Question 1

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\tan ( \arcsin 2 )$

Accepted Answer

A)  $0.8944$ 
B)  0 
C)  $- 0.8172$ 
D)  Undefined 
A)  $0.8944$ 
B)  0 
C)  $- 0.8172$ 
D)  Undefined

Question 2

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \tan ^ { - 1 } ( - 0.7002 )$$

Accepted Answer

A)  $- 0.5761905$ 
B)  $- 0.6634921$ 
C)  $2.5317483$ 
D)  $- 0.6108602$ 
A)  $- 0.5761905$ 
B)  $- 0.6634921$ 
C)  $2.5317483$ 
D)  $- 0.6108602$

Question 3

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$$

Accepted Answer

A)  1 
B)  $\sin ^ { 2 } x + 1$ 
C)  $\cot ^ { 2 } x - 1$ 
D)  $\cot ^ { 2 } x + 1$ 
A)  1 
B)  $\sin ^ { 2 } x + 1$ 
C)  $\cot ^ { 2 } x - 1$ 
D)  $\cot ^ { 2 } x + 1$

Question 4

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-The area of a triangle is given by
$$\mathrm { A } = \frac { 1 } { 2 } \mathrm { ab } \sin \mathrm { C } \text {, }$$
where $\mathrm { a }$ and $\mathrm { b }$ are the lengths of two of the sides and $\mathrm { C }$ is the included angle. If $\mathrm { A } = 28$ in. $2 , \mathrm { a } = 9 \mathrm { in }$., $\mathrm { b } = 8 \mathrm { in }$., and $\mathrm { C }$ is an acute angle, what must $\mathrm { C }$ be? Give your answer in degrees to the nearest hundredth.

Accepted Answer

A)  $128.94 ^ { \circ }$ 
B)  $51.06 ^ { \circ }$ 
C)  $25.53 ^ { \circ }$ 
D)  No such triangle exists. 
A)  $128.94 ^ { \circ }$ 
B)  $51.06 ^ { \circ }$ 
C)  $25.53 ^ { \circ }$ 
D)  No such triangle exists.

Question 5

Use a sum or difference identity to find the exact value.
-$\sin 125 ^ { \circ } \cos 65 ^ { \circ } - \cos 125 ^ { \circ } \sin 65 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 25 } { 12 }$ 
A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $\frac { \sqrt { 3 } } { 2 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 25 } { 12 }$

Question 6

Provide an appropriate response.
-Explain what is wrong with the following solution for the equation $\tan 3 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$.
$\tan 3 \theta = \sqrt { 3 }$
$\tan \theta = \frac { \sqrt { 3 } } { 3 }$
$\theta = \frac { \pi } { 6 }$ or $\theta = \frac { 7 \pi } { 6 }$

Accepted Answer

The answer of Provide an appropriate response.
-Explain what is wrong...

Question 7

Verify that the equation is an identity.
-$$\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \beta$$

Accepted Answer

\[\begin{array} { l } 
\sin ( \alpha - \

Question 8

Perform the indicated operations and simplify the result so there are no quotients.
-$$\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }$$

Accepted Answer

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

Question 9

Use the fundamental identities to simplify the expression.
-$$\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$$

Accepted Answer

A)  $\cos ^ { 3 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sin \theta$ 
D)  $\sec ^ { 2 } \theta$ 
A)  $\cos ^ { 3 } \theta$ 
B)  $\tan ^ { 2 } \theta$ 
C)  $\sin \theta$ 
D)  $\sec ^ { 2 } \theta$

Question 10

Use Identities to find the exact value.
-$$\cos \left( - \frac { 7 \pi } { 12 } \right)$$

Accepted Answer

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

Question 11

Verify that the equation is an identity.
-Two voltages, $\mathrm { V } _ { 1 } = 120 \sin ( 220 \pi \mathrm { t } )$ and $\mathrm { V } _ { 2 } = 90 \cos ( 220 \pi \mathrm { t } )$, are added in a circuit to produce the sum $\mathrm { V } = \mathrm { V } _ { 1 } + \mathrm { V } _ { 2 }$. Graph $\mathrm { V }$ in $[ - 0.02,0.02,0.002 ]$ by $[ - 250,250,20 ]$ and use the graph to estimate values for a and $\mathrm { k }$ so that $\mathrm { V } = \mathrm { a } \sin ( 220 \pi \mathrm { t } + \mathrm { k } )$. Round $\mathrm { k }$ to the nearest ten-thousandth, if necessary.

