Question 1

Choose the one alternative that best completes the statement or answers the question.
Complete the identity.
-$\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =$ ?

Accepted Answer

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

Question 2

Write the word or phrase that best completes each statement or answers the question.
-$$\sin 15 ^ { \circ }$$

Accepted Answer

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

Question 3

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-Find $\cos \frac { \theta } { 2 }$, given that $\sec \theta = 4$ and $\theta$ terminates in $0 < \theta < \pi / 2$.

Accepted Answer

A)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 10 } } { 4 }$ 
A)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
B)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 10 } } { 4 }$

Question 4

Find the inverse function f-1 of the function f.
-$$f ( x ) = 3 \sin x - 2$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = 3 \sin ^ { - 1 } x - 2$ 
B)  $f ^ { - 1 } ( x ) = \sin ^ { - 1 } \left( \frac { x + 2 } { 3 } \right)$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \cos \left( \frac { \mathrm { x } + 2 } { 3 } \right)$ 
D)  $f ^ { - 1 } ( x ) = \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right)$ 
A)  $f ^ { - 1 } ( x ) = 3 \sin ^ { - 1 } x - 2$ 
B)  $f ^ { - 1 } ( x ) = \sin ^ { - 1 } \left( \frac { x + 2 } { 3 } \right)$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \cos \left( \frac { \mathrm { x } + 2 } { 3 } \right)$ 
D)  $f ^ { - 1 } ( x ) = \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right)$

Question 5

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$$

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Write the word or phrase that best completes each statement or answers the question.
Establish the identity.
-$$\cos x \csc x \tan x = 1$$

Accepted Answer

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

Question 8

Express the sum or difference as a product of sines and/or cosines.
-$\sin ( 7 \theta ) + \sin ( 3 \theta )$

Accepted Answer

A)  $2 \sin ( 5 \theta ) \sin ( 2 \theta )$ 
B)  $2 \cos ( 5 \theta ) \sin ( 2 \theta )$ 
C)  $2 \sin ( 10 \theta )$ 
D)  $2 \sin ( 5 \theta ) \cos ( 2 \theta )$ 
A)  $2 \sin ( 5 \theta ) \sin ( 2 \theta )$ 
B)  $2 \cos ( 5 \theta ) \sin ( 2 \theta )$ 
C)  $2 \sin ( 10 \theta )$ 
D)  $2 \sin ( 5 \theta ) \cos ( 2 \theta )$

Question 9

Write the word or phrase that best completes each statement or answers the question.
-The two equal sides of an isosceles triangle measure three feet. Let the angle between the sides measure θ. Find
the area A of the triangle as a function of . The answer may include more than one trigonometric function. $$\frac { \theta } { 2 } .$$

Accepted Answer

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

Question 10

Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
-A product of two oscillations with different frequencies such as
$$f ( t ) = \sin ( 10 t ) \sin ( t )$$
is important in acoustics. The result is an oscillation with "oscillating amplitude."
(i) Writethe product $\mathrm { f } ( \mathrm { t } )$ of the two oscillations as a sum of two cosines and call it $\mathrm { g } ( \mathrm { t } )$.
(ii) Usinga graphing utility, graph the function $g ( t )$ on the interval $0 \leq t \leq 2 \pi$.
(iii) On the same system as your graph, graph $y = \sin t$ and $y = - \sin t$.
(iv) Thelast two functions constitute an "envelope" for the function $g ( t )$. For certain values of t, the two cosine functions in $g ( t )$ cancel each other out and near-silence occurs; between these values, the two functions combine in varying degrees. The phenomenon is known (and heard) as "beats." For what values of t do the functions cancel each other?

