Question 1

Solve the equation for solutions in the interval [0, 2 π).
-$$\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$$

Accepted Answer

A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 3 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$

Question 2

Use the fundamental identities to simplify the expression.
-$\frac { \csc \theta \cot \theta } { \sec \theta }$

Accepted Answer

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

Question 3

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

A)  $f ( - x ) = f ( x )$ 
B)  $f ( - x ) = - f ( x )$ 
A)  $f ( - x ) = f ( x )$ 
B)  $f ( - x ) = - f ( x )$

Question 4

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $\frac { \sqrt { 33 } } { 4 }$ 
B)  $- \frac { 5 \sqrt { 29 } } { 29 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $- \frac { \sqrt { 11 } } { 6 }$ 
A)  $\frac { \sqrt { 33 } } { 4 }$ 
B)  $- \frac { 5 \sqrt { 29 } } { 29 }$ 
C)  $\frac { 4 } { 5 }$ 
D)  $- \frac { \sqrt { 11 } } { 6 }$

Question 5

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\frac { \sin 80 ^ { \circ } } { 1 + \cos 80 ^ { \circ } }$$

Accepted Answer

A)  $\cos 40 ^ { \circ }$ 
B)  $\tan 40 ^ { \circ }$ 
C)  $\cot 40 ^ { \circ }$ 
D)  $\sin 40 ^ { \circ }$ 
A)  $\cos 40 ^ { \circ }$ 
B)  $\tan 40 ^ { \circ }$ 
C)  $\cot 40 ^ { \circ }$ 
D)  $\sin 40 ^ { \circ }$

Question 6

Provide an appropriate response.
-True or false? The statement $\cos \left( \cos ^ { - 1 } x \right) = x$ for all real numbers in the interval $0 \leq x \leq \pi$.

Accepted Answer

A)  True 
B)  False 
A)  True 
B)  False

Question 7

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$$

Accepted Answer

A)  $2 \cot ^ { 2 } \theta$ 
B)  0 
C)  $- 2 \cot ^ { 2 } \theta$ 
D)  2 
A)  $2 \cot ^ { 2 } \theta$ 
B)  0 
C)  $- 2 \cot ^ { 2 } \theta$ 
D)  2

Question 8

Write the word or phrase that best completes each statement or answers the question.
-Explain why you can find $\cos 1.219$ on your calculator, but $\cos ^ { - 1 } 1.219$ results in an error message.

Accepted Answer

1.219 is n

Question 9

Use identities to fill in the blank with the appropriate trigonometric function name.
-$$--- 56 ^ { \circ } = \tan 34 ^ { \circ }$$

Accepted Answer

A)  $\tan$ 
B)  $\cos$ 
C)  $\csc$ 
D)  $\cot$ 
A)  $\tan$ 
B)  $\cos$ 
C)  $\csc$ 
D)  $\cot$

Question 10

Use identities to find the indicated value for each angle measure.
-$\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $\frac { 527 } { 625 }$ 
B)  $- \frac { 336 } { 625 }$ 
C)  $\frac { 336 } { 625 }$ 
D)  $- \frac { 527 } { 625 }$ 
A)  $\frac { 527 } { 625 }$ 
B)  $- \frac { 336 } { 625 }$ 
C)  $\frac { 336 } { 625 }$ 
D)  $- \frac { 527 } { 625 }$

Question 11

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.
-$\sin ^ { 2 } x + \sin x = 0$

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sin \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant II.

Accepted Answer

A)  $\frac { 5 } { 4 }$ 
B)  $- \frac { 3 } { 2 }$ 
C)  $- \frac { \sqrt { 7 } } { 9 }$ 
D)  $- \frac { 3 \sqrt { 7 } } { 7 }$ 
A)  $\frac { 5 } { 4 }$ 
B)  $- \frac { 3 } { 2 }$ 
C)  $- \frac { \sqrt { 7 } } { 9 }$ 
D)  $- \frac { 3 \sqrt { 7 } } { 7 }$