Accepted Answer

A)  $a = 150 ; k = 640.8999$ 
B)  $\mathrm { a } = 15 ; \mathrm { k } = 0.6435$ 
C)  $\mathrm { a } = 150 ; \mathrm { k } = 444.7557$ 
D)  $\mathrm { a } = 150 ; \mathrm { k } = 0.6435$ 
A)  $a = 150 ; k = 640.8999$ 
B)  $\mathrm { a } = 15 ; \mathrm { k } = 0.6435$ 
C)  $\mathrm { a } = 150 ; \mathrm { k } = 444.7557$ 
D)  $\mathrm { a } = 150 ; \mathrm { k } = 0.6435$

Question 12

The statement tan-1(tan x) = x for all real numbers in the interval
-On the given graph of y $y = \csc x$ c x sketch the graph of y $y = \csc ^ { - 1 }$ x as defined in the text. Give
the domain and the range.

Accepted Answer

The answer of The statement tan-1(tan x) = x for...

Question 13

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$$

Accepted Answer

A)  0 
B)  $2 \cot ^ { 2 } \theta$ 
C)  2 
D)  $- 2 \cot ^ { 2 } \theta$ 
A)  0 
B)  $2 \cot ^ { 2 } \theta$ 
C)  2 
D)  $- 2 \cot ^ { 2 } \theta$

Question 14

Find the exact value of the expression using the provided information.
-Find $\cos ( s - t )$ given that $\cos s = - \frac { 3 } { 5 }$, with $s$ in quadrant II, and $\cos t = \frac { 5 } { 13 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $\frac { 63 } { 65 }$ 
B)  $- \frac { 33 } { 65 }$ 
C)  $\frac { 33 } { 65 }$ 
D)  $- \frac { 63 } { 65 }$ 
A)  $\frac { 63 } { 65 }$ 
B)  $- \frac { 33 } { 65 }$ 
C)  $\frac { 33 } { 65 }$ 
D)  $- \frac { 63 } { 65 }$

Question 15

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\sin \left( \cos ^ { - 1 } 0.8324 \right)$

Accepted Answer

A)  $0.5542$ 
B)  $0.1389$ 
C)  $0.0175$ 
D)  $0.6324$ 
A)  $0.5542$ 
B)  $0.1389$ 
C)  $0.0175$ 
D)  $0.6324$

Question 16

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$
-$\tan x + \sec x = 1$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { 5 \pi } { 4 } \right\}$ 
A)  $\varnothing$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { 5 \pi } { 4 } \right\}$

Question 17

Use the cofunction identities to find an angle $$\theta$$ that makes the statement true.
-$\sin \theta = \cos \left( 4 \theta + 60 ^ { \circ } \right)$

Accepted Answer

A)  $\theta = 16 ^ { \circ }$ 
B)  $\theta = 75 ^ { \circ }$ 
C)  $\theta = 10 ^ { \circ }$ 
D)  $\theta = 6 ^ { \circ }$ 
A)  $\theta = 16 ^ { \circ }$ 
B)  $\theta = 75 ^ { \circ }$ 
C)  $\theta = 10 ^ { \circ }$ 
D)  $\theta = 6 ^ { \circ }$

Question 18

Find the exact value of the expression using the provided information.
-Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$, with $s$ in quadrant $I$, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.

Accepted Answer

A)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 12 }$ 
B)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 12 }$ 
C)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 6 }$ 
D)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 6 }$ 
A)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 12 }$ 
B)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 12 }$ 
C)  $\frac { \sqrt { 15 } - 2 \sqrt { 2 } } { 6 }$ 
D)  $- \frac { \sqrt { 15 } + 2 \sqrt { 2 } } { 6 }$

Question 19

Give the exact value of the expression.
-$$\arccos \left( \cos \frac { \pi } { 2 } \right)$$

Accepted Answer

A)  $\frac { \pi } { 2 }$ 
B)  $\pi$ 
C)  $\frac { \pi } { 3 }$ 
D)  0 
A)  $\frac { \pi } { 2 }$ 
B)  $\pi$ 
C)  $\frac { \pi } { 3 }$ 
D)  0

Question 20

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$
-$$\csc ^ { 5 } x - 4 \csc x = 0$$

Accepted Answer

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

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\tan ( \arcsin 2 )$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \tan ^ { - 1 } ( - 0.7002 )$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$

Use a sum or difference identity to find the exact value. - $\sin 125 ^ { \circ } \cos 65 ^ { \circ } - \cos 125 ^ { \circ } \sin 65 ^ { \circ }$

Verify that the equation is an identity. - $\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \beta$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }$

Use the fundamental identities to simplify the expression. - $\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$

Use Identities to find the exact value. - $\cos \left( - \frac { 7 \pi } { 12 } \right)$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$

Find the exact value of the expression using the provided information. -Find $\cos ( s - t )$ given that $\cos s = - \frac { 3 } { 5 }$ , with $s$ in quadrant II, and $\cos t = \frac { 5 } { 13 }$ , with $t$ in quadrant IV.

