Question 1

Solve the equation. Give a general formula for all the solutions.
-$$\cos ( 2 \theta ) = \frac { \sqrt { 2 } } { 2 }$$

Accepted Answer

A)  $\theta = \frac { \pi } { 8 } + 2 \mathrm { k } \pi , \theta = \frac { 7 \pi } { 8 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { 2 \pi } { 3 } + \mathrm { k } \pi , \theta = \frac { 4 \pi } { 3 } + \mathrm { k } \pi$ 
C)  $\theta = \frac { \pi } { 4 } + \mathrm { k } \pi , \theta = \frac { 3 \pi } { 4 } + \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 8 } + \mathrm { k } \pi , \theta = \frac { 7 \pi } { 8 } + \mathrm { k } \pi$ 
A)  $\theta = \frac { \pi } { 8 } + 2 \mathrm { k } \pi , \theta = \frac { 7 \pi } { 8 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { 2 \pi } { 3 } + \mathrm { k } \pi , \theta = \frac { 4 \pi } { 3 } + \mathrm { k } \pi$ 
C)  $\theta = \frac { \pi } { 4 } + \mathrm { k } \pi , \theta = \frac { 3 \pi } { 4 } + \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 8 } + \mathrm { k } \pi , \theta = \frac { 7 \pi } { 8 } + \mathrm { k } \pi$

Question 2

Find the exact value of the expression.
-$\sin \left( \tan ^ { - 1 } 2 \right)$

Accepted Answer

A)  $\frac { 5 \sqrt { 2 } } { 2 }$ 
B)  $\frac { 2 \sqrt { 5 } } { 5 }$ 
C)  $2 \sqrt { 5 }$ 
D)  $5 \sqrt { 2 }$ 
A)  $\frac { 5 \sqrt { 2 } } { 2 }$ 
B)  $\frac { 2 \sqrt { 5 } } { 5 }$ 
C)  $2 \sqrt { 5 }$ 
D)  $5 \sqrt { 2 }$

Question 3

Complete the identity.
-$1 - \cos ( 2 \theta ) + \cos ( 6 \theta ) - \cos ( 8 \theta ) =$ ?

Accepted Answer

A)  $4 \sin \theta \sin ( 3 \theta ) \sin ( 4 \theta )$ 
B)  $4 \sin \theta \cos ( 3 \theta ) \sin ( 4 \theta )$ 
C)  $4 \cos \theta \cos ( 3 \theta ) \sin ( 4 \theta )$ 
D)  $4 \cos \theta \cos ( 3 \theta ) \cos ( 4 \theta )$ 
A)  $4 \sin \theta \sin ( 3 \theta ) \sin ( 4 \theta )$ 
B)  $4 \sin \theta \cos ( 3 \theta ) \sin ( 4 \theta )$ 
C)  $4 \cos \theta \cos ( 3 \theta ) \sin ( 4 \theta )$ 
D)  $4 \cos \theta \cos ( 3 \theta ) \cos ( 4 \theta )$

Question 4

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal
place.
-$2 + 13 \sin x = 14 \cos ^ { 2 } x$

Accepted Answer

A)  214.9°, 325.2° 
B)  34.9°, 214.9° 
C)  55.2°, 124.9° 
D)  34.9°, 145.2° 
A)  214.9°, 325.2° 
B)  34.9°, 214.9° 
C)  55.2°, 124.9° 
D)  34.9°, 145.2°

Question 5

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\cos \theta = - \frac { 3 } { 5 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\cos \frac { \theta } { 2 }$.

