Question 1

Find the exact value of the expression.
-$\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)$

Accepted Answer

A)  $- \frac { 3 } { 5 }$
В) $\frac { 1 } { 5 }$ 
C)  $- \frac { 4 } { 5 }$ 
D)  $\frac { 3 } { 5 }$ 
A)  $- \frac { 3 } { 5 }$
В) $\frac { 1 } { 5 }$ 
C)  $- \frac { 4 } { 5 }$ 
D)  $\frac { 3 } { 5 }$

Question 2

Write the trigonometric expression as an algebraic expression containing u and v.
-$\sin \left( \tan ^ { - 1 } \mathrm { u } - \tan ^ { - 1 } \mathrm { v } \right)$

Accepted Answer

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

Question 3

Solve the equation on the interval [0, 2π).
-Suppose $f ( x ) = \cos \theta - 1$. Solve $f ( x ) = 0$.

Accepted Answer

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

Question 4

Solve the problem.
-If $\tan \theta = \frac { 7 } { 24 }$, and $\pi < \theta < \frac { 3 \pi } { 2 }$, then find $\tan 2 \theta$.

Accepted Answer

A)  $\frac { 527 } { 336 }$ 
B)  $- \frac { 336 } { 527 }$ 
C)  $- \frac { 527 } { 336 }$ 
D)  $\frac { 336 } { 527 }$ 
A)  $\frac { 527 } { 336 }$ 
B)  $- \frac { 336 } { 527 }$ 
C)  $- \frac { 527 } { 336 }$ 
D)  $\frac { 336 } { 527 }$

Question 5

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\csc ( 3 \theta ) = 0$

Accepted Answer

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

Question 6

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sin \theta = - \frac { 21 } { 29 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\cos \frac { \theta } { 2 }$.

Accepted Answer

A)  $\frac { 10 } { 29 }$ 
B)  $\frac { 7 \sqrt { 58 } } { 58 }$ 
C)  $- \frac { 7 \sqrt { 58 } } { 58 }$ 
D)  $- \frac { 3 \sqrt { 58 } } { 58 }$ 
A)  $\frac { 10 } { 29 }$ 
B)  $\frac { 7 \sqrt { 58 } } { 58 }$ 
C)  $- \frac { 7 \sqrt { 58 } } { 58 }$ 
D)  $- \frac { 3 \sqrt { 58 } } { 58 }$

Question 7

Complete the identity.
-$\sin ( \alpha + \beta ) \cos \beta - \cos ( \alpha + \beta ) \sin \beta = ?$

Accepted Answer

A)  $\sin \alpha \cos \beta - \cos \alpha \sin \beta$ 
B)  $\sin \alpha \cos ^ { 2 } \beta - \sin \alpha \sin ^ { 2 } \beta$ 
C)  $\sin \alpha$ 
D)  $2 \sin \beta \cos \beta ( \sin \alpha - \cos \alpha )$ 
A)  $\sin \alpha \cos \beta - \cos \alpha \sin \beta$ 
B)  $\sin \alpha \cos ^ { 2 } \beta - \sin \alpha \sin ^ { 2 } \beta$ 
C)  $\sin \alpha$ 
D)  $2 \sin \beta \cos \beta ( \sin \alpha - \cos \alpha )$

Question 8

Complete the identity.
-$\sin \left( \tan ^ { - 1 } v \right) = ?$

Accepted Answer

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

Question 9

Use a calculator to solve the equation on the interval 0 $0 \leq x < 2 \pi$ . Round the answer to one decimal place if necessary.
-$e ^ { x } = \cos x$

Accepted Answer

The answer of Use a calculator to solve the equation...

Question 10

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\tan 165 ^ { \circ }$

Accepted Answer

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

Question 11

Find the exact value of the expression. Do not use a calculator.
-$$\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]$$

Accepted Answer

A)  $5 \pi$ 
B)  $\frac { \pi } { 5 }$ 
C)  $- 5 \pi$ 
D)  $- \frac { \pi } { 5 }$ 
A)  $5 \pi$ 
B)  $\frac { \pi } { 5 }$ 
C)  $- 5 \pi$ 
D)  $- \frac { \pi } { 5 }$

Question 12

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\sqrt { 2 } \cos ( 2 \theta ) = 1$

Accepted Answer

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

Question 13

Find the exact value under the given conditions.
-$\cos \alpha = \frac { 1 } { 3 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \sin \beta = - \frac { 1 } { 2 } , - \frac { \pi } { 2 } < \beta < 0$
Find $\tan ( \alpha + \beta )$

Accepted Answer

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

Question 14

Solve the equation on the interval [0, 2π).
-Suppose $f ( x ) = 2 \cos \theta + 1$. Solve $f ( x ) = 0$.

