Question 1

Establish the identity.
-$$\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$$

Accepted Answer

The answer of Establish the identity.
-\[\cot ^ { 2 }...

Question 2

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ .
-$$f ( x ) = - 5 \cos ( 9 x )$$

Accepted Answer

A)  Domain of $f : \left[ - \frac { 1 } { 9 } , \frac { 1 } { 9 } \right]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 5,5 ]$ 
C)  Domain of f: $( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 9,9 ]$ 
D)  Domain of f: $( \infty , \infty )$

Domain of $\mathrm { f } ^ { - 1 } : [ 4,14 ]$ 
A)  Domain of $f : \left[ - \frac { 1 } { 9 } , \frac { 1 } { 9 } \right]$
Domain of $\mathrm { f } ^ { - 1 } : ( \infty , \infty )$ 
B)  Domain of $f : ( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 5,5 ]$ 
C)  Domain of f: $( \infty , \infty )$
Domain of $\mathrm { f } ^ { - 1 } : [ - 9,9 ]$ 
D)  Domain of f: $( \infty , \infty )$

Domain of $\mathrm { f } ^ { - 1 } : [ 4,14 ]$

Question 3

Establish the identity.
-$$\csc \left( \frac { \pi } { 2 } + u \right) = \sec u$$

Accepted Answer

The answer of Establish the identity.
-\[\csc \left( \frac { \pi...

Question 4

Establish the identity.
-$$\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } x$$

Accepted Answer

The answer of Establish the identity.
-\[\sec ^ { 4 }...

Question 5

Complete the identity.
-$\sec ^ { 4 } \theta + \sec ^ { 2 } \theta \tan ^ { 2 } \theta - 2 \tan ^ { 4 } \theta =$ ?

Accepted Answer

A)  $4 \sec ^ { 2 } \theta$ 
B)  $\sec ^ { 2 } \theta + 2$ 
C)  $\tan ^ { 2 } \theta - 1$ 
D)  $3 \sec ^ { 2 } \theta - 2$ 
A)  $4 \sec ^ { 2 } \theta$ 
B)  $\sec ^ { 2 } \theta + 2$ 
C)  $\tan ^ { 2 } \theta - 1$ 
D)  $3 \sec ^ { 2 } \theta - 2$

Question 6

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

Accepted Answer

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

Question 7

Find the exact value of the expression.
-$\sin 25 ^ { \circ } \cos 95 ^ { \circ } + \cos 25 ^ { \circ } \sin 95 ^ { \circ }$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $- \frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 8

Complete the identity.
-$\sin ( \pi - \theta ) = ?$

Accepted Answer

A)  $\sin \theta$ 
B)  $\cos \theta$ 
C)  $- \cos \theta$ 
D)  $- \sin \theta$ 
A)  $\sin \theta$ 
B)  $\cos \theta$ 
C)  $- \cos \theta$ 
D)  $- \sin \theta$

Question 9

Use a calculator to find the value of the expression in radian measure rounded to two decimal places.
-$$\csc ^ { - 1 } \left( - \frac { 7 } { 4 } \right)$$

Accepted Answer

A)  $2.18$ 
B)  $124.85$ 
C)  $- 0.61$ 
D)  $- 34.85$ 
A)  $2.18$ 
B)  $124.85$ 
C)  $- 0.61$ 
D)  $- 34.85$

Question 10

Find the exact value of the expression.
-$$\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }$$

Accepted Answer

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

Question 11

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.
-$2 x ^ { 2 } - 3 x \sin x = 2$

Accepted Answer

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

Question 12

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

Accepted Answer

A)  $\frac { 527 } { 625 }$

В) $- \frac { 527 } { 625 }$ 
C)  $- \frac { 529 } { 625 }$ 
D)  $\frac { 336 } { 625 }$ 
A)  $\frac { 527 } { 625 }$

В) $- \frac { 527 } { 625 }$ 
C)  $- \frac { 529 } { 625 }$ 
D)  $\frac { 336 } { 625 }$

Question 13

Find the exact value of the expression.884:890
-$\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]$

Accepted Answer

A)  $\frac { 119 } { 169 }$ 
B)  $\frac { 10 } { 13 }$ 
C)  $- \frac { 12 } { 13 }$ 
D)  $\frac { 2 \sqrt { 5 } + 10 } { 13 }$ 
A)  $\frac { 119 } { 169 }$ 
B)  $\frac { 10 } { 13 }$ 
C)  $- \frac { 12 } { 13 }$ 
D)  $\frac { 2 \sqrt { 5 } + 10 } { 13 }$

Question 14

Solve the problem.
-A mass hangs from a spring which oscillates up and down. The position P (in feet) of the mass at time t (in seconds) is
given by $$P = 4 \cos ( 4 t )$$
For what values of $t , 0 \leq t < \pi$, will the position be $2 \sqrt { 2 }$ feet? Find the exact values. Do not use a calculator.

