Question 1

Use identities to write each expression as a function of .
-$\cos 2 x = \sin 2 x \tan x$

Accepted Answer

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

Question 2

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-Two low-frequency sound waves with similar frequencies, producing the pressures $\mathrm { P } _ { 1 } = 0.004 \sin ( 14 \pi \mathrm { t } )$ and $\mathrm { P } _ { 2 } = 0.003 \sin ( 20 \pi \mathrm { t } )$, combine to form a pressure $\mathrm { P } = \mathrm { P } _ { 1 } + \mathrm { P } _ { 2 }$. The variable $\mathrm { t }$ is time measured in seconds. Graph $\mathrm { P }$ in $[ 0,1.2 , .05 ]$ by $[ - 0.01,0.01,0.001 ]$ and use the graph to determine the frequency of the displayed beats.

Accepted Answer

A)  10 beats per second 
B)  3 beats per second 
C)  7 beats per second 
D)  6 beats per second 
A)  10 beats per second 
B)  3 beats per second 
C)  7 beats per second 
D)  6 beats per second

Question 3

Write the product as a sum or difference of trigonometric functions.
-$- 10 \cos ( - \theta ) \sin ( - 5 \theta )$

Accepted Answer

A)  $- 5 ( \cos 6 \theta - \cos 4 \theta )$ 
B)  $- 10 ( \cos 6 \theta + \cos 4 \theta )$ 
C)  $5 ( \sin 6 \theta + \sin 4 \theta )$ 
D)  $5 ( \sin 6 \theta - \sin 4 \theta )$ 
A)  $- 5 ( \cos 6 \theta - \cos 4 \theta )$ 
B)  $- 10 ( \cos 6 \theta + \cos 4 \theta )$ 
C)  $5 ( \sin 6 \theta + \sin 4 \theta )$ 
D)  $5 ( \sin 6 \theta - \sin 4 \theta )$

Question 4

Rewrite the following as a product of trigonometric functions.
-$\cos ( - x ) \cos x - \sin ( - x ) \sin x$

Accepted Answer

A)  1 
B)  2 
C)  $\cos ^ { 2 } x \sin ^ { 2 } x$ 
D)  0 
A)  1 
B)  2 
C)  $\cos ^ { 2 } x \sin ^ { 2 } x$ 
D)  0

Question 5

Solve the problem.
-Find $\cos ( s + t )$ given that $\sin s = - \frac { \sqrt { 3 } } { 3 }$, with $s$ in quadrant IV, and $\sin t = - \frac { \sqrt { 5 } } { 6 }$, with $t$ in quadrant IV.

Accepted Answer

A)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
A)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$

Question 6

Determine all solutions of the equation in radians.
-Find $\cos ( s + t )$ given that $\sin s = - \frac { 1 } { 2 }$, with $s$ in quadrant IV, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.

Accepted Answer

A)  $- \frac { 4 \sqrt { 3 } + \sqrt { 15 } } { 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
D)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
A)  $- \frac { 4 \sqrt { 3 } + \sqrt { 15 } } { 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
D)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$

Question 7

Use a sum or difference identity to find the exact value.
-$$--- = \frac { 1 } { \sin x }$$

Accepted Answer

A)  $\sec x$ 
B)  $\csc x$ 
C)  $\cot x$ 
D)  $\tan x$ 
A)  $\sec x$ 
B)  $\csc x$ 
C)  $\cot x$ 
D)  $\tan x$

Question 8

Use the fundamental identities to simplify the expression.
-$2 \cos 8 x \cos 2 x$

Accepted Answer

A)  $\sin 10 x + \sin 6 x$ 
B)  $\cos 10 x + \cos 6 x$ 
C)  $\cos 6 x - \cos 10 x$ 
D)  $\frac { 1 } { 2 } ( \cos 10 x + \cos 6 x )$ 
A)  $\sin 10 x + \sin 6 x$ 
B)  $\cos 10 x + \cos 6 x$ 
C)  $\cos 6 x - \cos 10 x$ 
D)  $\frac { 1 } { 2 } ( \cos 10 x + \cos 6 x )$

Question 9

Perform the indicated operations and simplify the result so there are no quotients.
-$$\cos 67 ^ { \circ } = \cos 13 ^ { \circ } \cos 54 ^ { \circ } - \sin 13 ^ { \circ } \sin 54 ^ { \circ }$$

Accepted Answer

A) True 
 B)False

Question 10

Write the product as a sum or difference of trigonometric functions.
-Find $\sin ( s + t )$ given that $\cos s = - \frac { 1 } { 4 }$, with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $t$ in quadrant III.