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\sin \left( \cos ^ { - 1 } 0.8324 \right)$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\tan x + \sec x = 1$

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\sin \theta = \cos \left( 4 \theta + 60 ^ { \circ } \right)$

Find the exact value of the expression using the provided information. -Find $\cos ( s + t )$ given that $\cos s = \frac { 1 } { 3 }$ , with $s$ in quadrant $I$ , and $\sin t = \frac { 1 } { 4 }$ , with $t$ in quadrant II.

Give the exact value of the expression. - $\arccos \left( \cos \frac { \pi } { 2 } \right)$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\csc ^ { 5 } x - 4 \csc x = 0$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

Exam 8: Trigonometric Identities and Equations

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - tan⁡(arcsin⁡2)\tan ( \arcsin 2 )tan(arcsin2)

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=tan⁡−1(−0.7002)y = \tan ^ { - 1 } ( - 0.7002 )y=tan−1(−0.7002)

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - sin⁡2x+sin⁡2xcot⁡2x\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } xsin2x+sin2xcot2x

Use a sum or difference identity to find the exact value. - sin⁡125∘cos⁡65∘−cos⁡125∘sin⁡65∘\sin 125 ^ { \circ } \cos 65 ^ { \circ } - \cos 125 ^ { \circ } \sin 65 ^ { \circ }sin125∘cos65∘−cos125∘sin65∘

Verify that the equation is an identity. - sin⁡(α−β)cos⁡(α+β)=sin⁡αcos⁡α−sin⁡βcos⁡β\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \betasin(α−β)cos(α+β)=sinαcosα−sinβcosβ

Perform the indicated operations and simplify the result so there are no quotients. - 1cos⁡2θ−1cot⁡2θ\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta }cos2θ1​−cot2θ1​

Use the fundamental identities to simplify the expression. - sin⁡2θ+tan⁡2θ+cos⁡2θ\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \thetasin2θ+tan2θ+cos2θ

Use Identities to find the exact value. - cos⁡(−7π12)\cos \left( - \frac { 7 \pi } { 12 } \right)cos(−127π​)

The statement tan-1(tan x) = x for all real numbers in the interval -On the given graph of y y=csc⁡xy = \csc xy=cscx c x sketch the graph of y y=csc⁡−1y = \csc ^ { - 1 }y=csc−1 x as defined in the text. Give the domain and the range.

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - (1+cot⁡θ)(1−cot⁡θ)−csc⁡2θ( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta(1+cotθ)(1−cotθ)−csc2θ

Find the exact value of the expression using the provided information. -Find cos⁡(s−t)\cos ( s - t )cos(s−t) given that cos⁡s=−35\cos s = - \frac { 3 } { 5 }coss=−53​ , with sss in quadrant II, and cos⁡t=513\cos t = \frac { 5 } { 13 }cost=135​ , with ttt in quadrant IV.

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - sin⁡(cos⁡−10.8324)\sin \left( \cos ^ { - 1 } 0.8324 \right)sin(cos−10.8324)

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - tan⁡x+sec⁡x=1\tan x + \sec x = 1tanx+secx=1

Use the cofunction identities to find an angle θ\thetaθ that makes the statement true. - sin⁡θ=cos⁡(4θ+60∘)\sin \theta = \cos \left( 4 \theta + 60 ^ { \circ } \right)sinθ=cos(4θ+60∘)

Find the exact value of the expression using the provided information. -Find cos⁡(s+t)\cos ( s + t )cos(s+t) given that cos⁡s=13\cos s = \frac { 1 } { 3 }coss=31​ , with sss in quadrant III , and sin⁡t=14\sin t = \frac { 1 } { 4 }sint=41​ , with ttt in quadrant II.

Give the exact value of the expression. - arccos⁡(cos⁡π2)\arccos \left( \cos \frac { \pi } { 2 } \right)arccos(cos2π​)

Solve the equation for exact solutions over the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - csc⁡5x−4csc⁡x=0\csc ^ { 5 } x - 4 \csc x = 0csc5x−4cscx=0