Question 1

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

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 2

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

Accepted Answer

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

Question 3

Choose the one alternative that best completes the statement or answers the question.
-$$\tan ^ { 2 } x + 5 \tan x + 3 = 0$$

Accepted Answer

A)  $103.1 ^ { \circ } , 145.1 ^ { \circ } , 283.1 ^ { \circ } , 325.1 ^ { \circ }$ 
B)  $70.5 ^ { \circ } , 109.5 ^ { \circ } , 180.0 ^ { \circ }$ 
C)  $51.8 ^ { \circ } , 128.2 ^ { \circ }$ 
D)  $49.8 ^ { \circ } , 130.2 ^ { \circ } , 229.8 ^ { \circ } , 310.2 ^ { \circ }$ 
A)  $103.1 ^ { \circ } , 145.1 ^ { \circ } , 283.1 ^ { \circ } , 325.1 ^ { \circ }$ 
B)  $70.5 ^ { \circ } , 109.5 ^ { \circ } , 180.0 ^ { \circ }$ 
C)  $51.8 ^ { \circ } , 128.2 ^ { \circ }$ 
D)  $49.8 ^ { \circ } , 130.2 ^ { \circ } , 229.8 ^ { \circ } , 310.2 ^ { \circ }$

Question 4

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\sin ( 4 \theta ) \sin ( 7 \theta ) \cos ( 4 \theta ) \cos ( 7 \theta ) = ?$

Accepted Answer

A)  $\cos ^ { 2 } ( 56 \theta )$ 
B)  $\frac { \sin ^ { 2 } ( 56 \theta ) } { 4 }$ 
C)  $\frac { \cos ^ { 2 } ( 11 \theta ) + \cos ^ { 2 } ( 3 \theta ) } { 4 }$ 
D)  $\frac { \cos ^ { 2 } ( 3 \theta ) - \cos ^ { 2 } ( 11 \theta ) } { 4 }$ 
A)  $\cos ^ { 2 } ( 56 \theta )$ 
B)  $\frac { \sin ^ { 2 } ( 56 \theta ) } { 4 }$ 
C)  $\frac { \cos ^ { 2 } ( 11 \theta ) + \cos ^ { 2 } ( 3 \theta ) } { 4 }$ 
D)  $\frac { \cos ^ { 2 } ( 3 \theta ) - \cos ^ { 2 } ( 11 \theta ) } { 4 }$

Question 5

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 6

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\cos ( x - y ) - \cos ( x + y ) = 2 \sin x \sin y$$

Accepted Answer

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

Question 7

Simplify the expression.
-$2 \cos ^ { 2 } 4 x - 1$

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
B)  $\frac { 2 \sqrt { 3 } } { 5 }$ 
C)  $\frac { 2 \sqrt { 3 } + 2 \sqrt { 10 } } { 9 }$ 
D)  $\frac { 7 \sqrt { 5 } + 8 \sqrt { 2 } } { 27 }$ 
A)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
B)  $\frac { 2 \sqrt { 3 } } { 5 }$ 
C)  $\frac { 2 \sqrt { 3 } + 2 \sqrt { 10 } } { 9 }$ 
D)  $\frac { 7 \sqrt { 5 } + 8 \sqrt { 2 } } { 27 }$

Question 9

Find the exact value of the expression.
-$\cos \left( \frac { 5 \pi } { 18 } \right) \cos \left( \frac { 2 \pi } { 9 } \right) - \sin \left( \frac { 5 \pi } { 18 } \right) \sin \left( \frac { 2 \pi } { 9 } \right)$

Accepted Answer

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

Question 10

Find the inverse function f-1 of the function f.
-$$f ( x ) = 7 \cos x + 4$$

Accepted Answer

A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 7 \cos ^ { - 1 } x + 4$ 
B)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } \left( \frac { x + 4 } { 7 } \right)$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sin \left( \frac { \mathrm { x } - 4 } { 7 } \right)$ 
D)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } \left( \frac { x - 4 } { 7 } \right)$ 
A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 7 \cos ^ { - 1 } x + 4$ 
B)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } \left( \frac { x + 4 } { 7 } \right)$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \sin \left( \frac { \mathrm { x } - 4 } { 7 } \right)$ 
D)  $f ^ { - 1 } ( x ) = \cos ^ { - 1 } \left( \frac { x - 4 } { 7 } \right)$

Question 11

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

Accepted Answer

A)  $\frac { \tan \alpha \tan \beta - 1 } { \tan \alpha \tan \beta + 1 }$ 
B)  $\frac { \tan \alpha - \tan \beta } { \tan \alpha + \tan \beta }$ 
C)  $- 1$ 
D)  $\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta }$ 
A)  $\frac { \tan \alpha \tan \beta - 1 } { \tan \alpha \tan \beta + 1 }$ 
B)  $\frac { \tan \alpha - \tan \beta } { \tan \alpha + \tan \beta }$ 
C)  $- 1$ 
D)  $\frac { \sin \alpha - \sin \beta } { \sin \alpha + \sin \beta }$

Question 12

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$\cos \left[ \tan ^ { - 1 } \left( \frac { \sqrt { 3 } } { 3 } \right) \right]$

Accepted Answer

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

Question 13

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

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 14

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

Accepted Answer

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

Question 15

Simplify the trigonometric expression by following the indicated direction.
-Multiply and simplify: $$\frac { ( \tan \theta + 1 ) ( \tan \theta + 1 ) - \sec ^ { 2 } \theta } { \tan \theta }$$

Accepted Answer

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

Question 16

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-Find $\cos \frac { \theta } { 2 }$, given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$.