Accepted Answer

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

Question 15

Solve the problem.
-If $\sin \theta = - \frac { 4 } { 5 }$, and $\theta$ terminates in quadrant IV, then find $\sin 2 \theta$.

Accepted Answer

A)  $- \frac { 24 } { 25 }$ 
B)  $\frac { 24 } { 25 }$ 
C)  $- \frac { 7 } { 25 }$ 
D)  $\frac { 7 } { 25 }$ 
A)  $- \frac { 24 } { 25 }$ 
B)  $\frac { 24 } { 25 }$ 
C)  $- \frac { 7 } { 25 }$ 
D)  $\frac { 7 } { 25 }$

Question 16

Find the exact value of the expression.
-$$\cos ^ { - 1 } \frac { \sqrt { 3 } } { 2 }$$

Accepted Answer

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

Question 17

Simplify the expression.
-$4 \sin 2 x \cos 2 x$

Accepted Answer

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

Question 18

Establish the identity.
-$$\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$$

Accepted Answer

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

Question 19

Solve the problem.
-Multiply and simplify: $\frac { ( \tan \theta + 1 ) ( \tan \theta + 1 ) - \sec ^ { 2 } \theta } { \tan \theta }$

Accepted Answer

The answer of Solve the problem.
-Multiply and simplify: \(\frac {...

Question 20

Complete the identity.
-$$\frac { \cos ( 3 \theta ) - \cos ( 9 \theta ) } { \cos ( 3 \theta ) + \cos ( 9 \theta ) } = ?$$

Accepted Answer

A)  0 
B)  $\tan ( 3 \theta ) \tan ( 6 \theta )$ 
C)  $\cot ( 6 \theta )$ 
D)  $- \tan ( 6 \theta )$ 
A)  0 
B)  $\tan ( 3 \theta ) \tan ( 6 \theta )$ 
C)  $\cot ( 6 \theta )$ 
D)  $- \tan ( 6 \theta )$

Find the exact value of the expression. - $\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)$

Write the trigonometric expression as an algebraic expression containing u and v. - $\sin \left( \tan ^ { - 1 } \mathrm { u } - \tan ^ { - 1 } \mathrm { v } \right)$

Solve the equation on the interval [0, 2π). -Suppose $f ( x ) = \cos \theta - 1$ . Solve $f ( x ) = 0$ .

Solve the problem. -If $\tan \theta = \frac { 7 } { 24 }$ , and $\pi < \theta < \frac { 3 \pi } { 2 }$ , then find $\tan 2 \theta$ .

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\csc ( 3 \theta ) = 0$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\sin \theta = - \frac { 21 } { 29 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\cos \frac { \theta } { 2 }$ .

Complete the identity. - $\sin ( \alpha + \beta ) \cos \beta - \cos ( \alpha + \beta ) \sin \beta = ?$

Complete the identity. - $\sin \left( \tan ^ { - 1 } v \right) = ?$

Use a calculator to solve the equation on the interval 0 $0 \leq x < 2 \pi$ . Round the answer to one decimal place if necessary. - $e ^ { x } = \cos x$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\tan 165 ^ { \circ }$

Find the exact value of the expression. Do not use a calculator. - $\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\sqrt { 2 } \cos ( 2 \theta ) = 1$

Find the exact value under the given conditions. - $\cos \alpha = \frac { 1 } { 3 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \sin \beta = - \frac { 1 } { 2 } , - \frac { \pi } { 2 } < \beta < 0$ Find $\tan ( \alpha + \beta )$

Solve the equation on the interval [0, 2π). -Suppose $f ( x ) = 2 \cos \theta + 1$ . Solve $f ( x ) = 0$ .

Solve the problem. -If $\sin \theta = - \frac { 4 } { 5 }$ , and $\theta$ terminates in quadrant IV, then find $\sin 2 \theta$ .

