Question 1

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

Accepted Answer

A)  $v \sqrt { 1 - u ^ { 2 } } - u \sqrt { 1 - v ^ { 2 } }$ 
B)  $u v + \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
C)  $u v - \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
D)  $v \sqrt { 1 - u ^ { 2 } } + u \sqrt { 1 - v ^ { 2 } }$ 
A)  $v \sqrt { 1 - u ^ { 2 } } - u \sqrt { 1 - v ^ { 2 } }$ 
B)  $u v + \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
C)  $u v - \left( \sqrt { 1 - u ^ { 2 } } \right) \left( \sqrt { 1 - v ^ { 2 } } \right)$ 
D)  $v \sqrt { 1 - u ^ { 2 } } + u \sqrt { 1 - v ^ { 2 } }$

Question 2

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\tan ^ { 5 } x = \tan x \left( \sec ^ { 4 } x - 2 \sec ^ { 2 } x + 1 \right)$$

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Complete the identity.
-$\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?$

Accepted Answer

A)  $\frac { \cos ( 6 \theta ) } { \sin ( 12 \theta ) }$ 
B)  $\frac { \sin ( 3 \theta ) } { \cos ( 6 \theta ) }$ 
C)  $\tan ( 3 \theta ) + \tan ( 6 \theta )$ 
D)  $\frac { \sin ( 6 \theta ) } { \cos ( 12 \theta ) }$ 
A)  $\frac { \cos ( 6 \theta ) } { \sin ( 12 \theta ) }$ 
B)  $\frac { \sin ( 3 \theta ) } { \cos ( 6 \theta ) }$ 
C)  $\tan ( 3 \theta ) + \tan ( 6 \theta )$ 
D)  $\frac { \sin ( 6 \theta ) } { \cos ( 12 \theta ) }$

Question 5

Write the word or phrase that best completes each statement or answers the question.
-Before exercising, an athlete measures her air flow and obtains
$$a = 0.65 \sin \left( \frac { 2 \pi } { 5 } t \right)$$ where a is measured in liters per second and t is the time in seconds. If a > 0, the athlete is inhaling; if a < 0, the
athlete is exhaling. The time to complete one complete inhalation/exhalation sequence is a respiratory cycle.
Find the values of t for which the athlete's air flow is zero. Find all values of t for t < 20 seconds.

Accepted Answer

The answer of Write the word or phrase that best...

Question 6

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

Accepted Answer

A)  $\frac { 1 + u v } { \sqrt { u ^ { 2 } + 1 } \cdot \sqrt { v ^ { 2 } + 1 } }$ 
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)  $\frac { \sqrt { \mathrm { u } ^ { 2 } + 1 } \cdot \sqrt { \mathrm { v } ^ { 2 } + 1 } } { 1 - \mathrm { uv } }$ 
A)  $\frac { 1 + u v } { \sqrt { u ^ { 2 } + 1 } \cdot \sqrt { v ^ { 2 } + 1 } }$ 
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)  $\frac { \sqrt { \mathrm { u } ^ { 2 } + 1 } \cdot \sqrt { \mathrm { v } ^ { 2 } + 1 } } { 1 - \mathrm { uv } }$

Question 7

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 8

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$$\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = ?$$

Accepted Answer

A)  0 
B)  $1 - \sin \theta$ 
C)  1 
D)  $- \sec ^ { 2 } \theta$ 
A)  0 
B)  $1 - \sin \theta$ 
C)  1 
D)  $- \sec ^ { 2 } \theta$

Question 9

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-A water wheel rotates through the angle $\theta$, the water level L behind the wheel changes according to the equation
$$\mathrm { L } = 1 - \sin \theta - 2 \cos ^ { 2 } \theta$$
where $\mathrm { L }$ is measured in inches. Determine the values of $\theta$ for which the water level is zero. Find the exact values. Do not use a calculator.

Accepted Answer

The answer of Write the word or phrase that best...