Accepted Answer

A)  $- \frac { \sqrt { 5 } } { 5 }$ 
B)  $\frac { \sqrt { 30 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$ 
A)  $- \frac { \sqrt { 5 } } { 5 }$ 
B)  $\frac { \sqrt { 30 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$

Question 6

Solve the problem.
-The ground movement of an earthquake near a fault line is modeled by the equation $$d = D \tan \left[ \frac { \pi } { 2 } \left( 1 - \frac { 2 M } { S } \right) \right]$$ where M is the horizontal movement (in meters) at a distance d (in kilometers) from the earthquake, D is the depth (also
in kilometers) below the surface of the center of the earthquake, and S is the total horizontal displacement (also in
meters) at the fault line. What is the horizontal movement 5 kilometers from an earthquake centered 3 kilometers below
the surface with a total horizontal displacement of 4 meters? Round the answer to the nearest 0.01 meter.

Accepted Answer

The answer of Solve the problem.
-The ground movement of an...

Question 7

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

Accepted Answer

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

Question 8

Find the exact value of the expression.
-$\sin 255 ^ { \circ } \cos 15 ^ { \circ } - \cos 255 ^ { \circ } \sin 15 ^ { \circ }$

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { 17 } { 4 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $- \frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { 17 } { 4 }$ 
C)  $- \frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 9

Find the exact solution of the equation.
-$$- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$$

Accepted Answer

A)  $x = - \frac { \sqrt { 2 } } { 8 }$ 
B)  $x = \frac { \sqrt { 2 } } { 8 }$ 
C)  $x = - \frac { \sqrt { 2 } } { 2 }$ 
D)  $x = 0$ 
A)  $x = - \frac { \sqrt { 2 } } { 8 }$ 
B)  $x = \frac { \sqrt { 2 } } { 8 }$ 
C)  $x = - \frac { \sqrt { 2 } } { 2 }$ 
D)  $x = 0$

Question 10

Solve the problem.
-The seasonal variation in the length of daylight can be represented by a sine function. For example, the daily
number of hours of daylight in a certain city in the U.S. can be given b $h = \frac { 41 } { 4 } + \frac { 5 } { 3 } \sin \frac { 2 \pi x } { 365 }$ , where x is the
number of days after March 21 ( disregarding leap year). On what day(s) will there be about 10 hours of
daylight?

Accepted Answer

191 days after March 21 and 35

Question 11

Solve the problem.
-If $\cos \theta = - \frac { 5 } { 13 }$, and $\frac { \pi } { 2 } < \theta < \pi$, then find $\tan 2 \theta$.

Accepted Answer

A)  $\frac { 120 } { 119 }$ 
B)  $\frac { 169 } { 119 }$ 
C)  $\frac { 169 } { 120 }$ 
D)  $\frac { 119 } { 120 }$ 
A)  $\frac { 120 } { 119 }$ 
B)  $\frac { 169 } { 119 }$ 
C)  $\frac { 169 } { 120 }$ 
D)  $\frac { 119 } { 120 }$

Question 12

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = 5 \tan ( 10 x - 9 )$$

Accepted Answer

A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 5 \tan ^ { - 1 } ( 10 \mathrm { x } - 9 )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 5 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) + 9 \right]$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) - 9 \right]$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) + 9 \right]$ 
A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 5 \tan ^ { - 1 } ( 10 \mathrm { x } - 9 )$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 5 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 10 } \right) + 9 \right]$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) - 9 \right]$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 10 } \left[ \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 5 } \right) + 9 \right]$

Question 13

Write the trigonometric expression as an algebraic expression in u.
-$\cos \left( \cot ^ { - 1 } \mathrm { u } \right)$

Accepted Answer

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

Question 14

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sec \theta = 4 , \quad 0 < \theta < \frac { \pi } { 2 }$
Find $\cos \frac { \theta } { 2 }$.

Accepted Answer

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

Question 15

Complete the identity.
-$\sec \theta - \frac { 1 } { \sec \theta } =$ ?