Accepted Answer

(i) @#LAT-DLM&
(ii),

Question 11

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

Accepted Answer

A)  $63.44$ 
B)  $1.11$ 
C)  $26.57$ 
D)  $0.46$ 
A)  $63.44$ 
B)  $1.11$ 
C)  $26.57$ 
D)  $0.46$

Question 12

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$\sec ^ { - 1 } ( \sqrt { 2 } )$

Accepted Answer

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

Question 13

Find the domain of the function f and of its inverse function f-1.
-$$f ( x ) = \cos ( x - 3 ) + 4$$

Accepted Answer

A)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5 , - 3 ]$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ 3,5 ]$ 
D)  Domain of $\mathrm { f } : [ - 3,3 ]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
A)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ - 5 , - 3 ]$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
C)  Domain of $f : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ 3,5 ]$ 
D)  Domain of $\mathrm { f } : [ - 3,3 ]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

Question 14

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

Accepted Answer

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

Question 15

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$

Accepted Answer

A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 5 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 3 }$ 
A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 4 \pi } { 5 }$ 
C)  $\frac { \pi } { 6 }$ 
D)  $\frac { \pi } { 3 }$

Question 16

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

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 17

Write the word or phrase that best completes each statement or answers the question.
Use a calculator to solve the equation on the interval 0 . Round the answer to one decimal place if necessary. $$0 \leq x < 2 \pi$$
-$$2 x ^ { 2 } - 3 x \sin x = 2$$

Accepted Answer

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

Question 18

Solve the equation. Give a general formula for all the solutions.
-$2 \cos \theta + 1 = 0$

Accepted Answer

A)  $\theta = \frac { \pi } { 2 } + 2 \mathrm { k } \pi , \theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { 2 \pi } { 3 } + 2 \mathrm { k } \pi , \theta = \frac { 4 \pi } { 3 } + 2 \mathrm { k } \pi$ 
C)  $\theta = \frac { 2 \pi } { 3 } + \mathrm { k } \pi , \theta = \frac { 4 \pi } { 3 } + \mathrm { k } \pi$ 
D)  $\theta = \frac { 3 \pi } { 2 } + \mathrm { k } \pi$ 
A)  $\theta = \frac { \pi } { 2 } + 2 \mathrm { k } \pi , \theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \frac { 2 \pi } { 3 } + 2 \mathrm { k } \pi , \theta = \frac { 4 \pi } { 3 } + 2 \mathrm { k } \pi$ 
C)  $\theta = \frac { 2 \pi } { 3 } + \mathrm { k } \pi , \theta = \frac { 4 \pi } { 3 } + \mathrm { k } \pi$ 
D)  $\theta = \frac { 3 \pi } { 2 } + \mathrm { k } \pi$

Question 19

Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression.
-$\sec ^ { - 1 } ( - 2 )$

Accepted Answer

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

Question 20

Find the domain of the function f and of its inverse function f-1.
-$$f ( x ) = 3 \sin x - 5$$

Accepted Answer

A)  Domain of $\mathrm { f } : [ 2,8 ]$
Domain of $\mathrm { f } ^ { - 1 } : [ - 8 , - 2 ]$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 8 , - 2 ]$ 
C)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ 2,8 ]$ 
D)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
A)  Domain of $\mathrm { f } : [ 2,8 ]$
Domain of $\mathrm { f } ^ { - 1 } : [ - 8 , - 2 ]$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 8 , - 2 ]$ 
C)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $f ^ { - 1 } : [ 2,8 ]$ 
D)  Domain of $\mathrm { f } : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =$ ?

Write the word or phrase that best completes each statement or answers the question. - $\sin 15 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find $\cos \frac { \theta } { 2 }$ , given that $\sec \theta = 4$ and $\theta$ terminates in $0 < \theta < \pi / 2$ .

Find the inverse function f-1 of the function f. - $f ( x ) = 3 \sin x - 2$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cos x \csc x \tan x = 1$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 7 \theta ) + \sin ( 3 \theta )$

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\sec ^ { - 1 } ( \sqrt { 2 } )$

Find the domain of the function f and of its inverse function f-1. - $f ( x ) = \cos ( x - 3 ) + 4$

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$

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

Write the word or phrase that best completes each statement or answers the question. Use a calculator to solve the equation on the interval 0 . Round the answer to one decimal place if necessary. $0 \leq x < 2 \pi$ - $2 x ^ { 2 } - 3 x \sin x = 2$

Solve the equation. Give a general formula for all the solutions. - $2 \cos \theta + 1 = 0$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\sec ^ { - 1 } ( - 2 )$

Find the domain of the function f and of its inverse function f-1. - $f ( x ) = 3 \sin x - 5$

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

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - csc⁡(2θ)−sec⁡(2θ)(tan⁡θ−1)=\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =csc(2θ)−sec(2θ)(tanθ−1)= ?

