Question 1

Find the exact value of the real number y.
-$$y = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)$$

Accepted Answer

A)  $\frac { 7 \pi } { 4 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { 11 \pi } { 6 }$ 
A)  $\frac { 7 \pi } { 4 }$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { 11 \pi } { 6 }$

Question 2

Factor the trigonometric expression and simplify.
-$1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

Accepted Answer

A)  $\sin ^ { 2 } x$ 
B)  $\left( 1 - \sin ^ { 2 } x \right)$ 
C)  $\left( 1 + \tan ^ { 2 } x \right)$ 
D)  $\cos ^ { 4 } x$ 
A)  $\sin ^ { 2 } x$ 
B)  $\left( 1 - \sin ^ { 2 } x \right)$ 
C)  $\left( 1 + \tan ^ { 2 } x \right)$ 
D)  $\cos ^ { 4 } x$

Question 3

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$, with $\mathrm { s }$ in quadrant $\mathrm { I }$, and $\sin t = - \frac { 1 } { 2 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 5 }$ 
B)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 3 }$ 
C)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 5 }$ 
D)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 3 }$ 
A)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 5 }$ 
B)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 3 }$ 
C)  $\frac { 9 \sqrt { 3 } + 8 \sqrt { 2 } } { 5 }$ 
D)  $\frac { 9 \sqrt { 3 } - 8 \sqrt { 2 } } { 3 }$

Question 4

Solve the equation for solutions in the interval [0, 2 π).
-$$\sin x \cos x = \frac { 1 } { 2 }$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
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)  $\{ 0 \}$ 
D)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
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)  $\{ 0 \}$ 
D)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } \right\}$

Question 5

Complete the sentence so the result is an identity. Let x be any real number.
-$$\sin x = ( ـــــــــ ) ( \cos x )$$

Accepted Answer

A)  $\cot x$ 
B)  $\csc x$ 
C)  $\tan x$ 
D)  $\sec x$ 
A)  $\cot x$ 
B)  $\csc x$ 
C)  $\tan x$ 
D)  $\sec x$

Question 6

Provide an appropriate response.
-True or false? The statement $\sin \left( \sin ^ { - 1 } x \right) = x$ for all real numbers in the interval $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.

Accepted Answer

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

Question 7

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 x$$

Accepted Answer

A)  $2 \sin 4 x$ 
B)  $\frac { 1 } { 2 } \sin 16 x$ 
C)  $\cos 4 x$ 
D)  $\cos 8 x$ 
A)  $2 \sin 4 x$ 
B)  $\frac { 1 } { 2 } \sin 16 x$ 
C)  $\cos 4 x$ 
D)  $\cos 8 x$

Question 8

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }$

Accepted Answer

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

Question 9

Use identities to fill in the blank with the appropriate trigonometric function name.
-

Accepted Answer

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

Question 10

Solve the equation for exact solutions.
-$$- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$$

Accepted Answer

A)  $\left\{ - \frac { \sqrt { 2 } } { 2 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ - \frac { \sqrt { 2 } } { 8 } \right\}$ 
D)  $\left\{ \frac { \sqrt { 2 } } { 8 } \right\}$ 
A)  $\left\{ - \frac { \sqrt { 2 } } { 2 } \right\}$ 
B)  $\{ 0 \}$ 
C)  $\left\{ - \frac { \sqrt { 2 } } { 8 } \right\}$ 
D)  $\left\{ \frac { \sqrt { 2 } } { 8 } \right\}$

Question 11

Use an identity to write the expression as a single trigonometric function or as a single number.
-$2 \cos ^ { 2 } 4 x - 1$

Accepted Answer

A)  $\cos 4 x$ 
B)  $2 \sin 4 x$ 
C)  $\frac { 1 } { 2 } \sin 16 x$ 
D)  $\cos 8 x$ 
A)  $\cos 4 x$ 
B)  $2 \sin 4 x$ 
C)  $\frac { 1 } { 2 } \sin 16 x$ 
D)  $\cos 8 x$

