Question 1

Use a calculator to give the value to the nearest degree.
-$$\theta = \cos ^ { - 1 } ( - 0.9397 )$$

Accepted Answer

A)  $- 20 ^ { \circ }$ 
B)  $162 ^ { \circ }$ 
C)  $160 ^ { \circ }$ 
D)  $157 ^ { \circ }$ 
A)  $- 20 ^ { \circ }$ 
B)  $162 ^ { \circ }$ 
C)  $160 ^ { \circ }$ 
D)  $157 ^ { \circ }$

Question 2

Use the fundamental identities to simplify the expression.
-$$\frac { \tan \theta } { \cot \theta }$$

Accepted Answer

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

Question 3

Use the fundamental identities to simplify the expression.
-$\sin \theta \cos \theta \sec \theta \csc \theta$

Accepted Answer

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

Question 4

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\csc x ( \sin x + \cos x )$

Accepted Answer

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

Question 5

Verify that the equation is an identity.
-The output voltage of a generator is given by $\mathrm { V } = 164 \cos \left( 2 \pi \mathrm { ft } - \frac { \pi } { 6 } \right)$. Express the voltage as the sum of a sine and a cosine function.

Accepted Answer

A)  $V = 82 \sqrt { 3 } \cos 2 \pi f t - 82 \sin 2 \pi f t$ 
B)  $\mathrm { V } = 82 \cos 2 \pi \mathrm { ft } + 82 \sin 2 \pi \mathrm { ft }$ 
C)  $\mathrm { V } = 82 \sqrt { 3 } \sin 2 \pi \mathrm { ft } + 82 \cos 2 \pi \mathrm { ft }$ 
D)  $V = 82 \sqrt { 3 } \cos 2 \pi f t + 82 \sin 2 \pi f t$ 
A)  $V = 82 \sqrt { 3 } \cos 2 \pi f t - 82 \sin 2 \pi f t$ 
B)  $\mathrm { V } = 82 \cos 2 \pi \mathrm { ft } + 82 \sin 2 \pi \mathrm { ft }$ 
C)  $\mathrm { V } = 82 \sqrt { 3 } \sin 2 \pi \mathrm { ft } + 82 \cos 2 \pi \mathrm { ft }$ 
D)  $V = 82 \sqrt { 3 } \cos 2 \pi f t + 82 \sin 2 \pi f t$

Question 6

Verify that each equation is an identity.
-$$\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\cot...

Question 7

Verify that each equation is an identity.
-$$( \sec \alpha + \tan \alpha ) ^ { 2 } = \frac { 1 + \sin \alpha } { 1 - \sin \alpha }$$

Accepted Answer

\[\begin{array} { l } 
( \sec \alpha + \

Question 8

Find the exact value by using a half-angle identity.
-$$\cos \frac { 5 \pi } { 12 }$$

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 9

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\sin 8 x \cos 8 x$

Accepted Answer

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

Question 10

Graph the inverse circular function.
-$y=\operatorname{arcsec} 2 x$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 11

Use Identities to find the exact value.
-$$\cos \left( \frac { 19 \pi } { 12 } \right)$$

Accepted Answer

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

Question 12

Solve the problem.

-Suppose that for an electrical appliance, voltage is given by $\mathrm { V } = 189 \sin 120 \pi \mathrm { t }$ and amperage by $\mathrm { I } = 1.07 \sin 120 \pi \mathrm { t }$, where $\mathrm { t }$ is time in seconds. Use identities to write the wattage $\mathrm { W } = \mathrm { VI }$ in the form $\mathrm { W } = \mathrm { a } \cos \omega \mathrm { t } + \mathrm { c }$

Accepted Answer

A)  $\mathrm { W } = 101.115 \cos 240 \pi t - 101.115$ 
B)  $W = - 101.115 \cos 120 \pi t + 101.115$ 
C)  $W = - 202.23 \cos 240 \pi t + 202.23$ 
D)  $W = - 101.115 \cos 240 \pi t + 101.115$ 
A)  $\mathrm { W } = 101.115 \cos 240 \pi t - 101.115$ 
B)  $W = - 101.115 \cos 120 \pi t + 101.115$ 
C)  $W = - 202.23 \cos 240 \pi t + 202.23$ 
D)  $W = - 101.115 \cos 240 \pi t + 101.115$

Question 13

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as
appropriate.
-In an electrical circuit, let $\mathrm { V }$ represent the electromagnetic force in volts at $\mathrm { t }$ seconds. Assume $\mathrm { V } = \cos 2 \pi \mathrm { t }$. Find the smallest positive value of $\mathrm { t }$ where $0 \leq \mathrm { t } \leq \frac { 1 } { 2 }$ for $\mathrm { V } = \frac { \sqrt { 2 } } { 2 }$.

