Question 1

Find the exact value of the expression.
-$\sin \left( \cos ^ { - 1 } \frac { 1 } { 2 } - \sin ^ { - 1 } \frac { \sqrt { 3 } } { 2 } \right)$

Accepted Answer

A)  0 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\frac { 2 \sqrt { 3 } } { 2 }$ 
D)  1 
A)  0 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\frac { 2 \sqrt { 3 } } { 2 }$ 
D)  1

Question 2

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\tan \theta + \sec \theta = 1$

Accepted Answer

A)  $\frac { 5 \pi } { 4 }$ 
B)  0 
C)  $\frac { \pi } { 4 }$ 
D)  No solution 
A)  $\frac { 5 \pi } { 4 }$ 
B)  0 
C)  $\frac { \pi } { 4 }$ 
D)  No solution

Question 3

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

Accepted Answer

A)  $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
C)  $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { 3 \pi } { 4 } , \frac { 7 \pi } { 2 }$ 
A)  $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
C)  $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { 3 \pi } { 4 } , \frac { 7 \pi } { 2 }$

Question 4

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$2 \cos \theta + 1 = 0$

Accepted Answer

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

Question 5

Complete the identity.
-$$\frac { \sin ( 3 \theta ) } { \sin \theta } = ?$$

Accepted Answer

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

Question 6

Solve the equation. Give a general formula for all the solutions.
-$$\csc \frac { \theta } { 3 } = \frac { 2 \sqrt { 3 } } { 3 }$$

Accepted Answer

A)  $\theta = \frac { \pi } { 3 } + 2 k \pi , \theta = \frac { \pi } { 9 } + 2 k \pi$ 
B)  $\theta = \pi + 6 \mathrm { k } \pi , \theta = 2 \pi + 6 \mathrm { k } \pi$ 
C)  $\theta = \frac { \pi } { 9 } + 2 \mathrm { k } \pi , \theta = \frac { \pi } { 18 } + 2 \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 2 } + 6 \mathrm { k } \pi , \theta = \pi + 6 \mathrm { k } \pi$ 
A)  $\theta = \frac { \pi } { 3 } + 2 k \pi , \theta = \frac { \pi } { 9 } + 2 k \pi$ 
B)  $\theta = \pi + 6 \mathrm { k } \pi , \theta = 2 \pi + 6 \mathrm { k } \pi$ 
C)  $\theta = \frac { \pi } { 9 } + 2 \mathrm { k } \pi , \theta = \frac { \pi } { 18 } + 2 \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 2 } + 6 \mathrm { k } \pi , \theta = \pi + 6 \mathrm { k } \pi$

Question 7

Establish the identity.
-$$\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$$

Accepted Answer

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

Question 8

Solve the problem.
-A consumer notes the sinusoidal nature of her monthly power bills. In winter when she uses electricity to heat her home
and in summer when she cools her home, the bills are high. In spring and fall, significantly less electricity is used and the
bills are much smaller. The following function models this behavior. $C = 60 + 40 \cos \left( \frac { \pi } { 3 } t - \frac { \pi } { 3 } \right)$ Here C is the cost of power in dollars for the month t, 1 $1 \leq t \leq 12$ 2, with t = 1 corresponding to January. For what
values of t, 1 $1 \leq t \leq 12$ 12, is the cost exactly $80?

Accepted Answer

The answer of Solve the problem.
-A consumer notes the sinusoidal...

Question 9

Solve the problem.
-The path of a projectile fired at an inclination $$\theta$$ (in degrees) to the horizontal with an initial speed v0 is a
parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the
formula $$R = \frac { \mathrm { v } _ { 0 } ^ { 2 } } { \mathrm {~g} } \sin ( 2 \theta )$$ where g is the acceleration due to gravity. The maximum height H of the projectile is $$\mathrm { H } = \frac { \mathrm { v } _ { 0 } ^ { 2 } } { 4 \mathrm {~g} } ( 1 - \cos ( 2 \theta ) )$$ Find the range R and the maximum height H in terms of g if the projectile is fired with an initial speed of 200 meters per
second at an angle of 15° and then at an angle of 22.5°. Do not use a calculator, but simplify the answers.

