Question 1

Find the exact value by using a half-angle identity.
-$\cos 22.5 ^ { \circ }$

Accepted Answer

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

Question 2

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.
-$$\cos ^ { 2 } x + 2 \cos x = - 1$$

Accepted Answer

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

Question 3

Use identities to fill in the blank with the appropriate trigonometric function name.
-$\sec \frac { \pi } { 6 } $= ____$ \frac { \pi } { 3 }$

Accepted Answer

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

Question 4

Use a sum or difference identity to find the exact value.
-$\tan \left( - 15 ^ { \circ } \right)$

Accepted Answer

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

Question 5

Use identities to write the expression as a single function of x or θ.
-$\sin \left( \frac { \pi } { 4 } - x \right)$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
C)  $- \cos x$ 
D)  $\sin x$ 
A)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
C)  $- \cos x$ 
D)  $\sin x$

Question 6

Give the degree measure of θ.
-$\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )$

Accepted Answer

A)  $45 ^ { \circ }$ 
B)  $315 ^ { \circ }$ 
C)  $135 ^ { \circ }$ 
D)  $120 ^ { \circ }$ 
A)  $45 ^ { \circ }$ 
B)  $315 ^ { \circ }$ 
C)  $135 ^ { \circ }$ 
D)  $120 ^ { \circ }$

Question 7

Give the exact value of the expression.
-$\arccos \left( \cos \frac { 7 \pi } { 6 } \right)$

Accepted Answer

A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 5 \pi } { 6 }$ 
C)  $\frac { 4 \pi } { 3 }$ 
D)  $\frac { \pi } { 6 }$ 
A)  $\frac { \pi } { 3 }$ 
B)  $\frac { 5 \pi } { 6 }$ 
C)  $\frac { 4 \pi } { 3 }$ 
D)  $\frac { \pi } { 6 }$

Question 8

Solve the equation for exact solutions.
-$\arcsin 2 x + 2 \arccos x = \pi$

Accepted Answer

A)  $\{ 1 \}$ 
B)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
C)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
D)  $\{ 0 \}$ 
A)  $\{ 1 \}$ 
B)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
C)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
D)  $\{ 0 \}$

Question 9

Provide an appropriate response.
-Graph the expression $\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } x$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Accepted Answer

The answer of Provide an appropriate response.
-Graph the expression \(\sec...

Question 10

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

Accepted Answer

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

Question 11

Solve the equation for solutions in the interval [0, 2 π).
-$2 \sqrt { 3 } \sin 4 x = 3$

Accepted Answer

A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
C)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
D)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
C)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
D)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$

Question 12

Solve the problem.
-In a particular situation, the pressure, $P$, exerted on a person's eardrum at a distance of 12 feet the source is given by the function: $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 12 } \cos ( 2 \mathrm { n } \pi - \mathrm { ct } )$, where $\mathrm { n }$ is a positive integer. Use difference identity for cosine to simplify $\mathrm { P }$ in this situation.

Accepted Answer

A)  $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 12 } \cos ( \mathrm { ct } ) - \frac { \mathrm { a } } { 12 } \sin ( \mathrm { ct } )$ 
B)  $P ( t ) = \frac { a } { 12 } \cos ( c t )$ 
C)  $P ( t ) = \frac { 12 } { 13 } \cos ( c t + 12 )$ 
D)  $\mathrm { P } ( \mathrm { t } ) = \cos ( \mathrm { ct } )$ 
A)  $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 12 } \cos ( \mathrm { ct } ) - \frac { \mathrm { a } } { 12 } \sin ( \mathrm { ct } )$ 
B)  $P ( t ) = \frac { a } { 12 } \cos ( c t )$ 
C)  $P ( t ) = \frac { 12 } { 13 } \cos ( c t + 12 )$ 
D)  $\mathrm { P } ( \mathrm { t } ) = \cos ( \mathrm { ct } )$

Question 13

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\tan ^ { 2 } 2 \theta = 5$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } , 327 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } , 327 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$

Question 14

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

Accepted Answer

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

Question 15

Provide an appropriate response.
-The equation $\tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1 = 0$ has no solution in the interval $[ 0,2 \pi )$. Explain what this tells you about the graph of $y = \tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1$.