Accepted Answer

A)  $\frac { 1 } { 8 } \mathrm { sec }$ 
B)  $0 \mathrm { sec }$ 
C)  $\frac { 1 } { 4 } \mathrm { sec }$ 
D)  $\frac { 1 } { 2 } \mathrm { sec }$ 
A)  $\frac { 1 } { 8 } \mathrm { sec }$ 
B)  $0 \mathrm { sec }$ 
C)  $\frac { 1 } { 4 } \mathrm { sec }$ 
D)  $\frac { 1 } { 2 } \mathrm { sec }$

Question 14

Solve the equation for x, where x is restricted to the given interval.
-$y = 8 \cos 3 x$, for $x$ in $\left[ 0 , \frac { \pi } { 3 } \right]$

Accepted Answer

A)  $x = 3 \arccos \frac { y } { 8 }$ 
B)  $x = 8 \arccos \frac { y } { 3 }$ 
C)  $x = \frac { 1 } { 3 } \arccos \frac { y } { 8 }$ 
D)  $x = \frac { 1 } { 8 } \arccos \frac { y } { 3 }$ 
A)  $x = 3 \arccos \frac { y } { 8 }$ 
B)  $x = 8 \arccos \frac { y } { 3 }$ 
C)  $x = \frac { 1 } { 3 } \arccos \frac { y } { 8 }$ 
D)  $x = \frac { 1 } { 8 } \arccos \frac { y } { 3 }$

Question 15

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 12 } { 13 }$, with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 15 } { 17 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $- \frac { 171 } { 221 }$ 
B)  $\frac { 21 } { 221 }$ 
C)  $- \frac { 21 } { 221 }$ 
D)  $\frac { 171 } { 221 }$ 
A)  $- \frac { 171 } { 221 }$ 
B)  $\frac { 21 } { 221 }$ 
C)  $- \frac { 21 } { 221 }$ 
D)  $\frac { 171 } { 221 }$

Question 16

Use a calculator to give the value to the nearest degree.
-$$\theta = \cos ^ { - 1 } ( - 0.3907 )$$

Accepted Answer

A)  $110 ^ { \circ }$ 
B)  $113 ^ { \circ }$ 
C)  $- 67 ^ { \circ }$ 
D)  $115 ^ { \circ }$ 
A)  $110 ^ { \circ }$ 
B)  $113 ^ { \circ }$ 
C)  $- 67 ^ { \circ }$ 
D)  $115 ^ { \circ }$

Question 17

Verify that each equation is an identity.
-$$\frac { \sec \theta - 1 } { \tan \theta } = \frac { \tan \theta } { \sec \theta + 1 }$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\frac...

Question 18

Solve the equation in the interval $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Give solutions to the nearest tenth, if necessary.
-$4 \sin ^ { 2 } \theta = 3$

Accepted Answer

A)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } \right\}$ 
B)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
A)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } \right\}$ 
B)  $\left\{ 60 ^ { \circ } , 120 ^ { \circ } , 240 ^ { \circ } , 300 ^ { \circ } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ 240 ^ { \circ } , 300 ^ { \circ } \right\}$

Question 19

Find the exact value of the expression using the provided information.
-Find $\sin ( s - t )$ given that $\sin s = - \frac { 1 } { 2 }$, with $s$ in quadrant $I V$, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.

Accepted Answer

A)  $\frac { - \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
B)  $\frac { \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
C)  $\frac { \sqrt { 15 } - \sqrt { 3 } } { 8 }$ 
D)  $\frac { - \sqrt { 15 } - \sqrt { 3 } } { 8 }$ 
A)  $\frac { - \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
B)  $\frac { \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
C)  $\frac { \sqrt { 15 } - \sqrt { 3 } } { 8 }$ 
D)  $\frac { - \sqrt { 15 } - \sqrt { 3 } } { 8 }$

Question 20

Use a calculator to give the value to the nearest degree.
-$$\theta = \tan ^ { - 1 } ( 0.5774 )$$

Accepted Answer

A)  $32 ^ { \circ }$ 
B)  $27 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $60 ^ { \circ }$ 
A)  $32 ^ { \circ }$ 
B)  $27 ^ { \circ }$ 
C)  $30 ^ { \circ }$ 
D)  $60 ^ { \circ }$

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( - 0.9397 )$

Use the fundamental identities to simplify the expression. - $\frac { \tan \theta } { \cot \theta }$

Use the fundamental identities to simplify the expression. - $\sin \theta \cos \theta \sec \theta \csc \theta$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\csc x ( \sin x + \cos x )$

Verify that the equation is an identity. -The output voltage of a generator is given by $\mathrm { V } = 164 \cos \left( 2 \pi \mathrm { ft } - \frac { \pi } { 6 } \right)$ . Express the voltage as the sum of a sine and a cosine function.

