Question 1

Complete the identity.
-$\ln | \csc x | = ?$

Accepted Answer

A)  $- \ln | \sin x |$ 
B)  $\ln | \sin x |$ 
C)  $- \ln | \cos x |$ 
D)  $\ln | \cos x |$ 
A)  $- \ln | \sin x |$ 
B)  $\ln | \sin x |$ 
C)  $- \ln | \cos x |$ 
D)  $\ln | \cos x |$

Question 2

Solve the equation on the interval $0 \leq \theta < 2 \pi .$
-$\csc ^ { 5 } \theta - 4 \csc \theta = 0$

Accepted Answer

A)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 6 }$ 
D)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
A)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 }$ 
C)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 6 }$ 
D)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$

Question 3

Simplify the expression.
-$$\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta$$

Accepted Answer

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

Question 4

Use a calculator to find the value of the expression rounded to two decimal places.
-$\tan ^ { - 1 } ( 0.2 )$

Accepted Answer

A)  78.69 
B)  11.31 
C)  0.20 
D)  1.37 
A)  78.69 
B)  11.31 
C)  0.20 
D)  1.37

Question 5

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

Accepted Answer

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

Question 6

Solve the problem using Snell's Law: $\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { v _ { 1 } } { v _ { 2 } }$
-A light beam in air travels at 2 $2.99 \times 10 ^ { 8 }$ meters per second. If its angle of incidence to a second medium is 77° and its angle of refraction in the second medium is 67°, what is its speed in the second medium (to two decimal
Places)? A) $2.91 \times 10 ^ { 8 } \mathrm { mps }$

Accepted Answer

B)  $3.16 \times 10 ^ { 8 } \mathrm { mps }$ 
C)  $2.82 \times 10 ^ { 8 } \mathrm { mps }$ 
D)  $2.75 \times 10 ^ { 8 } \mathrm { mps }$ 
B)  $3.16 \times 10 ^ { 8 } \mathrm { mps }$ 
C)  $2.82 \times 10 ^ { 8 } \mathrm { mps }$ 
D)  $2.75 \times 10 ^ { 8 } \mathrm { mps }$

Question 7

Solve the problem.
-The function $$\mathrm { I } ( \mathrm { t } ) = 40 \sin \left( 60 \pi t - \frac { \pi } { 2 } \right)$$ represents the amperes of current produced by an electric generator as a function of time t, where t is measured in seconds. Find the smallest value of t for which the current is 20 amperes. Round your answer to three decimal
Places, if necessary.

Accepted Answer

A)  0.008 
B)  0.011 
C)  0.017 
D)  0.033 
A)  0.008 
B)  0.011 
C)  0.017 
D)  0.033

Question 8

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.
-$- 11 + 24 \sin x = 16 \cos ^ { 2 } x$

Accepted Answer

The answer of Use a graphing utility to solve the...

Question 9

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\cos \theta = \frac { 24 } { 25 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\sin ( 2 \theta )$.

Accepted Answer

A)  $- \frac { 336 } { 625 }$ 
B)  $\frac { 527 } { 625 }$ 
C)  $- \frac { 527 } { 625 }$ 
D)  $\frac { 336 } { 625 }$ 
A)  $- \frac { 336 } { 625 }$ 
B)  $\frac { 527 } { 625 }$ 
C)  $- \frac { 527 } { 625 }$ 
D)  $\frac { 336 } { 625 }$

Question 10

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

Accepted Answer

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

Question 11

Find the exact value of the expression.
-$\tan \frac { 5 \pi } { 12 }$

Accepted Answer

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

Question 12

Complete the identity.
-$$\frac { \csc \theta \cot \theta } { \sec \theta } = \text { ? }$$

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\sin 22.5 ^ { \circ }$

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Complete the identity.
-$$\frac { 1 } { 2 } \sec \theta \csc \theta = \text { ? }$$

Accepted Answer

A)  $\frac { \tan ( 2 \theta ) } { \sin \theta }$ 
B)  $\sec ( 2 \theta )$ 
C)  $\csc ( 2 \theta )$ 
D)  $\frac { \cot ( 2 \theta ) } { \cos \theta }$ 
A)  $\frac { \tan ( 2 \theta ) } { \sin \theta }$ 
B)  $\sec ( 2 \theta )$ 
C)  $\csc ( 2 \theta )$ 
D)  $\frac { \cot ( 2 \theta ) } { \cos \theta }$

Question 17

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$$\sin 15 ^ { \circ }$$

Accepted Answer

The answer of Use the Half-angle Formulas to find the...

