Question 1

Find the exact value of the expression.
-$\csc ^ { - 1 } ( - 2 )$

Accepted Answer

A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $- \frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 6 }$ 
A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { \pi } { 3 }$ 
C)  $- \frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 6 }$

Question 2

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ .
-$$f ( x ) = 2 \sin ( 3 x - 1 )$$

Accepted Answer

A)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 3,3 ]$ 
B)  Domain of f: $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $f : [ - 2,2 ]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
D)  Domain of $\mathrm { f } : ( \infty , \infty )$

Domain of $f ^ { - 1 } : [ - 2,2 ]$ 
A)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 3,3 ]$ 
B)  Domain of f: $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $f : [ - 2,2 ]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
D)  Domain of $\mathrm { f } : ( \infty , \infty )$

Domain of $f ^ { - 1 } : [ - 2,2 ]$

Question 3

Solve the problem using Snell's Law: $\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { v _ { 1 } } { v _ { 2 } }$
-The index of refraction of light passing from air into a second medium is 1.67. If the angle of incidence is 87°, what is the angle of refraction (to two decimal places)?

Accepted Answer

A)  54.31° 
B)  53.27° 
C)  36.73° 
D)  35.69° 
A)  54.31° 
B)  53.27° 
C)  36.73° 
D)  35.69°

Question 4

Simplify the expression.
-$1 - 2 \sin ^ { 2 } 2 x$

Accepted Answer

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

Question 5

Find the exact solution of the equation.
-$2 \cos ^ { - 1 } x = \pi$

Accepted Answer

A)  $x = 0$ 
B)  $x = 1$ 
C)  $x = \frac { 3 \pi } { 2 }$ 
D)  $x = \frac { \pi } { 2 }$ 
A)  $x = 0$ 
B)  $x = 1$ 
C)  $x = \frac { 3 \pi } { 2 }$ 
D)  $x = \frac { \pi } { 2 }$

Question 6

Simplify the expression.
-$4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$

Accepted Answer

A)  $\sin 4 \theta$ 
B)  $4 \sin \theta \cos \theta \cos 2 \theta$ 
C)  $\sin 2 \theta \cos 2 \theta$ 
D)  $4 \sin \theta \cos \theta$ 
A)  $\sin 4 \theta$ 
B)  $4 \sin \theta \cos \theta \cos 2 \theta$ 
C)  $\sin 2 \theta \cos 2 \theta$ 
D)  $4 \sin \theta \cos \theta$

Question 7

Complete the identity.
-$$\frac { \sin \alpha + \sin \beta } { \cos \alpha + \cos \beta } = ?$$

Accepted Answer

A)  $\tan \frac { \alpha + \beta } { 2 }$ 
B)  $\tan \alpha + \tan \beta$ 
C)  $\cot \frac { \alpha - \beta } { 2 }$ 
D)  $\tan \frac { \alpha - \beta } { 2 }$ 
A)  $\tan \frac { \alpha + \beta } { 2 }$ 
B)  $\tan \alpha + \tan \beta$ 
C)  $\cot \frac { \alpha - \beta } { 2 }$ 
D)  $\tan \frac { \alpha - \beta } { 2 }$

Question 8

Establish the identity.
-$$\sin \left( \frac { \pi } { 4 } + x \right) = \sqrt { 2 } ( \cos x + \sin x )$$

Accepted Answer

The answer of Establish the identity.
-\[\sin \left( \frac { \pi...

Question 9

Express the product as a sum containing only sines or cosines.
-$\sin ( 4 \theta ) \sin ( 6 \theta )$

Accepted Answer

A)  $\sin ^ { 2 } \left( 24 \theta ^ { 2 } \right)$ 
B)  $\frac { 1 } { 2 } [ - \cos ( 2 \theta ) - \cos ( 10 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \cos ( 2 \theta ) - \cos ( 10 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 10 \theta ) - \sin ( 2 \theta ) ]$ 
A)  $\sin ^ { 2 } \left( 24 \theta ^ { 2 } \right)$ 
B)  $\frac { 1 } { 2 } [ - \cos ( 2 \theta ) - \cos ( 10 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \cos ( 2 \theta ) - \cos ( 10 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 10 \theta ) - \sin ( 2 \theta ) ]$

Question 10

Solve the problem.
-A weight suspended from a spring is vibrating vertically with up being the positive direction. The function $f ( t ) = 10 \sin \left( \frac { 3 \pi t } { 4 } - \frac { \pi } { 4 } \right)$ represents the distance in centimeters of the weight from its rest position as a function of time t, where t is measured in seconds. Find the smallest positive value of t for which the displacement of the weight above its rest position is 5 cm.
Round answer to three decimal places, if necessary.

