Question 1

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 2

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

Accepted Answer

\[\begin{array} { l } 
\sec 4 x - \tan 4

Question 3

Find the exact value of the expression using the provided information.
-Find $\sin ( s - t )$ given that $\cos s = \frac { 1 } { 3 }$, with $s$ in quadrant $I$, and $\sin t = - \frac { 1 } { 2 }$, with $t$ in quadrant IV.

Accepted Answer

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

Question 4

Use a sum or difference identity to find the exact value.
-$\sin 255 ^ { \circ }$

Accepted Answer

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

Question 5

Verify that each equation is an identity.
-$$\csc s - \sin s = \cos s \cot s$$

Accepted Answer

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

Question 6

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x } = \csc ^ { 2 } x$$

Accepted Answer

A)  Not an Identity 
B)  Identity 
A)  Not an Identity 
B)  Identity

Question 7

Give the exact value of the expression.
-$\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)$

Accepted Answer

A)  $\frac { 5 } { 8 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 3 } { 8 }$ 
D)  $\frac { 7 } { 8 }$ 
A)  $\frac { 5 } { 8 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 3 } { 8 }$ 
D)  $\frac { 7 } { 8 }$

Question 8

Perform the indicated operations and simplify the result so there are no quotients.
-$\csc \theta ( \sin \theta + \cos \theta )$

Accepted Answer

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

Question 9

Find the exact value by using a half-angle identity.
-$\cos 165 ^ { \circ }$

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 10

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

Accepted Answer

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

Question 11

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

A)  $f ( - x ) = f ( x )$ 
B)  $f ( - x ) = - f ( x )$ 
A)  $f ( - x ) = f ( x )$ 
B)  $f ( - x ) = - f ( x )$

Question 12

Use identities to find the indicated value for each angle measure.
-$\cos \theta = \frac { 3 } { 5 } , \quad \sin \theta < 0 \quad$ Find $\sin ( 2 \theta )$.

Accepted Answer

A)  $\frac { 24 } { 25 }$ 
B)  $- \frac { 7 } { 25 }$ 
C)  $- \frac { 24 } { 25 }$ 
D)  $\frac { 7 } { 25 }$ 
A)  $\frac { 24 } { 25 }$ 
B)  $- \frac { 7 } { 25 }$ 
C)  $- \frac { 24 } { 25 }$ 
D)  $\frac { 7 } { 25 }$

Question 13

Solve the problem.
-Suppose that a hair dryer operates on a voltage that can be represented by the relation $\mathrm { V } ( \mathrm { t } ) = 160 \cos ( 120 \pi \mathrm { t } )$ and that it draws a current represented by the relation $\mathrm { I } ( \mathrm { t } ) = 7 \cos ( 120 \pi \mathrm { t } )$, where $t$ is time measured in seconds. The power consumed by the appliance is $P =$ VI. Graph the power in $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$ and use an identity to write the expression for the power in the form $\mathrm { P } = \mathrm { a } \cos ( \mathrm { k } \pi \mathrm { t } ) + \mathrm { d }$, where $\mathrm { a } , \mathrm { k }$, and $\mathrm { d }$ are constants.

Accepted Answer

A)  $P = 560 \cos ( 240 \pi t ) + 560$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
B)  $P = 1120 \cos ( 240 \pi t ) + 1120$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
C)  $\mathrm { P } = - 560 \cos ( 240 \pi \mathrm { t } ) + 560$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
D)  $P = 560 \cos ( 120 \pi t ) + 560$
 $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
A)  $P = 560 \cos ( 240 \pi t ) + 560$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
B)  $P = 1120 \cos ( 240 \pi t ) + 1120$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
C)  $\mathrm { P } = - 560 \cos ( 240 \pi \mathrm { t } ) + 560$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
D)  $P = 560 \cos ( 120 \pi t ) + 560$
 $[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$

Question 14

Use a sum or difference identity to find the exact value.
-$$\frac { \tan \frac { \pi } { 15 } + \tan \frac { 4 \pi } { 15 } } { 1 - \tan \frac { \pi } { 15 } \tan \frac { 4 \pi } { 15 } }$$

Accepted Answer

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

Question 15

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$\frac { \tan \theta } { \sec \theta }$$

Accepted Answer

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

Question 16

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\sin 2 \theta + \sin \theta = 0$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } , 120 ^ { \circ } , 180 ^ { \circ } , 240 ^ { \circ } \right\}$ 
B)  $\left\{ 105 ^ { \circ } , 165 ^ { \circ } , 285 ^ { \circ } , 345 ^ { \circ } \right\}$ 
C)  $\left\{ 30 ^ { \circ } , 90 ^ { \circ } , 150 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 15 ^ { \circ } , 165 ^ { \circ } , 195 ^ { \circ } , 345 ^ { \circ } \right\}$

Question 17

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$\mathrm { y } = \sin ^ { - 1 } ( 0.6561 )$$

