 } }21−cos66∘​​

Use a graphing calculator to make a conjecture as to whether each equation is an identity. -Find sin⁡θ\sin \thetasinθ if cos⁡θ=23\cos \theta = \frac { 2 } { 3 }cosθ=32​ and θ\thetaθ is in quadrant IV.

Solve the problem. - sin⁡2x+cos⁡2x=1\sqrt { \sin ^ { 2 } x + \cos ^ { 2 } x } = 1sin2x+cos2x​=1

Use an identity to write the expression as a single trigonometric function or as a single number. -Find sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ , given that cos⁡θ=14\cos \theta = \frac { 1 } { 4 }cosθ=41​ and θ\thetaθ terminates in 0<θ<90∘0 < \theta < 90 ^ { \circ }0<θ<90∘ .

Determine all solutions of the equation in radians. - cos⁡2θ=56\cos 2 \theta = \frac { 5 } { 6 }cos2θ=65​ and θ\thetaθ terminates in quadrant I\mathrm { I } \quadI Find sin⁡θ\sin \thetasinθ .

Use the fundamental identities to simplify the expression. - tan⁡7π12\tan \frac { 7 \pi } { 12 }tan127π​

Solve the problem. - tan⁡θsec⁡θ\frac { \tan \theta } { \sec \theta }secθtanθ​

Solve the problem. - sin⁡x2cos⁡x2=sin⁡x2\sin \frac { x } { 2 } \cos \frac { x } { 2 } = \frac { \sin x } { 2 }sin2x​cos2x​=2sinx​

Use a sum or difference identity to find the exact value. - sin⁡14∘=176∘\sin 14 ^ { \circ } = \frac { 1 } { 76 ^ { \circ } }sin14∘=76∘1​

Verify that the equation is an identity. - tan⁡(x−π4)=tan⁡x−11+tan⁡x\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }tan(x−4π​)=1+tanxtanx−1​

Use an identity to write the expression as a single trigonometric function or as a single number. - cos⁡(θ+π2)\cos \left( \theta + \frac { \pi } { 2 } \right)cos(θ+2π​)

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - sin⁡52∘1+cos⁡52∘\frac { \sin 52 ^ { \circ } } { 1 + \cos 52 ^ { \circ } }1+cos52∘sin52∘​

Use a sum or difference identity to find the exact value. - sin⁡7π12\sin \frac { 7 \pi } { 12 }sin127π​

Decide whether the expression is or is not an identity. - 11−cos⁡x+11+cos⁡x=csc⁡2x\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x } = \csc ^ { 2 } x1−cosx1​+1+cosx1​=csc2x

Find the exact value of the expression using the provided information. -Find tan⁡(s+t)\tan ( s + t )tan(s+t) given that sin⁡s=14\sin s = \frac { 1 } { 4 }sins=41​ , with sss in quadrant II, and sin⁡t=−12\sin t = - \frac { 1 } { 2 }sint=−21​ , with ttt in quadrant IV.

Verify that the equation is an identity. - sin⁡(3π2−θ)=−cos⁡θ\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \thetasin(23π​−θ)=−cosθ

Find the exact value of the expression using the provided information. -Find tan⁡(s+t)\tan ( s + t )tan(s+t) given that cos⁡s=−513\cos s = - \frac { 5 } { 13 }coss=−135​ , with sss in quadrant II, and sin⁡t=817\sin t = \frac { 8 } { 17 }sint=178​ , with ttt in quadrant II.

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - cos⁡2x=2cos⁡2x−1\cos 2 x = 2 \cos ^ { 2 } x - 1cos2x=2cos2x−1