Accepted Answer

\[\begin{array} { l } 
\theta = 15 ^ { \

Question 10

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\cos \frac { 5 \pi } { 12 }$

Accepted Answer

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

Question 11

Solve the equation. Give a general formula for all the solutions.
-$\sin \theta = 1$

Accepted Answer

A)  $\theta = 0 + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
C)  $\theta = \frac { \pi } { 2 } + 2 \mathrm { k } \pi$ 
D)  $\theta = \pi + 2 \mathrm { k } \pi$ 
A)  $\theta = 0 + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
C)  $\theta = \frac { \pi } { 2 } + 2 \mathrm { k } \pi$ 
D)  $\theta = \pi + 2 \mathrm { k } \pi$

Question 12

Complete the identity.
-$\cos \theta - \cos \theta \sin ^ { 2 } \theta =$ ?

Accepted Answer

A)  $\cos ^ { 3 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sin \theta$ 
D)  $\sec ^ { 2 } \theta$ 
A)  $\cos ^ { 3 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sin \theta$ 
D)  $\sec ^ { 2 } \theta$

Question 13

Find the exact solution of the equation.
-$$\sin ^ { - 1 } x = \frac { \pi } { 6 }$$

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places.
-$\cos \theta = 0.92$

Accepted Answer

A)  0.40, 2.74 
B)  0.40, 5.88 
C)  0.40, 1.97 
D)  0.40, 3.54 
A)  0.40, 2.74 
B)  0.40, 5.88 
C)  0.40, 1.97 
D)  0.40, 3.54

Question 16

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

Accepted Answer

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

Question 17

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\cos ( 2 \theta ) = \sqrt { 2 } - \cos ( 2 \theta )$

Accepted Answer

A)  $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
D)  No solution 
A)  $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 15 \pi } { 8 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
D)  No solution

Question 18

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\cos \theta = - \frac { 5 } { 7 } , \quad \csc \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Accepted Answer

A)  $- \frac { 1 } { 49 }$ 
B)  $\frac { 1 } { 49 }$ 
C)  $\frac { 20 \sqrt { 6 } } { 49 }$ 
D)  $\frac { - 20 \sqrt { 6 } } { 49 }$ 
A)  $- \frac { 1 } { 49 }$ 
B)  $\frac { 1 } { 49 }$ 
C)  $\frac { 20 \sqrt { 6 } } { 49 }$ 
D)  $\frac { - 20 \sqrt { 6 } } { 49 }$

Question 19

Solve the problem.
-Wildlife management personnel use predator-prey equations to model the populations of certain predators and
their prey in the wild. Suppose the population M of a predator after t months is given by $M = 750 + 125 \sin \frac { \pi } { 6 } t$ while the population N of its primary prey is given by $N = 12,250 + 3050 \cos \frac { \pi } { 6 } t$ Find the values of t $0 \leq t < 12$ or which the predator population is 875. Find the values of t, $0 \leq t < 12$ for which
the prey population is 10,725.

Accepted Answer

M = 875, t

Question 20

Find the exact value of the expression. Do not use a calculator.
-$$\sin ^ { - 1 } \left[ \sin \left( \frac { 6 \pi } { 7 } \right) \right]$$

Accepted Answer

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

Find the exact value of the expression. - $\sin \left( \cos ^ { - 1 } \frac { 1 } { 2 } - \sin ^ { - 1 } \frac { \sqrt { 3 } } { 2 } \right)$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\tan \theta + \sec \theta = 1$

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

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $2 \cos \theta + 1 = 0$

Complete the identity. - $\frac { \sin ( 3 \theta ) } { \sin \theta } = ?$

Solve the equation. Give a general formula for all the solutions. - $\csc \frac { \theta } { 3 } = \frac { 2 \sqrt { 3 } } { 3 }$

Establish the identity. - $\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\cos \frac { 5 \pi } { 12 }$

Solve the equation. Give a general formula for all the solutions. - $\sin \theta = 1$

Complete the identity. - $\cos \theta - \cos \theta \sin ^ { 2 } \theta =$ ?