Verify that each equation is an identity. - $\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$

Verify that each equation is an identity. - $( \sec \alpha + \tan \alpha ) ^ { 2 } = \frac { 1 + \sin \alpha } { 1 - \sin \alpha }$

Find the exact value by using a half-angle identity. - $\cos \frac { 5 \pi } { 12 }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sin 8 x \cos 8 x$

Graph the inverse circular function. - $y=\operatorname{arcsec} 2 x$ $Graph the inverse circular function. - y=\operatorname{arcsec} 2 x$

Use Identities to find the exact value. - $\cos \left( \frac { 19 \pi } { 12 } \right)$

Solve the equation for x, where x is restricted to the given interval. - $y = 8 \cos 3 x$ , for $x$ in $\left[ 0 , \frac { \pi } { 3 } \right]$

Find the exact value of the expression using the provided information. -Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 12 } { 13 }$ , with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 15 } { 17 }$ , with $\mathrm { t }$ in quadrant II.

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( - 0.3907 )$

Verify that each equation is an identity. - $\frac { \sec \theta - 1 } { \tan \theta } = \frac { \tan \theta } { \sec \theta + 1 }$

Solve the equation in the interval $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Give solutions to the nearest tenth, if necessary. - $4 \sin ^ { 2 } \theta = 3$

Find the exact value of the expression using the provided information. -Find $\sin ( s - t )$ given that $\sin s = - \frac { 1 } { 2 }$ , with $s$ in quadrant $I V$ , and $\sin t = \frac { 1 } { 4 }$ , with $t$ in quadrant II.

Use a calculator to give the value to the nearest degree. - $\theta = \tan ^ { - 1 } ( 0.5774 )$

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 give the value to the nearest degree. - θ=cos⁡−1(−0.9397)\theta = \cos ^ { - 1 } ( - 0.9397 )θ=cos−1(−0.9397)

Use the fundamental identities to simplify the expression. - tan⁡θcot⁡θ\frac { \tan \theta } { \cot \theta }cotθtanθ​

Use the fundamental identities to simplify the expression. - sin⁡θcos⁡θsec⁡θcsc⁡θ\sin \theta \cos \theta \sec \theta \csc \thetasinθcosθsecθcscθ

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - csc⁡x(sin⁡x+cos⁡x)\csc x ( \sin x + \cos x )cscx(sinx+cosx)

Verify that the equation is an identity. -The output voltage of a generator is given by V=164cos⁡(2πft−π6)\mathrm { V } = 164 \cos \left( 2 \pi \mathrm { ft } - \frac { \pi } { 6 } \right)V=164cos(2πft−6π​) . Express the voltage as the sum of a sine and a cosine function.

Verify that each equation is an identity. - cot⁡xsec⁡4x=cot⁡x+2tan⁡x+tan⁡3x\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } xcotxsec4x=cotx+2tanx+tan3x

Verify that each equation is an identity. - (sec⁡α+tan⁡α)2=1+sin⁡α1−sin⁡α( \sec \alpha + \tan \alpha ) ^ { 2 } = \frac { 1 + \sin \alpha } { 1 - \sin \alpha }(secα+tanα)2=1−sinα1+sinα​

Find the exact value by using a half-angle identity. - cos⁡5π12\cos \frac { 5 \pi } { 12 }cos125π​

Use an identity to write the expression as a single trigonometric function or as a single number. - sin⁡8xcos⁡8x\sin 8 x \cos 8 xsin8xcos8x

Graph the inverse circular function. - y=arcsec⁡2xy=\operatorname{arcsec} 2 xy=arcsec2x

Use Identities to find the exact value. - cos⁡(19π12)\cos \left( \frac { 19 \pi } { 12 } \right)cos(1219π​)

Solve the equation for x, where x is restricted to the given interval. - y=8cos⁡3xy = 8 \cos 3 xy=8cos3x , for xxx in [0,π3]\left[ 0 , \frac { \pi } { 3 } \right][0,3π​]

Use a calculator to give the value to the nearest degree. - θ=cos⁡−1(−0.3907)\theta = \cos ^ { - 1 } ( - 0.3907 )θ=cos−1(−0.3907)

Verify that each equation is an identity. - sec⁡θ−1tan⁡θ=tan⁡θsec⁡θ+1\frac { \sec \theta - 1 } { \tan \theta } = \frac { \tan \theta } { \sec \theta + 1 }tanθsecθ−1​=secθ+1tanθ​

Solve the equation in the interval [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Give solutions to the nearest tenth, if necessary. - 4sin⁡2θ=34 \sin ^ { 2 } \theta = 34sin2θ=3

Find the exact value of the expression using the provided information. -Find sin⁡(s−t)\sin ( s - t )sin(s−t) given that sin⁡s=−12\sin s = - \frac { 1 } { 2 }sins=−21​ , with sss in quadrant IVI VIV , and sin⁡t=14\sin t = \frac { 1 } { 4 }sint=41​ , with ttt in quadrant II.

Use a calculator to give the value to the nearest degree. - θ=tan⁡−1(0.5774)\theta = \tan ^ { - 1 } ( 0.5774 )θ=tan−1(0.5774)