Question 18

Find the exact value of the expression.
-$\tan 105 ^ { \circ }$

Accepted Answer

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

Question 19

Find the exact value of the expression. Do not use a calculator.
-$\tan \left[ \tan ^ { - 1 } ( - 0.4 ) \right]$

Accepted Answer

A)  -0.6 
B)  -0.4 
C)  -2.3652 
D)  -2.5 
A)  -0.6 
B)  -0.4 
C)  -2.3652 
D)  -2.5

Question 20

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\tan \theta = \frac { 8 } { 15 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { 161 } { 289 }$ 
B)  $- \frac { 161 } { 289 }$ 
C)  $\frac { 240 } { 289 }$ 
D)  $- \frac { 240 } { 289 }$ 
A)  $\frac { 161 } { 289 }$ 
B)  $- \frac { 161 } { 289 }$ 
C)  $\frac { 240 } { 289 }$ 
D)  $- \frac { 240 } { 289 }$

Complete the identity. - $\ln | \csc x | = ?$

Solve the equation on the interval $0 \leq \theta < 2 \pi .$ - $\csc ^ { 5 } \theta - 4 \csc \theta = 0$

Simplify the expression. - $\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta$

Use a calculator to find the value of the expression rounded to two decimal places. - $\tan ^ { - 1 } ( 0.2 )$

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

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. - $- 11 + 24 \sin x = 16 \cos ^ { 2 } x$

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\cos \theta = \frac { 24 } { 25 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\sin ( 2 \theta )$ .

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

Find the exact value of the expression. - $\tan \frac { 5 \pi } { 12 }$

Complete the identity. - $\frac { \csc \theta \cot \theta } { \sec \theta } = \text { ? }$

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

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 22.5 ^ { \circ }$

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

Complete the identity. - $\frac { 1 } { 2 } \sec \theta \csc \theta = \text { ? }$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 15 ^ { \circ }$

Find the exact value of the expression. - $\tan 105 ^ { \circ }$

Find the exact value of the expression. Do not use a calculator. - $\tan \left[ \tan ^ { - 1 } ( - 0.4 ) \right]$

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\tan \theta = \frac { 8 } { 15 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

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

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Review

Filters

Exam 6: Analytic Trigonometry

Complete the identity. - ln⁡∣csc⁡x∣=?\ln | \csc x | = ?ln∣cscx∣=?

Solve the equation on the interval 0≤θ<2π.0 \leq \theta < 2 \pi .0≤θ<2π. - csc⁡5θ−4csc⁡θ=0\csc ^ { 5 } \theta - 4 \csc \theta = 0csc5θ−4cscθ=0

Simplify the expression. - cos⁡θ1+sin⁡θ+tan⁡θ\frac { \cos \theta } { 1 + \sin \theta } + \tan \theta1+sinθcosθ​+tanθ

Use a calculator to find the value of the expression rounded to two decimal places. - tan⁡−1(0.2)\tan ^ { - 1 } ( 0.2 )tan−1(0.2)

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

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. - −11+24sin⁡x=16cos⁡2x- 11 + 24 \sin x = 16 \cos ^ { 2 } x−11+24sinx=16cos2x

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

Find the exact value of the expression. - tan⁡5π12\tan \frac { 5 \pi } { 12 }tan125π​

Complete the identity. - csc⁡θcot⁡θsec⁡θ= ? \frac { \csc \theta \cot \theta } { \sec \theta } = \text { ? }secθcscθcotθ​= ?

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

Use the Half-angle Formulas to find the exact value of the trigonometric function. - sin⁡22.5∘\sin 22.5 ^ { \circ }sin22.5∘

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

Complete the identity. - 12sec⁡θcsc⁡θ= ? \frac { 1 } { 2 } \sec \theta \csc \theta = \text { ? }21​secθcscθ= ?

Use the Half-angle Formulas to find the exact value of the trigonometric function. - sin⁡15∘\sin 15 ^ { \circ }sin15∘

Find the exact value of the expression. - tan⁡105∘\tan 105 ^ { \circ }tan105∘

Find the exact value of the expression. Do not use a calculator. - tan⁡[tan⁡−1(−0.4)]\tan \left[ \tan ^ { - 1 } ( - 0.4 ) \right]tan[tan−1(−0.4)]