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 17

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

Accepted Answer

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

Question 18

Use a calculator to find the value of the expression rounded to two decimal places.
-$$\cos ^ { - 1 } \left( - \frac { 4 } { 5 } \right)$$

Accepted Answer

A)  $- 0.93$ 
B)  $143.13$ 
C)  $2.50$ 
D)  $- 53.13$ 
A)  $- 0.93$ 
B)  $143.13$ 
C)  $2.50$ 
D)  $- 53.13$

Question 19

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)$

Accepted Answer

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

Question 20

Choose the one alternative that best completes the statement or answers the question.
Express the product as a sum containing only sines or cosines.
-$\cos \frac { 11 \theta } { 2 } \cos \frac { \theta } { 2 }$

Accepted Answer

A)  $\frac { 1 } { 4 } \cos ^ { 2 } ( 5 \theta )$ 
B)  $\frac { 1 } { 4 } [ \cos ( 6 \theta ) - \sin ( 4 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \cos ( 2 \theta ) + \cos ( 3 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 3 \theta ) - \sin ( 2 \theta ) ]$ 
A)  $\frac { 1 } { 4 } \cos ^ { 2 } ( 5 \theta )$ 
B)  $\frac { 1 } { 4 } [ \cos ( 6 \theta ) - \sin ( 4 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \cos ( 2 \theta ) + \cos ( 3 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 3 \theta ) - \sin ( 2 \theta ) ]$

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

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

Choose the one alternative that best completes the statement or answers the question. - $\tan ^ { 2 } x + 5 \tan x + 3 = 0$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sin ( 4 \theta ) \sin ( 7 \theta ) \cos ( 4 \theta ) \cos ( 7 \theta ) = ?$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cos ( x - y ) - \cos ( x + y ) = 2 \sin x \sin y$

Simplify the expression. - $2 \cos ^ { 2 } 4 x - 1$

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

Find the exact value of the expression. - $\cos \left( \frac { 5 \pi } { 18 } \right) \cos \left( \frac { 2 \pi } { 9 } \right) - \sin \left( \frac { 5 \pi } { 18 } \right) \sin \left( \frac { 2 \pi } { 9 } \right)$

Find the inverse function f-1 of the function f. - $f ( x ) = 7 \cos x + 4$

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos \left[ \tan ^ { - 1 } \left( \frac { \sqrt { 3 } } { 3 } \right) \right]$

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

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

Simplify the trigonometric expression by following the indicated direction. -Multiply and simplify: $\frac { ( \tan \theta + 1 ) ( \tan \theta + 1 ) - \sec ^ { 2 } \theta } { \tan \theta }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find $\cos \frac { \theta } { 2 }$ , given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$ .

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

Use a calculator to find the value of the expression rounded to two decimal places. - $\cos ^ { - 1 } \left( - \frac { 4 } { 5 } \right)$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)$

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $\cos \frac { 11 \theta } { 2 } \cos \frac { \theta } { 2 }$

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

Use the Half-angle Formulas to find the exact value of the trigonometric function. - sin⁡5π12\sin \frac { 5 \pi } { 12 }sin125π​

Choose the one alternative that best completes the statement or answers the question. - tan⁡2x+5tan⁡x+3=0\tan ^ { 2 } x + 5 \tan x + 3 = 0tan2x+5tanx+3=0

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - sin⁡(4θ)sin⁡(7θ)cos⁡(4θ)cos⁡(7θ)=?\sin ( 4 \theta ) \sin ( 7 \theta ) \cos ( 4 \theta ) \cos ( 7 \theta ) = ?sin(4θ)sin(7θ)cos(4θ)cos(7θ)=?

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

Write the word or phrase that best completes each statement or answers the question. 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

Simplify the expression. - 2cos⁡24x−12 \cos ^ { 2 } 4 x - 12cos24x−1

Find the exact value of the expression. - cos⁡[sin⁡−123+2sin⁡−1(−13)]\cos \left[ \sin ^ { - 1 } \frac { 2 } { 3 } + 2 \sin ^ { - 1 } \left( - \frac { 1 } { 3 } \right) \right]cos[sin−132​+2sin−1(−31​)]

Find the inverse function f-1 of the function f. - f(x)=7cos⁡x+4f ( x ) = 7 \cos x + 4f(x)=7cosx+4

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - cos⁡[tan⁡−1(33)]\cos \left[ \tan ^ { - 1 } \left( \frac { \sqrt { 3 } } { 3 } \right) \right]cos[tan−1(33​​)]

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

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

Simplify the trigonometric expression by following the indicated direction. -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​

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

Use a calculator to find the value of the expression rounded to two decimal places. - cos⁡−1(−45)\cos ^ { - 1 } \left( - \frac { 4 } { 5 } \right)cos−1(−54​)

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - cos⁡(sin⁡−145)\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)cos(sin−154​)

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - cos⁡11θ2cos⁡θ2\cos \frac { 11 \theta } { 2 } \cos \frac { \theta } { 2 }cos211θ​cos2θ​