Write the word or phrase that best completes each statement or answers the question. - sin⁡15∘\sin 15 ^ { \circ }sin15∘

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that sec⁡θ=4\sec \theta = 4secθ=4 and θ\thetaθ terminates in 0<θ<π/20 < \theta < \pi / 20<θ<π/2 .

Find the inverse function f-1 of the function f. - f(x)=3sin⁡x−2f ( x ) = 3 \sin x - 2f(x)=3sinx−2

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - tan⁡2x=sec⁡2x−sin⁡2x−cos⁡2x\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } xtan2x=sec2x−sin2x−cos2x

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - csc⁡θ(sin⁡θ+cos⁡θ)=\csc \theta ( \sin \theta + \cos \theta ) =cscθ(sinθ+cosθ)= ?

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - cos⁡xcsc⁡xtan⁡x=1\cos x \csc x \tan x = 1cosxcscxtanx=1

Express the sum or difference as a product of sines and/or cosines. - sin⁡(7θ)+sin⁡(3θ)\sin ( 7 \theta ) + \sin ( 3 \theta )sin(7θ)+sin(3θ)

Use a calculator to find the value of the expression rounded to two decimal places. - sin⁡−1(55)\sin ^ { - 1 } \left( \frac { \sqrt { 5 } } { 5 } \right)sin−1(55​​)

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - sec⁡−1(2)\sec ^ { - 1 } ( \sqrt { 2 } )sec−1(2​)

Find the domain of the function f and of its inverse function f-1. - f(x)=cos⁡(x−3)+4f ( x ) = \cos ( x - 3 ) + 4f(x)=cos(x−3)+4

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - cos⁡−1(sin⁡7π6)\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)cos−1(sin67π​)

Find the exact value of the composition. - cos⁡(sin⁡−145)\cos \left( \sin ^ { - 1 } \frac { 4 } { 5 } \right)cos(sin−154​)

Write the word or phrase that best completes each statement or answers the question. Use a calculator to solve the equation on the interval 0 . Round the answer to one decimal place if necessary. 0≤x<2π0 \leq x < 2 \pi0≤x<2π - 2x2−3xsin⁡x=22 x ^ { 2 } - 3 x \sin x = 22x2−3xsinx=2

Solve the equation. Give a general formula for all the solutions. - 2cos⁡θ+1=02 \cos \theta + 1 = 02cosθ+1=0

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - sec⁡−1(−2)\sec ^ { - 1 } ( - 2 )sec−1(−2)

Find the domain of the function f and of its inverse function f-1. - f(x)=3sin⁡x−5f ( x ) = 3 \sin x - 5f(x)=3sinx−5

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

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =$ ?

Write the word or phrase that best completes each statement or answers the question. - $\sin 15 ^ { \circ }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find $\cos \frac { \theta } { 2 }$ , given that $\sec \theta = 4$ and $\theta$ terminates in $0 < \theta < \pi / 2$ .

Find the inverse function f-1 of the function f. - $f ( x ) = 3 \sin x - 2$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$

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

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\cos x \csc x \tan x = 1$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 7 \theta ) + \sin ( 3 \theta )$

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\sec ^ { - 1 } ( \sqrt { 2 } )$

Find the domain of the function f and of its inverse function f-1. - $f ( x ) = \cos ( x - 3 ) + 4$

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

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\cos ^ { - 1 } \left( \sin \frac { 7 \pi } { 6 } \right)$

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

Write the word or phrase that best completes each statement or answers the question. Use a calculator to solve the equation on the interval 0 . Round the answer to one decimal place if necessary. $0 \leq x < 2 \pi$ - $2 x ^ { 2 } - 3 x \sin x = 2$

Solve the equation. Give a general formula for all the solutions. - $2 \cos \theta + 1 = 0$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\sec ^ { - 1 } ( - 2 )$

Find the domain of the function f and of its inverse function f-1. - $f ( x ) = 3 \sin x - 5$