Find the exact solution of the equation. - $\sin ^ { - 1 } x = \frac { \pi } { 6 }$

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

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places. - $\cos \theta = 0.92$

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

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\cos ( 2 \theta ) = \sqrt { 2 } - \cos ( 2 \theta )$

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\cos \theta = - \frac { 5 } { 7 } , \quad \csc \theta < 0 \quad$ Find $\cos ( 2 \theta )$

Find the exact value of the expression. Do not use a calculator. - $\sin ^ { - 1 } \left[ \sin \left( \frac { 6 \pi } { 7 } \right) \right]$

Functions and Their Graphs

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Exponential and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometric Functions

Polar Coordinates; Vectors

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Systems of Equations and Inequalities

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Exam 6: Analytic Trigonometry

Find the exact value of the expression. - sin⁡(cos⁡−112−sin⁡−132)\sin \left( \cos ^ { - 1 } \frac { 1 } { 2 } - \sin ^ { - 1 } \frac { \sqrt { 3 } } { 2 } \right)sin(cos−121​−sin−123​​)

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - tan⁡θ+sec⁡θ=1\tan \theta + \sec \theta = 1tanθ+secθ=1

Solve the equation on the interval 0≤θ<2π. 0 \leq \theta < 2 \pi \text {. }0≤θ<2π. - cos⁡θ=sin⁡θ\cos \theta = \sin \thetacosθ=sinθ

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - 2cos⁡θ+1=02 \cos \theta + 1 = 02cosθ+1=0

Complete the identity. - sin⁡(3θ)sin⁡θ=?\frac { \sin ( 3 \theta ) } { \sin \theta } = ?sinθsin(3θ)​=?

Solve the equation. Give a general formula for all the solutions. - csc⁡θ3=233\csc \frac { \theta } { 3 } = \frac { 2 \sqrt { 3 } } { 3 }csc3θ​=323​​

Establish the identity. - tan⁡(π2+x)=−cot⁡x\tan \left( \frac { \pi } { 2 } + x \right) = - \cot xtan(2π​+x)=−cotx

Use the Half-angle Formulas to find the exact value of the trigonometric function. - cos⁡5π12\cos \frac { 5 \pi } { 12 }cos125π​

Solve the equation. Give a general formula for all the solutions. - sin⁡θ=1\sin \theta = 1sinθ=1

Complete the identity. - cos⁡θ−cos⁡θsin⁡2θ=\cos \theta - \cos \theta \sin ^ { 2 } \theta =cosθ−cosθsin2θ= ?

Find the exact solution of the equation. - sin⁡−1x=π6\sin ^ { - 1 } x = \frac { \pi } { 6 }sin−1x=6π​

Complete the identity. - sin⁡(2θ)+2sin⁡2θcos⁡(2θ)=?\frac { \sin ( 2 \theta ) + 2 \sin ^ { 2 } \theta } { \cos ( 2 \theta ) } = ?cos(2θ)sin(2θ)+2sin2θ​=?

Use a calculator to solve the equation on the interval 0 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . Round the answer to two decimal places. - cos⁡θ=0.92\cos \theta = 0.92cosθ=0.92

Find the exact value of the expression. - sin⁡15∘\sin 15 ^ { \circ }sin15∘

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - cos⁡(2θ)=2−cos⁡(2θ)\cos ( 2 \theta ) = \sqrt { 2 } - \cos ( 2 \theta )cos(2θ)=2​−cos(2θ)

Find the exact value of the expression. Do not use a calculator. - sin⁡−1[sin⁡(6π7)]\sin ^ { - 1 } \left[ \sin \left( \frac { 6 \pi } { 7 } \right) \right]sin−1[sin(76π​)]