Accepted Answer

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

Question 16

Complete the identity.
-$$\csc ^ { 2 } \frac { \theta } { 2 } = ?$$

Accepted Answer

A)  $\frac { 2 } { 1 - \cos \theta }$ 
B)  $\frac { 1 - \sec \theta } { 2 }$ 
C)  $\frac { 2 } { 1 + \sin \theta }$ 
D)  $\frac { 1 + \sec \theta } { 2 }$ 
A)  $\frac { 2 } { 1 - \cos \theta }$ 
B)  $\frac { 1 - \sec \theta } { 2 }$ 
C)  $\frac { 2 } { 1 + \sin \theta }$ 
D)  $\frac { 1 + \sec \theta } { 2 }$

Question 17

Complete the identity.
-$\tan ^ { 4 } \theta - \sec ^ { 4 } \theta =$ ?

Accepted Answer

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

Question 18

Solve the equation on the interval $0 \leq \theta < 2 \pi .$
-$\sin ^ { 2 } \theta + \sin \theta = 0$

Accepted Answer

A)  $0 , \pi , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
B)  $0 , \pi , \frac { 3 \pi } { 2 }$ 
C)  $0 , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  $0 , \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
A)  $0 , \pi , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
B)  $0 , \pi , \frac { 3 \pi } { 2 }$ 
C)  $0 , \pi , \frac { 4 \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
D)  $0 , \pi , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$

Question 19

Establish the identity.
-$$\cos ( x - y ) - \cos ( x + y ) = 2 \sin x \sin y$$

Accepted Answer

The answer of Establish the identity.
-\[\cos ( x - y...

Question 20

Complete the identity.
-$\sin ( 4 \theta ) = ?$

Accepted Answer

A)  $1 - 8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta$ 
B)  $4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$ 
C)  $3 \sin \theta - 4 \sin ^ { 3 } \theta$ 
D)  $\cos ^ { 3 } \theta - 3 \sin ^ { 2 } \theta \cos \theta$ 
A)  $1 - 8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta$ 
B)  $4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta$ 
C)  $3 \sin \theta - 4 \sin ^ { 3 } \theta$ 
D)  $\cos ^ { 3 } \theta - 3 \sin ^ { 2 } \theta \cos \theta$

Solve the equation. Give a general formula for all the solutions. - $\cos ( 2 \theta ) = \frac { \sqrt { 2 } } { 2 }$

Find the exact value of the expression. - $\sin \left( \tan ^ { - 1 } 2 \right)$

Complete the identity. - $1 - \cos ( 2 \theta ) + \cos ( 6 \theta ) - \cos ( 8 \theta ) =$ ?

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal place. - $2 + 13 \sin x = 14 \cos ^ { 2 } x$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\cos \theta = - \frac { 3 } { 5 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\cos \frac { \theta } { 2 }$ .

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

Find the exact value of the expression. - $\sin 255 ^ { \circ } \cos 15 ^ { \circ } - \cos 255 ^ { \circ } \sin 15 ^ { \circ }$

Find the exact solution of the equation. - $- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }$

Solve the problem. -If $\cos \theta = - \frac { 5 } { 13 }$ , and $\frac { \pi } { 2 } < \theta < \pi$ , then find $\tan 2 \theta$ .

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f. - $f ( x ) = 5 \tan ( 10 x - 9 )$

Write the trigonometric expression as an algebraic expression in u. - $\cos \left( \cot ^ { - 1 } \mathrm { u } \right)$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\sec \theta = 4 , \quad 0 < \theta < \frac { \pi } { 2 }$ Find $\cos \frac { \theta } { 2 }$ .

Complete the identity. - $\sec \theta - \frac { 1 } { \sec \theta } =$ ?

Complete the identity. - $\csc ^ { 2 } \frac { \theta } { 2 } = ?$

Complete the identity. - $\tan ^ { 4 } \theta - \sec ^ { 4 } \theta =$ ?