Question 14

Solve the equation for solutions in the interval [0, 2 π).
-$\csc 3 x = 0$

Accepted Answer

A)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
B)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
B)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\varnothing$

Question 15

Solve the equation for exact solutions.
-$6 \cos ^ { - 1 } x = \pi$

Accepted Answer

A)  $\left\{ \frac { 1 } { 2 } \right\}$ 
B)  $\left\{ \frac { \sqrt { 3 } } { 2 } \right\}$ 
C)  $\left\{ \frac { \pi } { 6 } \right\}$ 
D)  $\left\{ \frac { \sqrt { 2 } } { 2 } \right\}$ 
A)  $\left\{ \frac { 1 } { 2 } \right\}$ 
B)  $\left\{ \frac { \sqrt { 3 } } { 2 } \right\}$ 
C)  $\left\{ \frac { \pi } { 6 } \right\}$ 
D)  $\left\{ \frac { \sqrt { 2 } } { 2 } \right\}$

Question 16

Solve the equation for solutions in the interval [0, 2 π).
-$\cos 2 x = \sqrt { 2 } - \cos 2 x$

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
A)  $\varnothing$ 
B)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$

Question 17

Give the exact value of the expression.
-$\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)$

Accepted Answer

A)  $\frac { 48 } { 65 }$ 
B)  $\frac { 16 } { 65 }$ 
C)  $\frac { 72 } { 65 }$ 
D)  $\frac { 56 } { 65 }$ 
A)  $\frac { 48 } { 65 }$ 
B)  $\frac { 16 } { 65 }$ 
C)  $\frac { 72 } { 65 }$ 
D)  $\frac { 56 } { 65 }$

Question 18

Determine all solutions of the equation in radians.
-Find $\tan \frac { x } { 2 }$, given that $\tan x = 3$ and $x$ terminates in $\pi < x < \frac { 3 \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } + 1 } { 3 }$ 
B)  $\frac { \sqrt { 10 } - 1 } { - 3 }$ 
C)  $\frac { \sqrt { 10 } - 1 } { 3 }$ 
D)  $\frac { \sqrt { 10 } + 1 } { - 3 }$ 
A)  $\frac { \sqrt { 10 } + 1 } { 3 }$ 
B)  $\frac { \sqrt { 10 } - 1 } { - 3 }$ 
C)  $\frac { \sqrt { 10 } - 1 } { 3 }$ 
D)  $\frac { \sqrt { 10 } + 1 } { - 3 }$

Question 19

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$$\tan 4 \theta = 0$$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 147 ^ { \circ } , 237 ^ { \circ } , 327 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 147 ^ { \circ } , 237 ^ { \circ } , 327 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$

Question 20

Use identities to write the expression as a single function of x or θ.
-$\tan \left( x - 360 ^ { \circ } \right)$

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)  $\tan x$

Solve the equation for solutions in the interval [0, 2 π). - $\sec \frac { x } { 2 } = \cos \frac { x } { 2 }$

Use the fundamental identities to simplify the expression. - $\frac { \csc \theta \cot \theta } { \sec \theta }$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { \sin 80 ^ { \circ } } { 1 + \cos 80 ^ { \circ } }$

Provide an appropriate response. -True or false? The statement $\cos \left( \cos ^ { - 1 } x \right) = x$ for all real numbers in the interval $0 \leq x \leq \pi$ .

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$

Write the word or phrase that best completes each statement or answers the question. -Explain why you can find $\cos 1.219$ on your calculator, but $\cos ^ { - 1 } 1.219$ results in an error message.

Use identities to fill in the blank with the appropriate trigonometric function name. - $--- 56 ^ { \circ } = \tan 34 ^ { \circ }$

Use identities to find the indicated value for each angle measure. - $\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quad$ Find $\cos ( 2 \theta )$

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. - $\sin ^ { 2 } x + \sin x = 0$

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

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\sin \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant II.