Accepted Answer

The answer of Solve the problem.
-A mass hangs from a...

Question 15

Solve the problem.
-Find $\sin \frac { \theta } { 2 }$, given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$.

Accepted Answer

A)  $- \frac { \sqrt { 5 } } { 5 }$

В) $- \frac { \sqrt { 30 } } { 10 }$ 
C)  $\frac { \sqrt { 10 } } { 10 }$ 
D)  $\frac { \sqrt { 5 } } { 5 }$ 
A)  $- \frac { \sqrt { 5 } } { 5 }$

В) $- \frac { \sqrt { 30 } } { 10 }$ 
C)  $\frac { \sqrt { 10 } } { 10 }$ 
D)  $\frac { \sqrt { 5 } } { 5 }$

Question 16

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places.
-$\sin \theta = - 0.41$

Accepted Answer

A)  3.56, 5.86 
B)  0.42, 3.56 
C)  0.42, 1.99 
D)  0.42, 5.86 
A)  3.56, 5.86 
B)  0.42, 3.56 
C)  0.42, 1.99 
D)  0.42, 5.86

Question 17

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

Accepted Answer

A)  $\theta = \frac { \pi } { 2 } + 2 k \pi , \theta = \frac { 3 \pi } { 2 } + 2 k \pi$ 
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)  $\theta = \frac { \pi } { 2 } + 2 k \pi , \theta = \frac { 3 \pi } { 2 } + 2 k \pi$ 
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 18

Establish the identity.
-$$\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$$

Accepted Answer

The answer of Establish the identity.
-\[\cot ( x + y...

Question 19

Express the sum or difference as a product of sines and/or cosines.
-$\sin ( 4 \theta ) - \sin ( 2 \theta )$

Accepted Answer

A)  $\sin ( 3 \theta ) \cos \theta$ 
B)  $\sin \theta \cos ( 3 \theta )$ 
C)  $2 \sin ( 3 \theta ) \cos \theta$ 
D)  $2 \sin \theta \cos ( 3 \theta )$ 
A)  $\sin ( 3 \theta ) \cos \theta$ 
B)  $\sin \theta \cos ( 3 \theta )$ 
C)  $2 \sin ( 3 \theta ) \cos \theta$ 
D)  $2 \sin \theta \cos ( 3 \theta )$

Question 20

Solve the equation on the interval $0 \leq \theta < 2 \pi .$
-$\cos ^ { 2 } \theta + 2 \cos \theta + 1 = 0$

Accepted Answer

A)  $2 \pi$ 
B)  $\pi$ 
C)  $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$ 
A)  $2 \pi$ 
B)  $\pi$ 
C)  $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
D)  $\frac { \pi } { 2 } , \frac { 3 \pi } { 2 }$

Establish the identity. - $\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )$

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ . - $f ( x ) = - 5 \cos ( 9 x )$

Establish the identity. - $\csc \left( \frac { \pi } { 2 } + u \right) = \sec u$

Establish the identity. - $\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } x$

Complete the identity. - $\sec ^ { 4 } \theta + \sec ^ { 2 } \theta \tan ^ { 2 } \theta - 2 \tan ^ { 4 } \theta =$ ?

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

Find the exact value of the expression. - $\sin 25 ^ { \circ } \cos 95 ^ { \circ } + \cos 25 ^ { \circ } \sin 95 ^ { \circ }$

Complete the identity. - $\sin ( \pi - \theta ) = ?$

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - $\csc ^ { - 1 } \left( - \frac { 7 } { 4 } \right)$

Find the exact value of the expression. - $\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }$

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. - $2 x ^ { 2 } - 3 x \sin x = 2$

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

Find the exact value of the expression.884:890 - $\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]$

Solve the problem. -Find $\sin \frac { \theta } { 2 }$ , given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$ .