Accepted Answer

A)  1.586 
B)  2.293 
C)  0.556 
D)  0.222 
A)  1.586 
B)  2.293 
C)  0.556 
D)  0.222

Question 11

Complete the identity.
-$$1 - \frac { 1 } { 2 } \sin ( 2 \theta ) = ?$$

Accepted Answer

A)  $\frac { \sin \theta \cos ( 2 \theta ) } { \sin \theta + \cos \theta }$ 
B)  $\frac { \sin ^ { 3 } \theta + \cos ^ { 3 } \theta } { \sin \theta + \cos \theta }$ 
C)  $\frac { \sin ^ { 3 } \theta - \cos ^ { 3 } \theta } { \sin \theta + \cos \theta }$ 
D)  $\frac { \sin \theta \cos \theta \cos ( 2 \theta ) } { \sin \theta + \cos \theta }$ 
A)  $\frac { \sin \theta \cos ( 2 \theta ) } { \sin \theta + \cos \theta }$ 
B)  $\frac { \sin ^ { 3 } \theta + \cos ^ { 3 } \theta } { \sin \theta + \cos \theta }$ 
C)  $\frac { \sin ^ { 3 } \theta - \cos ^ { 3 } \theta } { \sin \theta + \cos \theta }$ 
D)  $\frac { \sin \theta \cos \theta \cos ( 2 \theta ) } { \sin \theta + \cos \theta }$

Question 12

Establish the identity.
-$$\frac { 5 \csc ^ { 2 } x + 4 \csc x - 1 } { \cot ^ { 2 } x } = \frac { 5 \csc x - 1 } { \csc x - 1 }$$

Accepted Answer

The answer of Establish the identity.
-\[\frac { 5 \csc ^...

Question 13

Find the exact value under the given conditions.
-$\cos \alpha = - \frac { 4 } { 5 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \sin \beta = - \frac { \sqrt { 21 } } { 5 } , \pi < \beta < \frac { 3 \pi } { 2 } \quad$ Find $\cos ( \alpha + \beta )$.

Accepted Answer

A)  $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$ 
B)  $\frac { 8 + 3 \sqrt { 21 } } { 25 }$ 
C)  $\frac { 6 - 4 \sqrt { 21 } } { 25 }$ 
D)  $\frac { - 8 - 3 \sqrt { 21 } } { 25 }$ 
A)  $\frac { - 6 + 4 \sqrt { 21 } } { 25 }$ 
B)  $\frac { 8 + 3 \sqrt { 21 } } { 25 }$ 
C)  $\frac { 6 - 4 \sqrt { 21 } } { 25 }$ 
D)  $\frac { - 8 - 3 \sqrt { 21 } } { 25 }$

Question 14

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\sec \theta = - \frac { 4 \sqrt { 7 } } { 7 } , \quad \csc \theta > 0 \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { 4 \sqrt { 5 } } { 9 }$ 
B)  $- \frac { 1 } { 9 }$ 
C)  $\frac { 1 } { 9 }$ 
D)  $\frac { - 4 \sqrt { 5 } } { 9 }$ 
A)  $\frac { 4 \sqrt { 5 } } { 9 }$ 
B)  $- \frac { 1 } { 9 }$ 
C)  $\frac { 1 } { 9 }$ 
D)  $\frac { - 4 \sqrt { 5 } } { 9 }$

Question 15

Use a calculator to find the value of the expression rounded to two decimal places.
-$\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)$

Accepted Answer

A)  -0.62 
B)  125.26 
C)  2.19 
D)  -35.26 
A)  -0.62 
B)  125.26 
C)  2.19 
D)  -35.26

Question 16

Express the sum or difference as a product of sines and/or cosines.
-$\cos ( 6 \theta ) + \cos ( 4 \theta )$

Accepted Answer

A)  $2 \cos ( 5 \theta ) \sin \theta$ 
B)  $2 \cos ( 5 \theta ) \cos \theta$ 
C)  $2 \cos ( 5 \theta )$ 
D)  $2 \sin ( 5 \theta ) \sin \theta$ 
A)  $2 \cos ( 5 \theta ) \sin \theta$ 
B)  $2 \cos ( 5 \theta ) \cos \theta$ 
C)  $2 \cos ( 5 \theta )$ 
D)  $2 \sin ( 5 \theta ) \sin \theta$

Question 17

Simplify the expression.
-$2 \cos ^ { 2 } 4 x - 1$

Accepted Answer

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

Question 18

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = - 4 \cos ( 10 x + 7 )$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = - 4 \cos ^ { - 1 } ( 10 x + 7 )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) - 7 \right]$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right) - 7 \right]$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = - \frac { 1 } { 10 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right) + 7 \right]$ 
A)  $f ^ { - 1 } ( x ) = - 4 \cos ^ { - 1 } ( 10 x + 7 )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) - 7 \right]$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right) - 7 \right]$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = - \frac { 1 } { 10 } \left[ \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right) + 7 \right]$

Question 19

Complete the identity.
-$\csc \theta ( \sin \theta + \cos \theta ) =$ ?