Solve the equation for solutions in the interval [0, 2 π). - $\csc 3 x = 0$

Solve the equation for exact solutions. - $6 \cos ^ { - 1 } x = \pi$

Solve the equation for solutions in the interval [0, 2 π). - $\cos 2 x = \sqrt { 2 } - \cos 2 x$

Give the exact value of the expression. - $\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)$

Determine all solutions of the equation in radians. -Find $\tan \frac { x } { 2 }$ , given that $\tan x = 3$ and $x$ terminates in $\pi < x < \frac { 3 \pi } { 2 }$ .

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\tan 4 \theta = 0$

Use identities to write the expression as a single function of x or θ. - $\tan \left( x - 360 ^ { \circ } \right)$

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

Solve the equation for solutions in the interval [0, 2 π). - sec⁡x2=cos⁡x2\sec \frac { x } { 2 } = \cos \frac { x } { 2 }sec2x​=cos2x​

Use the fundamental identities to simplify the expression. - csc⁡θcot⁡θsec⁡θ\frac { \csc \theta \cot \theta } { \sec \theta }secθcscθcotθ​

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Use the fundamental identities to find the value of the trigonometric function. -Find cos⁡θ\cos \thetacosθ if tan⁡θ=25\tan \theta = \frac { 2 } { 5 }tanθ=52​ and θ\thetaθ is in quadrant III.

Use an identity to write the expression as a single trigonometric function or as a single number. - sin⁡80∘1+cos⁡80∘\frac { \sin 80 ^ { \circ } } { 1 + \cos 80 ^ { \circ } }1+cos80∘sin80∘​

Provide an appropriate response. -True or false? The statement cos⁡(cos⁡−1x)=x\cos \left( \cos ^ { - 1 } x \right) = xcos(cos−1x)=x for all real numbers in the interval 0≤x≤π0 \leq x \leq \pi0≤x≤π .

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - (1+cot⁡θ)(1−cot⁡θ)−csc⁡2θ( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta(1+cotθ)(1−cotθ)−csc2θ

Write the word or phrase that best completes each statement or answers the question. -Explain why you can find cos⁡1.219\cos 1.219cos1.219 on your calculator, but cos⁡−11.219\cos ^ { - 1 } 1.219cos−11.219 results in an error message.

Use identities to fill in the blank with the appropriate trigonometric function name. - −−−56∘=tan⁡34∘--- 56 ^ { \circ } = \tan 34 ^ { \circ }−−−56∘=tan34∘

Use identities to find the indicated value for each angle measure. - tan⁡θ=724,sin⁡θ<0\tan \theta = \frac { 7 } { 24 } , \quad \sin \theta < 0 \quadtanθ=247​,sinθ<0 Find cos⁡(2θ)\cos ( 2 \theta )cos(2θ)

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. - sin⁡2x+sin⁡x=0\sin ^ { 2 } x + \sin x = 0sin2x+sinx=0

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

Use the fundamental identities to find the value of the trigonometric function. -Find tan⁡θ\tan \thetatanθ if sin⁡θ=34\sin \theta = \frac { 3 } { 4 }sinθ=43​ and θ\thetaθ is in quadrant II.

Solve the equation for solutions in the interval [0, 2 π). - csc⁡3x=0\csc 3 x = 0csc3x=0

Solve the equation for exact solutions. - 6cos⁡−1x=π6 \cos ^ { - 1 } x = \pi6cos−1x=π

Solve the equation for solutions in the interval [0, 2 π). - cos⁡2x=2−cos⁡2x\cos 2 x = \sqrt { 2 } - \cos 2 xcos2x=2​−cos2x

Give the exact value of the expression. - cos⁡(arcsin⁡513+arccos⁡35)\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)cos(arcsin135​+arccos53​)

Determine all solutions of the equation in radians. -Find tan⁡x2\tan \frac { x } { 2 }tan2x​ , given that tan⁡x=3\tan x = 3tanx=3 and xxx terminates in π<x<3π2\pi < x < \frac { 3 \pi } { 2 }π<x<23π​ .

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - tan⁡4θ=0\tan 4 \theta = 0tan4θ=0

Use identities to write the expression as a single function of x or θ. - tan⁡(x−360∘)\tan \left( x - 360 ^ { \circ } \right)tan(x−360∘)