Question 1

Verify that each equation is an identity.
-$$\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta$$

Accepted Answer

\[\begin{array} { l } 
\frac { 1 - \cot

Question 2

Use the fundamental identities to simplify the expression.
-$\sin \frac { \pi } { 15 } \cos \frac { 4 \pi } { 15 } + \cos \frac { \pi } { 15 } \sin \frac { 4 \pi } { 15 }$

Accepted Answer

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

Question 3

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 $\mathrm { I } ( \mathrm { t } ) = 8 \cos ( 120$ where $t$ is time measured in seconds. The power consumed by the appliance is $\mathrm { P } = \mathrm { VI }$. 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)  $P = 640 \cos ( 120 \pi t ) + 640$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
B)  $P = 640 \cos ( 240 \pi t ) + 640$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
C)  $\mathrm { P } = 1280 \cos ( 240 \pi \mathrm { t } ) + 1280$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
D)  $\mathrm { P } = - 640 \cos ( 240 \pi \mathrm { t } ) + 640$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
A)  $P = 640 \cos ( 120 \pi t ) + 640$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
B)  $P = 640 \cos ( 240 \pi t ) + 640$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
C)  $\mathrm { P } = 1280 \cos ( 240 \pi \mathrm { t } ) + 1280$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$
  
D)  $\mathrm { P } = - 640 \cos ( 240 \pi \mathrm { t } ) + 640$
$[ 0,0.04,0.01 ]$ by $[ - 200,2000,200 ]$

Question 4

Use Identities to find the exact value.
-$$\frac { \tan 25 ^ { \circ } + \tan 5 ^ { \circ } } { 1 - \tan 25 ^ { \circ } \tan 5 ^ { \circ } }$$

Accepted Answer

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

Question 5

Use the cofunction identities to find an angle that makes the statement true.
-$$\frac { \tan 80 ^ { \circ } + \tan 70 ^ { \circ } } { 1 - \tan 80 ^ { \circ } \tan 70 ^ { \circ } }$$

Accepted Answer

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

Question 6

Use identities to write each expression as a function of .
-Find $\csc \theta$ if $\cot \theta = - \sqrt { 15 }$ and $\theta$ is in quadrant II.

Accepted Answer

A)  4 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $- 4$ 
A)  4 
B)  $- \frac { 1 } { 4 }$ 
C)  $\frac { 1 } { 4 }$ 
D)  $- 4$

Question 7

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

Question 8

Verify that the equation is an identity.
-cos(x - y)- cos(x + y)= 2 sin x sin y

Accepted Answer

cos (x - y)- cos (x

Question 9

Write the product as a sum or difference of trigonometric functions.
-$\sin 8 x \cos 8 x$

Accepted Answer

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

Question 10

Verify that the equation is an identity.
-$$\sec ^ { 2 } \frac { u } { 2 } = \frac { 2 \sec u } { \sec u + 1 }$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\sec...

Question 11

Find the exact value of the expression using the provided information.
-$$\frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta } + \csc \theta \sin \theta$$

Accepted Answer

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

Question 12

Write in terms of the cofunction of a complementary angle.
-$\cos \frac { \pi } { 24 }$

Accepted Answer

A)  $\sin \frac { 23 \pi } { 24 }$ 
B)  $\sin \frac { 11 \pi } { 24 }$ 
C)  $\sec \frac { 11 \pi } { 24 }$ 
D)  $\sec \frac { 23 \pi } { 24 }$ 
A)  $\sin \frac { 23 \pi } { 24 }$ 
B)  $\sin \frac { 11 \pi } { 24 }$ 
C)  $\sec \frac { 11 \pi } { 24 }$ 
D)  $\sec \frac { 23 \pi } { 24 }$

Question 13

Find the exact value of the expression using the provided information.
-Find $\sin \theta$ if $\tan \theta = - \frac { 12 } { 5 }$ and $\cos \theta > 0$.

Accepted Answer

A)  $- \frac { 5 } { 13 }$ 
B)  $\frac { 5 } { 13 }$ 
C)  $- \frac { 13 } { 12 }$ 
D)  $- \frac { 12 } { 13 }$ 
A)  $- \frac { 5 } { 13 }$ 
B)  $\frac { 5 } { 13 }$ 
C)  $- \frac { 13 } { 12 }$ 
D)  $- \frac { 12 } { 13 }$

Question 14

Determine all solutions of the equation in radians.
-Find $\tan \frac { x } { 2 }$, given that $\tan x = - 3$ and $x$ terminates in $90 ^ { \circ } < x < 180 ^ { \circ }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } + 1 } { - 3 }$ 
B)  $\frac { \sqrt { 10 } - 1 } { 3 }$ 
C)  $\frac { \sqrt { 10 } + 1 } { 3 }$ 
D)  $\frac { \sqrt { 10 } } { - 3 }$ 
A)  $\frac { \sqrt { 10 } + 1 } { - 3 }$ 
B)  $\frac { \sqrt { 10 } - 1 } { 3 }$ 
C)  $\frac { \sqrt { 10 } + 1 } { 3 }$ 
D)  $\frac { \sqrt { 10 } } { - 3 }$

Question 15

Verify that the equation is an 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 Verify that the equation is an identity.
-\[\cot...