Question 12

Write the following as an algebraic expression in u, u > 0.
-$\sin \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 25 } } { \mathrm { u } } \right)$

Accepted Answer

A)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 5 } } { \mathrm { u } ^ { 2 } + 5 }$ 
B)  $u \sqrt { 5 }$ 
C)  $\frac { \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$ 
D)  $\frac { 5 \sqrt { u ^ { 2 } + 25 } } { u ^ { 2 } + 25 }$ 
A)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 5 } } { \mathrm { u } ^ { 2 } + 5 }$ 
B)  $u \sqrt { 5 }$ 
C)  $\frac { \sqrt { u ^ { 2 } + 5 } } { u ^ { 2 } + 5 }$ 
D)  $\frac { 5 \sqrt { u ^ { 2 } + 25 } } { u ^ { 2 } + 25 }$

Question 13

Find the exact value of the expression using the provided information.
-Find $\sin ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$, with $s$ in quadrant II, and $\sin t = \frac { 15 } { 17 }$, with $t$ in quadrant II.

Accepted Answer

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

Question 14

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$\cos ( x + y ) = \cos x + \cos y$$

Accepted Answer

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

Find the exact value of the real number y. - $y = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)$

Factor the trigonometric expression and simplify. - $1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x$

Find the exact value of the expression using the provided information. -Find $\tan ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = \frac { 1 } { 3 }$ , with $\mathrm { s }$ in quadrant $\mathrm { I }$ , and $\sin t = - \frac { 1 } { 2 }$ , with $\mathrm { t }$ in quadrant IV.

Solve the equation for solutions in the interval [0, 2 π). - $\sin x \cos x = \frac { 1 } { 2 }$

Complete the sentence so the result is an identity. Let x be any real number. - $\sin x = ( ـــــــــ ) ( \cos x )$

Provide an appropriate response. -True or false? The statement $\sin \left( \sin ^ { - 1 } x \right) = x$ for all real numbers in the interval $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ .

Use an identity to write the expression as a single trigonometric function or as a single number. - $\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 x$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }$

Use identities to fill in the blank with the appropriate trigonometric function name. -

Solve the equation for exact solutions. - $- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $2 \cos ^ { 2 } 4 x - 1$

Write the following as an algebraic expression in u, u > 0. - $\sin \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 25 } } { \mathrm { u } } \right)$

Find the exact value of the expression using the provided information. -Find $\sin ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$ , with $s$ in quadrant II, and $\sin t = \frac { 15 } { 17 }$ , with $t$ in quadrant II.

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\cos ( x + y ) = \cos x + \cos y$

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Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

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Applications of Trigonometry

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Exam 7: Trigonometric Identities and Equations

Find the exact value of the real number y. - y=cos⁡−1(22)y = \cos ^ { - 1 } \left( \frac { \sqrt { 2 } } { 2 } \right)y=cos−1(22​​)

Factor the trigonometric expression and simplify. - 1−2sin⁡2x+sin⁡4x1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x1−2sin2x+sin4x

Solve the equation for solutions in the interval [0, 2 π). - sin⁡xcos⁡x=12\sin x \cos x = \frac { 1 } { 2 }sinxcosx=21​

Complete the sentence so the result is an identity. Let x be any real number. - sin⁡x=(ـــــــــ)(cos⁡x)\sin x = ( ـــــــــ ) ( \cos x )sinx=(ـــــــــ)(cosx)

Provide an appropriate response. -True or false? The statement sin⁡(sin⁡−1x)=x\sin \left( \sin ^ { - 1 } x \right) = xsin(sin−1x)=x for all real numbers in the interval −π2<x<π2- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }−2π​<x<2π​ .

Use an identity to write the expression as a single trigonometric function or as a single number. - cos⁡24x−sin⁡24x\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 xcos24x−sin24x

Use an identity to write the expression as a single trigonometric function or as a single number. - sin⁡22.5∘cos⁡22.5∘\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }sin22.5∘cos22.5∘