Accepted Answer

A)  $\frac { 3 + 4 \sqrt { 15 } } { 20 }$ 
B)  $\frac { 3 - 4 \sqrt { 15 } } { 20 }$ 
C)  $\frac { 3 \sqrt { 15 } + 4 } { 20 }$ 
D)  $\frac { 3 \sqrt { 15 } - 4 } { 20 }$ 
A)  $\frac { 3 + 4 \sqrt { 15 } } { 20 }$ 
B)  $\frac { 3 - 4 \sqrt { 15 } } { 20 }$ 
C)  $\frac { 3 \sqrt { 15 } + 4 } { 20 }$ 
D)  $\frac { 3 \sqrt { 15 } - 4 } { 20 }$

Question 11

Use the fundamental identities to find the value of the trigonometric function.
-$\sin 165 ^ { \circ }$

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 12

Use a sum or difference identity to find the exact value.
-$$\frac { \tan 100 ^ { \circ } - \tan ( - 20 ) ^ { \circ } } { 1 + \tan 100 ^ { \circ } \tan ( - 20 ) ^ { \circ } }$$

Accepted Answer

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

Question 13

Use a sum or difference identity to find the exact value.
-Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 5 } } { 2 }$ and $\sin \theta < 0$.

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  2 
C)  $- 2$ 
D)  $- \sqrt { 5 }$ 
A)  $- \frac { 1 } { 2 }$ 
B)  2 
C)  $- 2$ 
D)  $- \sqrt { 5 }$

Question 14

Complete the sentence so the result is an identity. Let x be any real number.
-Find $\sin ( s + t )$ given that $\cos s = - \frac { 12 } { 13 }$, with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$, with $t$ in quadrant II.

Accepted Answer

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

Question 15

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

Accepted Answer

sin (x + y)- sin (x

Question 16

Verify that the equation is an identity.
-$$\sec \left( \frac { \pi } { 2 } + x \right) = - \csc x$$

Accepted Answer

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

Question 17

Verify that each equation is an identity.
-$$\sin ^ { 3 } ( 3 x ) = \frac { 1 } { 2 } ( \sin ( 3 x ) ) ( 1 - \cos ( 6 x ) )$$

Accepted Answer

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

Question 18

Use the cofunction identities to find an angle that makes the statement true.
-tan (2ϴ - 140°)= cot (ϴ + 5°)

Accepted Answer

A) ϴ = 6° 
B) ϴ = 75° 
C) ϴ = 10° 
D) ϴ = 16° 
A) ϴ = 6° 
B) ϴ = 75° 
C) ϴ = 10° 
D) ϴ = 16°

Question 19

Verify that the equation is an identity.
-$$\cot ( \pi - \theta ) = - \cot \theta$$

Accepted Answer

\[\begin{array} { l } 
\cot ( \pi - \the

Question 20

Perform the indicated operations and simplify the result so there are no quotients.
-$\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

Accepted Answer

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

Use identities to write each expression as a function of . - $\cos 2 x = \sin 2 x \tan x$

Write the product as a sum or difference of trigonometric functions. - $- 10 \cos ( - \theta ) \sin ( - 5 \theta )$

Rewrite the following as a product of trigonometric functions. - $\cos ( - x ) \cos x - \sin ( - x ) \sin x$

Solve the problem. -Find $\cos ( s + t )$ given that $\sin s = - \frac { \sqrt { 3 } } { 3 }$ , with $s$ in quadrant IV, and $\sin t = - \frac { \sqrt { 5 } } { 6 }$ , with $t$ in quadrant IV.

Determine all solutions of the equation in radians. -Find $\cos ( s + t )$ given that $\sin s = - \frac { 1 } { 2 }$ , with $s$ in quadrant IV, and $\sin t = \frac { 1 } { 4 }$ , with $t$ in quadrant II.

Use a sum or difference identity to find the exact value. - $--- = \frac { 1 } { \sin x }$

Use the fundamental identities to simplify the expression. - $2 \cos 8 x \cos 2 x$

Perform the indicated operations and simplify the result so there are no quotients. - $\cos 67 ^ { \circ } = \cos 13 ^ { \circ } \cos 54 ^ { \circ } - \sin 13 ^ { \circ } \sin 54 ^ { \circ }$

Write the product as a sum or difference of trigonometric functions. -Find $\sin ( s + t )$ given that $\cos s = - \frac { 1 } { 4 }$ , with $s$ in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $t$ in quadrant III.

Use the fundamental identities to find the value of the trigonometric function. - $\sin 165 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\frac { \tan 100 ^ { \circ } - \tan ( - 20 ) ^ { \circ } } { 1 + \tan 100 ^ { \circ } \tan ( - 20 ) ^ { \circ } }$

Use a sum or difference identity to find the exact value. -Find $\tan \theta$ if $\sec \theta = \frac { \sqrt { 5 } } { 2 }$ and $\sin \theta < 0$ .