Accepted Answer

The graph does not c

Question 16

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\cot ( - \arcsin \pi )$

Accepted Answer

A)  $- 0.0009$ 
B)  $- 1094.771$ 
C)  Undefined 
D)  0 
A)  $- 0.0009$ 
B)  $- 1094.771$ 
C)  Undefined 
D)  0

Question 17

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

Accepted Answer

A)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
B)  $3 \sin x - 4 \sin ^ { 3 } x$ 
C)  $\cos ^ { 4 } x - 6 \sin ^ { 2 } x \cos ^ { 2 } x + \sin ^ { 4 } x$ 
D)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$ 
A)  $\cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$ 
B)  $3 \sin x - 4 \sin ^ { 3 } x$ 
C)  $\cos ^ { 4 } x - 6 \sin ^ { 2 } x \cos ^ { 2 } x + \sin ^ { 4 } x$ 
D)  $4 \sin x \cos ^ { 3 } x - 4 \sin ^ { 3 } x \cos x$

Question 18

Find the exact value of the real number y.
-$$y = \sin ^ { - 1 } ( 0 )$$

Accepted Answer

A)  0 
B)  $\frac { \pi } { 2 }$ 
C)  $- \frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 3 }$ 
A)  0 
B)  $\frac { \pi } { 2 }$ 
C)  $- \frac { \pi } { 2 }$ 
D)  $\frac { \pi } { 3 }$

Question 19

Verify that each equation is an identity.
-$$\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$$

Accepted Answer

\[\begin{array} { l } 
\cos ( 4 \theta )

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

Find the exact value by using a half-angle identity. - $\cos 22.5 ^ { \circ }$

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. - $\cos ^ { 2 } x + 2 \cos x = - 1$

Use identities to fill in the blank with the appropriate trigonometric function name. - $\sec \frac { \pi } { 6 }$ = ____ $\frac { \pi } { 3 }$

Use a sum or difference identity to find the exact value. - $\tan \left( - 15 ^ { \circ } \right)$

Use identities to write the expression as a single function of x or θ. - $\sin \left( \frac { \pi } { 4 } - x \right)$

Give the degree measure of θ. - $\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )$

Give the exact value of the expression. - $\arccos \left( \cos \frac { 7 \pi } { 6 } \right)$

Solve the equation for exact solutions. - $\arcsin 2 x + 2 \arccos x = \pi$

Provide an appropriate response. -Graph the expression $\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } x$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

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

Solve the equation for solutions in the interval [0, 2 π). - $2 \sqrt { 3 } \sin 4 x = 3$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\tan ^ { 2 } 2 \theta = 5$

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

Provide an appropriate response. -The equation $\tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1 = 0$ has no solution in the interval $[ 0,2 \pi )$ . Explain what this tells you about the graph of $y = \tan \frac { x } { 2 } + \sec \frac { x } { 2 } + 1$ .

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\cot ( - \arcsin \pi )$

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

Find the exact value of the real number y. - $y = \sin ^ { - 1 } ( 0 )$

Verify that each equation is an identity. - $\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$

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.

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

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Exam 7: Trigonometric Identities and Equations

Find the exact value by using a half-angle identity. - cos⁡22.5∘\cos 22.5 ^ { \circ }cos22.5∘

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+2cos⁡x=−1\cos ^ { 2 } x + 2 \cos x = - 1cos2x+2cosx=−1

Use identities to fill in the blank with the appropriate trigonometric function name. - sec⁡π6\sec \frac { \pi } { 6 } sec6π​ = ____ π3 \frac { \pi } { 3 }3π​

Use a sum or difference identity to find the exact value. - tan⁡(−15∘)\tan \left( - 15 ^ { \circ } \right)tan(−15∘)

Use identities to write the expression as a single function of x or θ. - sin⁡(π4−x)\sin \left( \frac { \pi } { 4 } - x \right)sin(4π​−x)

Give the degree measure of θ. - θ=sec⁡−1(−2)\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )θ=sec−1(−2​)

Give the exact value of the expression. - arccos⁡(cos⁡7π6)\arccos \left( \cos \frac { 7 \pi } { 6 } \right)arccos(cos67π​)

Solve the equation for exact solutions. - arcsin⁡2x+2arccos⁡x=π\arcsin 2 x + 2 \arccos x = \piarcsin2x+2arccosx=π

Provide an appropriate response. -Graph the expression sec⁡2x−sec⁡2xsin⁡2x\sec ^ { 2 } x - \sec ^ { 2 } x \sin ^ { 2 } xsec2x−sec2xsin2x on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Factor the trigonometric expression and simplify. - tan⁡4x−sec⁡4x\tan ^ { 4 } x - \sec ^ { 4 } xtan4x−sec4x

Solve the equation for solutions in the interval [0, 2 π). - 23sin⁡4x=32 \sqrt { 3 } \sin 4 x = 323​sin4x=3

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - tan⁡22θ=5\tan ^ { 2 } 2 \theta = 5tan22θ=5

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

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - cot⁡(−arcsin⁡π)\cot ( - \arcsin \pi )cot(−arcsinπ)

Express the function as a trigonometric function of x. - cos⁡4x\cos 4 xcos4x

Find the exact value of the real number y. - y=sin⁡−1(0)y = \sin ^ { - 1 } ( 0 )y=sin−1(0)

Verify that each equation is an identity. - cos⁡(4θ)=cos⁡4θ−6sin⁡2θcos⁡2θ+sin⁡4θ\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \thetacos(4θ)=cos4θ−6sin2θcos2θ+sin4θ

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.