Question 16

Use the fundamental identities to find the value of the trigonometric function.
-$$\cos \frac { \pi } { 72 } = \cos \frac { \pi } { 9 } \cos \frac { \pi } { 8 } - \sin \frac { \pi } { 9 } \sin \frac { \pi } { 8 }$$

Accepted Answer

A) True 
 B)False

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$\frac { \cos ^ { 2 } x } { \sin ^ { 2 } x } + \cos x \sec x$$

Accepted Answer

A)  $\csc x$ 
B)  $\csc ^ { 2 } x$ 
C)  $\sec ^ { 2 } x$ 
D)  $\cot ^ { 2 } x$ 
A)  $\csc x$ 
B)  $\csc ^ { 2 } x$ 
C)  $\sec ^ { 2 } x$ 
D)  $\cot ^ { 2 } x$

Question 19

Use the cofunction identities to find an angle that makes the statement true.
-Find $\csc \theta$ if $\sin \theta = - \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  $- \frac { \sqrt { 7 } } { 9 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $- \frac { 3 } { 2 }$ 
D)  $\frac { 3 \sqrt { 7 } } { 7 }$ 
A)  $- \frac { \sqrt { 7 } } { 9 }$ 
B)  $\frac { 5 } { 4 }$ 
C)  $- \frac { 3 } { 2 }$ 
D)  $\frac { 3 \sqrt { 7 } } { 7 }$

Question 20

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

Accepted Answer

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

Verify that each equation is an identity. - $\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta$

Use the fundamental identities to simplify the expression. - $\sin \frac { \pi } { 15 } \cos \frac { 4 \pi } { 15 } + \cos \frac { \pi } { 15 } \sin \frac { 4 \pi } { 15 }$

Use Identities to find the exact value. - $\frac { \tan 25 ^ { \circ } + \tan 5 ^ { \circ } } { 1 - \tan 25 ^ { \circ } \tan 5 ^ { \circ } }$

Use the cofunction identities to find an angle that makes the statement true. - $\frac { \tan 80 ^ { \circ } + \tan 70 ^ { \circ } } { 1 - \tan 80 ^ { \circ } \tan 70 ^ { \circ } }$

Use identities to write each expression as a function of . -Find $\csc \theta$ if $\cot \theta = - \sqrt { 15 }$ and $\theta$ is in quadrant II.

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. -cos(x - y)- cos(x + y)= 2 sin x sin y

Write the product as a sum or difference of trigonometric functions. - $\sin 8 x \cos 8 x$

Verify that the equation is an identity. - $\sec ^ { 2 } \frac { u } { 2 } = \frac { 2 \sec u } { \sec u + 1 }$

Find the exact value of the expression using the provided information. - $\frac { \cos ^ { 2 } \theta } { \sin ^ { 2 } \theta } + \csc \theta \sin \theta$

Write in terms of the cofunction of a complementary angle. - $\cos \frac { \pi } { 24 }$

Find the exact value of the expression using the provided information. -Find $\sin \theta$ if $\tan \theta = - \frac { 12 } { 5 }$ and $\cos \theta > 0$ .

Determine all solutions of the equation in radians. -Find $\tan \frac { x } { 2 }$ , given that $\tan x = - 3$ and $x$ terminates in $90 ^ { \circ } < x < 180 ^ { \circ }$ .

Verify that the equation is an identity. - $\cot ( x + y ) \cot ( x - y ) = \frac { 1 - \tan ^ { 2 } x \tan ^ { 2 } y } { \tan ^ { 2 } x - \tan ^ { 2 } y }$

Use the fundamental identities to find the value of the trigonometric function. - $\cos \frac { \pi } { 72 } = \cos \frac { \pi } { 9 } \cos \frac { \pi } { 8 } - \sin \frac { \pi } { 9 } \sin \frac { \pi } { 8 }$

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\frac { \cos ^ { 2 } x } { \sin ^ { 2 } x } + \cos x \sec x$

Use the cofunction identities to find an angle that makes the statement true. -Find $\csc \theta$ if $\sin \theta = - \frac { 2 } { 3 }$ and $\theta$ is in quadrant IV.

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

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Inverse Circular Functions and Trigonometric Equations

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 5: Trigonometric Identities

Verify that each equation is an identity. - 1−cot⁡2θ1+cot⁡2θ+1=2sin⁡2θ\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta1+cot2θ1−cot2θ​+1=2sin2θ

Use the fundamental identities to simplify the expression. - sin⁡π15cos⁡4π15+cos⁡π15sin⁡4π15\sin \frac { \pi } { 15 } \cos \frac { 4 \pi } { 15 } + \cos \frac { \pi } { 15 } \sin \frac { 4 \pi } { 15 }sin15π​cos154π​+cos15π​sin154π​

Use Identities to find the exact value. - tan⁡25∘+tan⁡5∘1−tan⁡25∘tan⁡5∘\frac { \tan 25 ^ { \circ } + \tan 5 ^ { \circ } } { 1 - \tan 25 ^ { \circ } \tan 5 ^ { \circ } }1−tan25∘tan5∘tan25∘+tan5∘​

Use the cofunction identities to find an angle that makes the statement true. - tan⁡80∘+tan⁡70∘1−tan⁡80∘tan⁡70∘\frac { \tan 80 ^ { \circ } + \tan 70 ^ { \circ } } { 1 - \tan 80 ^ { \circ } \tan 70 ^ { \circ } }1−tan80∘tan70∘tan80∘+tan70∘​

Use identities to write each expression as a function of . -Find csc⁡θ\csc \thetacscθ if cot⁡θ=−15\cot \theta = - \sqrt { 15 }cotθ=−15​ and θ\thetaθ is in quadrant II.

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)