Solve the equation on the interval $0 \leq \theta < 2 \pi .$ - $\sin ^ { 2 } \theta + \sin \theta = 0$

Establish the identity. - $\cos ( x - y ) - \cos ( x + y ) = 2 \sin x \sin y$

Complete the identity. - $\sin ( 4 \theta ) = ?$

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Exam 6: Analytic Trigonometry

Solve the equation. Give a general formula for all the solutions. - cos⁡(2θ)=22\cos ( 2 \theta ) = \frac { \sqrt { 2 } } { 2 }cos(2θ)=22​​

Find the exact value of the expression. - sin⁡(tan⁡−12)\sin \left( \tan ^ { - 1 } 2 \right)sin(tan−12)

Complete the identity. - 1−cos⁡(2θ)+cos⁡(6θ)−cos⁡(8θ)=1 - \cos ( 2 \theta ) + \cos ( 6 \theta ) - \cos ( 8 \theta ) =1−cos(2θ)+cos(6θ)−cos(8θ)= ?

Use a graphing utility to solve the equation on the interval 0 0∘≤x<360∘0 ^ { \circ } \leq x < 360 ^ { \circ }0∘≤x<360∘ . Express the solution(s) rounded to one decimal place. - 2+13sin⁡x=14cos⁡2x2 + 13 \sin x = 14 \cos ^ { 2 } x2+13sinx=14cos2x

Find the exact value of the expression. - tan⁡−1[tan⁡(6π7)]\tan ^ { - 1 } \left[ \tan \left( \frac { 6 \pi } { 7 } \right) \right]tan−1[tan(76π​)]

Find the exact value of the expression. - sin⁡255∘cos⁡15∘−cos⁡255∘sin⁡15∘\sin 255 ^ { \circ } \cos 15 ^ { \circ } - \cos 255 ^ { \circ } \sin 15 ^ { \circ }sin255∘cos15∘−cos255∘sin15∘

Find the exact solution of the equation. - −sin⁡−1(4x)=π4- \sin ^ { - 1 } ( 4 x ) = \frac { \pi } { 4 }−sin−1(4x)=4π​

Solve the problem. -If cos⁡θ=−513\cos \theta = - \frac { 5 } { 13 }cosθ=−135​ , and π2<θ<π\frac { \pi } { 2 } < \theta < \pi2π​<θ<π , then find tan⁡2θ\tan 2 \thetatan2θ .

Find the inverse function f−1\mathrm { f } ^ { - 1 }f−1 of the function f. - f(x)=5tan⁡(10x−9)f ( x ) = 5 \tan ( 10 x - 9 )f(x)=5tan(10x−9)

Write the trigonometric expression as an algebraic expression in u. - cos⁡(cot⁡−1u)\cos \left( \cot ^ { - 1 } \mathrm { u } \right)cos(cot−1u)

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - sec⁡θ=4,0<θ<π2\sec \theta = 4 , \quad 0 < \theta < \frac { \pi } { 2 }secθ=4,0<θ<2π​ Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ .

Complete the identity. - sec⁡θ−1sec⁡θ=\sec \theta - \frac { 1 } { \sec \theta } =secθ−secθ1​= ?

Complete the identity. - csc⁡2θ2=?\csc ^ { 2 } \frac { \theta } { 2 } = ?csc22θ​=?

Complete the identity. - tan⁡4θ−sec⁡4θ=\tan ^ { 4 } \theta - \sec ^ { 4 } \theta =tan4θ−sec4θ= ?

Solve the equation on the interval 0≤θ<2π.0 \leq \theta < 2 \pi .0≤θ<2π. - sin⁡2θ+sin⁡θ=0\sin ^ { 2 } \theta + \sin \theta = 0sin2θ+sinθ=0

Establish the identity. - cos⁡(x−y)−cos⁡(x+y)=2sin⁡xsin⁡y\cos ( x - y ) - \cos ( x + y ) = 2 \sin x \sin ycos(x−y)−cos(x+y)=2sinxsiny

Complete the identity. - sin⁡(4θ)=?\sin ( 4 \theta ) = ?sin(4θ)=?