Accepted Answer

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

Question 20

Find the exact value of the expression.
-$\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$

Accepted Answer

A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 5 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 3 }$ 
A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 5 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 3 }$

Find the exact value of the expression. - $\csc ^ { - 1 } ( - 2 )$

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ . - $f ( x ) = 2 \sin ( 3 x - 1 )$

Simplify the expression. - $1 - 2 \sin ^ { 2 } 2 x$

Find the exact solution of the equation. - $2 \cos ^ { - 1 } x = \pi$

Simplify the expression. - $4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$

Complete the identity. - $\frac { \sin \alpha + \sin \beta } { \cos \alpha + \cos \beta } = ?$

Establish the identity. - $\sin \left( \frac { \pi } { 4 } + x \right) = \sqrt { 2 } ( \cos x + \sin x )$

Express the product as a sum containing only sines or cosines. - $\sin ( 4 \theta ) \sin ( 6 \theta )$

Complete the identity. - $1 - \frac { 1 } { 2 } \sin ( 2 \theta ) = ?$

Establish the identity. - $\frac { 5 \csc ^ { 2 } x + 4 \csc x - 1 } { \cot ^ { 2 } x } = \frac { 5 \csc x - 1 } { \csc x - 1 }$

Find the exact value under the given conditions. - $\cos \alpha = - \frac { 4 } { 5 } , \frac { \pi } { 2 } < \alpha < \pi ; \quad \sin \beta = - \frac { \sqrt { 21 } } { 5 } , \pi < \beta < \frac { 3 \pi } { 2 } \quad$ Find $\cos ( \alpha + \beta )$ .

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\sec \theta = - \frac { 4 \sqrt { 7 } } { 7 } , \quad \csc \theta > 0 \quad$ Find $\sin ( 2 \theta )$

Use a calculator to find the value of the expression rounded to two decimal places. - $\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)$

Express the sum or difference as a product of sines and/or cosines. - $\cos ( 6 \theta ) + \cos ( 4 \theta )$

Simplify the expression. - $2 \cos ^ { 2 } 4 x - 1$

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f. - $f ( x ) = - 4 \cos ( 10 x + 7 )$

Complete the identity. - $\csc \theta ( \sin \theta + \cos \theta ) =$ ?

Find the exact value of the expression. - $\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$

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Exam 6: Analytic Trigonometry

Find the exact value of the expression. - csc⁡−1(−2)\csc ^ { - 1 } ( - 2 )csc−1(−2)

Find the domain of the function f and of its inverse function f−1\mathrm { f } ^ { - 1 }f−1 . - f(x)=2sin⁡(3x−1)f ( x ) = 2 \sin ( 3 x - 1 )f(x)=2sin(3x−1)

Simplify the expression. - 1−2sin⁡22x1 - 2 \sin ^ { 2 } 2 x1−2sin22x

Find the exact solution of the equation. - 2cos⁡−1x=π2 \cos ^ { - 1 } x = \pi2cos−1x=π

Simplify the expression. - 4sin⁡θcos⁡3θ−4sin⁡3θcos⁡θ4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta4sinθcos3θ−4sin3θcosθ

Complete the identity. - sin⁡α+sin⁡βcos⁡α+cos⁡β=?\frac { \sin \alpha + \sin \beta } { \cos \alpha + \cos \beta } = ?cosα+cosβsinα+sinβ​=?

Establish the identity. - sin⁡(π4+x)=2(cos⁡x+sin⁡x)\sin \left( \frac { \pi } { 4 } + x \right) = \sqrt { 2 } ( \cos x + \sin x )sin(4π​+x)=2​(cosx+sinx)

Express the product as a sum containing only sines or cosines. - sin⁡(4θ)sin⁡(6θ)\sin ( 4 \theta ) \sin ( 6 \theta )sin(4θ)sin(6θ)

Complete the identity. - 1−12sin⁡(2θ)=?1 - \frac { 1 } { 2 } \sin ( 2 \theta ) = ?1−21​sin(2θ)=?

Establish the identity. - 5csc⁡2x+4csc⁡x−1cot⁡2x=5csc⁡x−1csc⁡x−1\frac { 5 \csc ^ { 2 } x + 4 \csc x - 1 } { \cot ^ { 2 } x } = \frac { 5 \csc x - 1 } { \csc x - 1 }cot2x5csc2x+4cscx−1​=cscx−15cscx−1​

Use a calculator to find the value of the expression rounded to two decimal places. - cos⁡−1(−33)\cos ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 3 } \right)cos−1(−33​​)

Express the sum or difference as a product of sines and/or cosines. - cos⁡(6θ)+cos⁡(4θ)\cos ( 6 \theta ) + \cos ( 4 \theta )cos(6θ)+cos(4θ)

Simplify the expression. - 2cos⁡24x−12 \cos ^ { 2 } 4 x - 12cos24x−1

Find the inverse function f−1\mathrm { f } ^ { - 1 }f−1 of the function f. - f(x)=−4cos⁡(10x+7)f ( x ) = - 4 \cos ( 10 x + 7 )f(x)=−4cos(10x+7)

Complete the identity. - csc⁡θ(sin⁡θ+cos⁡θ)=\csc \theta ( \sin \theta + \cos \theta ) =cscθ(sinθ+cosθ)= ?

Find the exact value of the expression. - cos⁡−1(sin⁡7π6)\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)cos−1(sin67π​)