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places. - $\sin \theta = - 0.41$

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

Establish the identity. - $\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 4 \theta ) - \sin ( 2 \theta )$

Solve the equation on the interval $0 \leq \theta < 2 \pi .$ - $\cos ^ { 2 } \theta + 2 \cos \theta + 1 = 0$

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

Sequences; Induction; the Binomial Theorem

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Exam 6: Analytic Trigonometry

Establish the identity. - cot⁡2x=(csc⁡x−1)(csc⁡x+1)\cot ^ { 2 } x = ( \csc x - 1 ) ( \csc x + 1 )cot2x=(cscx−1)(cscx+1)

Find the domain of the function f and of its inverse function f−1\mathrm { f } ^ { - 1 }f−1 . - f(x)=−5cos⁡(9x)f ( x ) = - 5 \cos ( 9 x )f(x)=−5cos(9x)

Establish the identity. - csc⁡(π2+u)=sec⁡u\csc \left( \frac { \pi } { 2 } + u \right) = \sec ucsc(2π​+u)=secu

Establish the identity. - sec⁡4x−tan⁡4x=sec⁡2x+tan⁡2x\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan ^ { 2 } xsec4x−tan4x=sec2x+tan2x

Complete the identity. - sec⁡4θ+sec⁡2θtan⁡2θ−2tan⁡4θ=\sec ^ { 4 } \theta + \sec ^ { 2 } \theta \tan ^ { 2 } \theta - 2 \tan ^ { 4 } \theta =sec4θ+sec2θtan2θ−2tan4θ= ?

Solve the equation. Give a general formula for all the solutions. - tan⁡θ=−1\tan \theta = - 1tanθ=−1

Find the exact value of the expression. - sin⁡25∘cos⁡95∘+cos⁡25∘sin⁡95∘\sin 25 ^ { \circ } \cos 95 ^ { \circ } + \cos 25 ^ { \circ } \sin 95 ^ { \circ }sin25∘cos95∘+cos25∘sin95∘

Complete the identity. - sin⁡(π−θ)=?\sin ( \pi - \theta ) = ?sin(π−θ)=?

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - csc⁡−1(−74)\csc ^ { - 1 } \left( - \frac { 7 } { 4 } \right)csc−1(−47​)

Find the exact value of the expression. - tan⁡175∘−tan⁡55∘1+tan⁡175∘tan⁡55∘\frac { \tan 175 ^ { \circ } - \tan 55 ^ { \circ } } { 1 + \tan 175 ^ { \circ } \tan 55 ^ { \circ } }1+tan175∘tan55∘tan175∘−tan55∘​

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. - 2x2−3xsin⁡x=22 x ^ { 2 } - 3 x \sin x = 22x2−3xsinx=2

Find the exact value of the expression.884:890 - cos⁡[2sin⁡−1(−513)]\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]cos[2sin−1(−135​)]

Solve the problem. -Find sin⁡θ2\sin \frac { \theta } { 2 }sin2θ​ , given that sin⁡θ=−35\sin \theta = - \frac { 3 } { 5 }sinθ=−53​ and θ\thetaθ terminates in 270∘<θ<360∘270 ^ { \circ } < \theta < 360 ^ { \circ }270∘<θ<360∘ .

Use a calculator to solve the equation on the interval 0 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . Round the answer to two decimal places. - sin⁡θ=−0.41\sin \theta = - 0.41sinθ=−0.41

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

Establish the identity. - cot⁡(x+y)cot⁡(x−y)=1−tan⁡2xtan⁡2ytan⁡2x−tan⁡2y\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }cot(x+y)cot(x−y)=tan2x−tan2y1−tan2xtan2y​

Express the sum or difference as a product of sines and/or cosines. - sin⁡(4θ)−sin⁡(2θ)\sin ( 4 \theta ) - \sin ( 2 \theta )sin(4θ)−sin(2θ)

Solve the equation on the interval 0≤θ<2π.0 \leq \theta < 2 \pi .0≤θ<2π. - cos⁡2θ+2cos⁡θ+1=0\cos ^ { 2 } \theta + 2 \cos \theta + 1 = 0cos2θ+2cosθ+1=0