Question 10

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$\cos ^ { - 1 } [ \cos ( - 0.9372 ) ]$

Accepted Answer

A)  $- 0.9372$ 
B)  $0.4686$ 
C)  $- 0.4686$ 
D)  $0.9372$ 
A)  $- 0.9372$ 
B)  $0.4686$ 
C)  $- 0.4686$ 
D)  $0.9372$

Question 11

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?$

Accepted Answer

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

Question 12

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$$\csc ( \alpha - \beta ) = ?$$

Accepted Answer

A)  $\frac { \sec \alpha \sec \beta } { \tan \alpha \tan \beta - 1 }$ 
B)  $\frac { \sec \alpha \sec \beta } { \tan \alpha - \tan \beta }$ 
C)  $\csc \alpha \sec \beta - \sec \alpha \csc \beta$ 
D)  $\csc \alpha - \csc \beta$ 
A)  $\frac { \sec \alpha \sec \beta } { \tan \alpha \tan \beta - 1 }$ 
B)  $\frac { \sec \alpha \sec \beta } { \tan \alpha - \tan \beta }$ 
C)  $\csc \alpha \sec \beta - \sec \alpha \csc \beta$ 
D)  $\csc \alpha - \csc \beta$

Question 13

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\sec ( \alpha + \beta ) \sec ( \alpha - \beta ) = ?$

Accepted Answer

A)  $\frac { \sec \alpha \sec \beta } { \cos ^ { 2 } \alpha - \sin ^ { 2 } \beta }$ 
B)  $\frac { 1 } { \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta }$ 
C)  $\frac { \sec ^ { 2 } \alpha \sec ^ { 2 } \beta } { 1 - \tan ^ { 2 } \alpha \tan ^ { 2 } \beta }$ 
D)  $\sec ^ { 2 } \alpha - \sec ^ { 2 } \beta$ 
A)  $\frac { \sec \alpha \sec \beta } { \cos ^ { 2 } \alpha - \sin ^ { 2 } \beta }$ 
B)  $\frac { 1 } { \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta }$ 
C)  $\frac { \sec ^ { 2 } \alpha \sec ^ { 2 } \beta } { 1 - \tan ^ { 2 } \alpha \tan ^ { 2 } \beta }$ 
D)  $\sec ^ { 2 } \alpha - \sec ^ { 2 } \beta$

Question 14

Simplify the trigonometric expression by following the indicated direction.
-Rewrite over a common denominator: $$\frac { 1 } { 1 - \sin \theta } + \frac { 1 } { 1 + \sin \theta }$$

Accepted Answer

The answer of Simplify the trigonometric expression by following the...

Question 15

Simplify the trigonometric expression by following the indicated direction.
-$$\text { Multiply } \frac { \sin \theta } { 1 - \cos \theta } \text { by } \frac { 1 + \cos \theta } { 1 + \cos \theta }$$

Accepted Answer

The answer of Simplify the trigonometric expression by following the...

Question 16

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

Accepted Answer

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

Question 17

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 18

Choose the one alternative that best completes the statement or answers the question.
-$7 \cot ^ { 2 } x - 5 = 0$

Accepted Answer

A)  $103.2 ^ { \circ } , 145.2 ^ { \circ } , 283.2 ^ { \circ } , 325.2 ^ { \circ }$ 
B)  $51.8 ^ { \circ } , 128.2 ^ { \circ }$ 
C)  $70.5 ^ { \circ } , 109.5 ^ { \circ } , 180.0 ^ { \circ }$ 
D)  $49.8 ^ { \circ } , 130.2 ^ { \circ } , 229.8 ^ { \circ } , 310.2 ^ { \circ }$ 
A)  $103.2 ^ { \circ } , 145.2 ^ { \circ } , 283.2 ^ { \circ } , 325.2 ^ { \circ }$ 
B)  $51.8 ^ { \circ } , 128.2 ^ { \circ }$ 
C)  $70.5 ^ { \circ } , 109.5 ^ { \circ } , 180.0 ^ { \circ }$ 
D)  $49.8 ^ { \circ } , 130.2 ^ { \circ } , 229.8 ^ { \circ } , 310.2 ^ { \circ }$

Question 19

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$$\frac { \csc \theta \cot \theta } { \sec \theta } = ?$$

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 20

Find the exact value of the expression. Do not use a calculator.
-$\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]$

Accepted Answer

A)  $2.7$ 
B)  $- 2.7$ 
C)  $0.3$ 
D)  $- 0.3$ 
A)  $2.7$ 
B)  $- 2.7$ 
C)  $0.3$ 
D)  $- 0.3$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\tan ^ { 5 } x = \tan x \left( \sec ^ { 4 } x - 2 \sec ^ { 2 } x + 1 \right)$

Complete the identity. - $\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?$

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

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

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = ?$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos ^ { - 1 } [ \cos ( - 0.9372 ) ]$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\csc ( \alpha - \beta ) = ?$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sec ( \alpha + \beta ) \sec ( \alpha - \beta ) = ?$

Simplify the trigonometric expression by following the indicated direction. -Rewrite over a common denominator: $\frac { 1 } { 1 - \sin \theta } + \frac { 1 } { 1 + \sin \theta }$