Accepted Answer

A)  $0.6809524$ 
B)  $0.7156393$ 
C)  $- 0.7158730$ 
D)  $0.5806515$ 
A)  $0.6809524$ 
B)  $0.7156393$ 
C)  $- 0.7158730$ 
D)  $0.5806515$

Question 18

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

Accepted Answer

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

Question 19

Use identities to write the expression as a single function of x or θ.
-$\tan \left( x + \frac { \pi } { 6 } \right)$

Accepted Answer

A)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
B)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
C)  $\tan x$ 
D)  $- \tan x$ 
A)  $\frac { 1 + \sqrt { 3 } \tan x } { \sqrt { 3 } - \tan x }$ 
B)  $\frac { \tan x - \sqrt { 3 } } { 1 + \sqrt { 3 } \tan x }$ 
C)  $\tan x$ 
D)  $- \tan x$

Question 20

Use the cofunction identities to find an angle θ that makes the statement true.
-$\cot \left( 2 \theta - 20 ^ { \circ } \right) = \tan \left( \theta - 10 ^ { \circ } \right)$

Accepted Answer

A)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
D)  $\theta = 40 ^ { \circ }$ 
A)  $\theta = \frac { 17 ^ { \circ } } { 7 }$ 
B)  $\theta = 10 ^ { \circ }$ 
C)  $\theta = \frac { 83 ^ { \circ } } { 7 }$ 
D)  $\theta = 40 ^ { \circ }$

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

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

Find the exact value of the expression using the provided information. -Find $\sin ( s - t )$ given that $\cos s = \frac { 1 } { 3 }$ , with $s$ in quadrant $I$ , and $\sin t = - \frac { 1 } { 2 }$ , with $t$ in quadrant IV.

Use a sum or difference identity to find the exact value. - $\sin 255 ^ { \circ }$

Verify that each equation is an identity. - $\csc s - \sin s = \cos s \cot s$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x } = \csc ^ { 2 } x$

Give the exact value of the expression. - $\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)$

Perform the indicated operations and simplify the result so there are no quotients. - $\csc \theta ( \sin \theta + \cos \theta )$

Find the exact value by using a half-angle identity. - $\cos 165 ^ { \circ }$

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

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Use identities to find the indicated value for each angle measure. - $\cos \theta = \frac { 3 } { 5 } , \quad \sin \theta < 0 \quad$ Find $\sin ( 2 \theta )$ .

Use a sum or difference identity to find the exact value. - $\frac { \tan \frac { \pi } { 15 } + \tan \frac { 4 \pi } { 15 } } { 1 - \tan \frac { \pi } { 15 } \tan \frac { 4 \pi } { 15 } }$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\frac { \tan \theta } { \sec \theta }$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\sin 2 \theta + \sin \theta = 0$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \sin ^ { - 1 } ( 0.6561 )$

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

Use identities to write the expression as a single function of x or θ. - $\tan \left( x + \frac { \pi } { 6 } \right)$

Use the cofunction identities to find an angle θ that makes the statement true. - $\cot \left( 2 \theta - 20 ^ { \circ } \right) = \tan \left( \theta - 10 ^ { \circ } \right)$

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

Review of Basic Concepts

Filters

Exam 7: Trigonometric Identities and Equations

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​

Verify that each equation is an identity. - sec⁡4x−tan⁡4x=sec⁡2x+tan⁡2x\sec ^ { 4 } x - \tan 4 x = \sec ^ { 2 } x + \tan 2 xsec4x−tan4x=sec2x+tan2x

Find the exact value of the expression using the provided information. -Find sin⁡(s−t)\sin ( s - t )sin(s−t) given that cos⁡s=13\cos s = \frac { 1 } { 3 }coss=31​ , with sss in quadrant III , and sin⁡t=−12\sin t = - \frac { 1 } { 2 }sint=−21​ , with ttt in quadrant IV.

Use a sum or difference identity to find the exact value. - sin⁡255∘\sin 255 ^ { \circ }sin255∘

Verify that each equation is an identity. - csc⁡s−sin⁡s=cos⁡scot⁡s\csc s - \sin s = \cos s \cot scscs−sins=cosscots

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - 11−cos⁡x+11+cos⁡x=csc⁡2x\frac { 1 } { 1 - \cos x } + \frac { 1 } { 1 + \cos x } = \csc ^ { 2 } x1−cosx1​+1+cosx1​=csc2x

Give the exact value of the expression. - cos⁡(2arcsin⁡14)\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)cos(2arcsin41​)

Perform the indicated operations and simplify the result so there are no quotients. - csc⁡θ(sin⁡θ+cos⁡θ)\csc \theta ( \sin \theta + \cos \theta )cscθ(sinθ+cosθ)

Find the exact value by using a half-angle identity. - cos⁡165∘\cos 165 ^ { \circ }cos165∘

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