Find the exact value of the expression. - $\cos ^ { - 1 } \frac { \sqrt { 3 } } { 2 }$

Simplify the expression. - $4 \sin 2 x \cos 2 x$

Establish the identity. - $\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }$

Solve the problem. -Multiply and simplify: $\frac { ( \tan \theta + 1 ) ( \tan \theta + 1 ) - \sec ^ { 2 } \theta } { \tan \theta }$

Complete the identity. - $\frac { \cos ( 3 \theta ) - \cos ( 9 \theta ) } { \cos ( 3 \theta ) + \cos ( 9 \theta ) } = ?$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

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. - cos⁡(sin⁡−145)\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)cos(sin−154​)

Write the trigonometric expression as an algebraic expression containing u and v. - sin⁡(tan⁡−1u−tan⁡−1v)\sin \left( \tan ^ { - 1 } \mathrm { u } - \tan ^ { - 1 } \mathrm { v } \right)sin(tan−1u−tan−1v)

Solve the equation on the interval [0, 2π). -Suppose f(x)=cos⁡θ−1f ( x ) = \cos \theta - 1f(x)=cosθ−1 . Solve f(x)=0f ( x ) = 0f(x)=0 .

Solve the problem. -If tan⁡θ=724\tan \theta = \frac { 7 } { 24 }tanθ=247​ , and π<θ<3π2\pi < \theta < \frac { 3 \pi } { 2 }π<θ<23π​ , then find tan⁡2θ\tan 2 \thetatan2θ .

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - csc⁡(3θ)=0\csc ( 3 \theta ) = 0csc(3θ)=0

Complete the identity. - sin⁡(α+β)cos⁡β−cos⁡(α+β)sin⁡β=?\sin ( \alpha + \beta ) \cos \beta - \cos ( \alpha + \beta ) \sin \beta = ?sin(α+β)cosβ−cos(α+β)sinβ=?

Complete the identity. - sin⁡(tan⁡−1v)=?\sin \left( \tan ^ { - 1 } v \right) = ?sin(tan−1v)=?

Use a calculator to solve the equation on the interval 0 0≤x<2π0 \leq x < 2 \pi0≤x<2π . Round the answer to one decimal place if necessary. - ex=cos⁡xe ^ { x } = \cos xex=cosx

Use the Half-angle Formulas to find the exact value of the trigonometric function. - tan⁡165∘\tan 165 ^ { \circ }tan165∘

Find the exact value of the expression. Do not use a calculator. - sin⁡−1[sin⁡(π5)]\sin ^ { - 1 } \left[ \sin \left( \frac { \pi } { 5 } \right) \right]sin−1[sin(5π​)]

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - 2cos⁡(2θ)=1\sqrt { 2 } \cos ( 2 \theta ) = 12​cos(2θ)=1

Solve the equation on the interval [0, 2π). -Suppose f(x)=2cos⁡θ+1f ( x ) = 2 \cos \theta + 1f(x)=2cosθ+1 . Solve f(x)=0f ( x ) = 0f(x)=0 .

Solve the problem. -If sin⁡θ=−45\sin \theta = - \frac { 4 } { 5 }sinθ=−54​ , and θ\thetaθ terminates in quadrant IV, then find sin⁡2θ\sin 2 \thetasin2θ .

Find the exact value of the expression. - cos⁡−132\cos ^ { - 1 } \frac { \sqrt { 3 } } { 2 }cos−123​​

Simplify the expression. - 4sin⁡2xcos⁡2x4 \sin 2 x \cos 2 x4sin2xcos2x

Establish the identity. - tan⁡(x−π4)=tan⁡x−11+tan⁡x\tan \left( x - \frac { \pi } { 4 } \right) = \frac { \tan x - 1 } { 1 + \tan x }tan(x−4π​)=1+tanxtanx−1​

Solve the problem. -Multiply and simplify: (tan⁡θ+1)(tan⁡θ+1)−sec⁡2θtan⁡θ\frac { ( \tan \theta + 1 ) ( \tan \theta + 1 ) - \sec ^ { 2 } \theta } { \tan \theta }tanθ(tanθ+1)(tanθ+1)−sec2θ​

Complete the identity. - cos⁡(3θ)−cos⁡(9θ)cos⁡(3θ)+cos⁡(9θ)=?\frac { \cos ( 3 \theta ) - \cos ( 9 \theta ) } { \cos ( 3 \theta ) + \cos ( 9 \theta ) } = ?cos(3θ)+cos(9θ)cos(3θ)−cos(9θ)​=?