Complete the sentence so the result is an identity. Let x be any real number. -Find $\sin ( s + t )$ given that $\cos s = - \frac { 12 } { 13 }$ , with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$ , with $t$ in quadrant II.

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

Verify that the equation is an identity. - $\sec \left( \frac { \pi } { 2 } + x \right) = - \csc x$

Verify that each equation is an identity. - $\sin ^ { 3 } ( 3 x ) = \frac { 1 } { 2 } ( \sin ( 3 x ) ) ( 1 - \cos ( 6 x ) )$

Use the cofunction identities to find an angle that makes the statement true. -tan (2ϴ - 140°)= cot (ϴ + 5°)

Verify that the equation is an identity. - $\cot ( \pi - \theta ) = - \cot \theta$

Perform the indicated operations and simplify the result so there are no quotients. - $\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

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

Use identities to write each expression as a function of . - cos⁡2x=sin⁡2xtan⁡x\cos 2 x = \sin 2 x \tan xcos2x=sin2xtanx

Write the product as a sum or difference of trigonometric functions. - −10cos⁡(−θ)sin⁡(−5θ)- 10 \cos ( - \theta ) \sin ( - 5 \theta )−10cos(−θ)sin(−5θ)

Rewrite the following as a product of trigonometric functions. - cos⁡(−x)cos⁡x−sin⁡(−x)sin⁡x\cos ( - x ) \cos x - \sin ( - x ) \sin xcos(−x)cosx−sin(−x)sinx

Solve the problem. -Find cos⁡(s+t)\cos ( s + t )cos(s+t) given that sin⁡s=−33\sin s = - \frac { \sqrt { 3 } } { 3 }sins=−33​​ , with sss in quadrant IV, and sin⁡t=−56\sin t = - \frac { \sqrt { 5 } } { 6 }sint=−65​​ , with ttt in quadrant IV.

Determine all solutions of the equation in radians. -Find cos⁡(s+t)\cos ( s + t )cos(s+t) given that sin⁡s=−12\sin s = - \frac { 1 } { 2 }sins=−21​ , with sss in quadrant IV, and sin⁡t=14\sin t = \frac { 1 } { 4 }sint=41​ , with ttt in quadrant II.

Use a sum or difference identity to find the exact value. - −−−=1sin⁡x--- = \frac { 1 } { \sin x }−−−=sinx1​

Use the fundamental identities to simplify the expression. - 2cos⁡8xcos⁡2x2 \cos 8 x \cos 2 x2cos8xcos2x

Perform the indicated operations and simplify the result so there are no quotients. - cos⁡67∘=cos⁡13∘cos⁡54∘−sin⁡13∘sin⁡54∘\cos 67 ^ { \circ } = \cos 13 ^ { \circ } \cos 54 ^ { \circ } - \sin 13 ^ { \circ } \sin 54 ^ { \circ }cos67∘=cos13∘cos54∘−sin13∘sin54∘

Write the product as a sum or difference of trigonometric functions. -Find sin⁡(s+t)\sin ( s + t )sin(s+t) given that cos⁡s=−14\cos s = - \frac { 1 } { 4 }coss=−41​ , with sss in quadrant III, and cos⁡t=−35\cos t = - \frac { 3 } { 5 }cost=−53​ , with ttt in quadrant III.

Use the fundamental identities to find the value of the trigonometric function. - sin⁡165∘\sin 165 ^ { \circ }sin165∘

Use a sum or difference identity to find the exact value. - tan⁡100∘−tan⁡(−20)∘1+tan⁡100∘tan⁡(−20)∘\frac { \tan 100 ^ { \circ } - \tan ( - 20 ) ^ { \circ } } { 1 + \tan 100 ^ { \circ } \tan ( - 20 ) ^ { \circ } }1+tan100∘tan(−20)∘tan100∘−tan(−20)∘​

Use a sum or difference identity to find the exact value. -Find tan⁡θ\tan \thetatanθ if sec⁡θ=52\sec \theta = \frac { \sqrt { 5 } } { 2 }secθ=25​​ and sin⁡θ<0\sin \theta < 0sinθ<0 .

Complete the sentence so the result is an identity. Let x be any real number. -Find sin⁡(s+t)\sin ( s + t )sin(s+t) given that cos⁡s=−1213\cos s = - \frac { 12 } { 13 }coss=−1312​ , with sss in quadrant II, and sin⁡t=817\sin t = \frac { 8 } { 17 }sint=178​ , with ttt in quadrant II.