Simplify the trigonometric expression by following the indicated direction. - $\text { Multiply } \frac { \sin \theta } { 1 - \cos \theta } \text { by } \frac { 1 + \cos \theta } { 1 + \cos \theta }$

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

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

Choose the one alternative that best completes the statement or answers the question. - $7 \cot ^ { 2 } x - 5 = 0$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { \csc \theta \cot \theta } { \sec \theta } = ?$

Find the exact value of the expression. Do not use a calculator. - $\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]$

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

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 6: Analytic Trigonometry

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - cot⁡2x+csc⁡2x=2csc⁡2x−1\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1cot2x+csc2x=2csc2x−1

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - tan⁡5x=tan⁡x(sec⁡4x−2sec⁡2x+1)\tan ^ { 5 } x = \tan x \left( \sec ^ { 4 } x - 2 \sec ^ { 2 } x + 1 \right)tan5x=tanx(sec4x−2sec2x+1)

Complete the identity. - sin⁡(2θ)+sin⁡(4θ)cos⁡(5θ)+cos⁡(7θ)=?\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?cos(5θ)+cos(7θ)sin(2θ)+sin(4θ)​=?

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

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)

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - (sin⁡θ+cos⁡θ)21+2sin⁡θcos⁡θ=?\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = ?1+2sinθcosθ(sinθ+cosθ)2​=?

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - cos⁡−1[cos⁡(−0.9372)]\cos ^ { - 1 } [ \cos ( - 0.9372 ) ]cos−1[cos(−0.9372)]

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - tan⁡2θ−3sin⁡θtan⁡θsec⁡θ=?\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?tan2θ−3sinθtanθsecθ=?

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - csc⁡(α−β)=?\csc ( \alpha - \beta ) = ?csc(α−β)=?

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - sec⁡(α+β)sec⁡(α−β)=?\sec ( \alpha + \beta ) \sec ( \alpha - \beta ) = ?sec(α+β)sec(α−β)=?

Simplify the trigonometric expression by following the indicated direction. -Rewrite over a common denominator: 11−sin⁡θ+11+sin⁡θ\frac { 1 } { 1 - \sin \theta } + \frac { 1 } { 1 + \sin \theta }1−sinθ1​+1+sinθ1​

Simplify the trigonometric expression by following the indicated direction. - Multiply sin⁡θ1−cos⁡θ by 1+cos⁡θ1+cos⁡θ\text { Multiply } \frac { \sin \theta } { 1 - \cos \theta } \text { by } \frac { 1 + \cos \theta } { 1 + \cos \theta } Multiply 1−cosθsinθ​ by 1+cosθ1+cosθ​

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

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

Choose the one alternative that best completes the statement or answers the question. - 7cot⁡2x−5=07 \cot ^ { 2 } x - 5 = 07cot2x−5=0

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - csc⁡θcot⁡θsec⁡θ=?\frac { \csc \theta \cot \theta } { \sec \theta } = ?secθcscθcotθ​=?

Find the exact value of the expression. Do not use a calculator. - sin⁡[sin⁡−1(−0.3)]\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]sin[sin−1(−0.3)]

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

Foundations: a Prelude to Functions

Graphing Utilities

Filters

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cot ^ { 2 } x + \csc ^ { 2 } x = 2 \csc ^ { 2 } x - 1$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\tan ^ { 5 } x = \tan x \left( \sec ^ { 4 } x - 2 \sec ^ { 2 } x + 1 \right)$

Complete the identity. - $\frac { \sin ( 2 \theta ) + \sin ( 4 \theta ) } { \cos ( 5 \theta ) + \cos ( 7 \theta ) } = ?$

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

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

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = ?$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos ^ { - 1 } [ \cos ( - 0.9372 ) ]$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\tan ^ { 2 } \theta - 3 \sin \theta \tan \theta \sec \theta = ?$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\csc ( \alpha - \beta ) = ?$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sec ( \alpha + \beta ) \sec ( \alpha - \beta ) = ?$

Simplify the trigonometric expression by following the indicated direction. -Rewrite over a common denominator: $\frac { 1 } { 1 - \sin \theta } + \frac { 1 } { 1 + \sin \theta }$

Simplify the trigonometric expression by following the indicated direction. - $\text { Multiply } \frac { \sin \theta } { 1 - \cos \theta } \text { by } \frac { 1 + \cos \theta } { 1 + \cos \theta }$

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

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

Choose the one alternative that best completes the statement or answers the question. - $7 \cot ^ { 2 } x - 5 = 0$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { \csc \theta \cot \theta } { \sec \theta } = ?$

Find the exact value of the expression. Do not use a calculator. - $\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]$