Question 1

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\sin \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant II.

Accepted Answer

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

Question 2

Use an identity to write the expression as a single trigonometric function or as a single number.
-sin (3ϴ - 17°)= cos (ϴ + 43°)

Accepted Answer

A) ϴ = 90° 
B) ϴ = 6° 
C) ϴ = 16° 
D) ϴ = 10° 
A) ϴ = 90° 
B) ϴ = 6° 
C) ϴ = 16° 
D) ϴ = 10°

Question 3

Find the exact value of the expression using the provided information.
-$\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
A)  $\frac { \sqrt { 3 } } { 3 }$ 
B)  $\frac { 5 } { 12 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$

Question 4

Perform the transformation.
-$\sin \theta \cos \theta \sec \theta \csc \theta$

Accepted Answer

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

Question 5

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 1 } { 2 }$, with s in quadrant III, and $\cos t = - \frac { 3 } { 5 }$, with $\mathrm { t }$ in quadrant III.

Accepted Answer

A)  $\frac { - 3 - 4 \sqrt { 3 } } { 10 }$ 
B)  $\frac { 3 - 4 \sqrt { 3 } } { 10 }$ 
C)  $\frac { 3 + 4 \sqrt { 3 } } { 10 }$ 
D)  $\frac { 3 - 4 \sqrt { } } { 10 }$ 
A)  $\frac { - 3 - 4 \sqrt { 3 } } { 10 }$ 
B)  $\frac { 3 - 4 \sqrt { 3 } } { 10 }$ 
C)  $\frac { 3 + 4 \sqrt { 3 } } { 10 }$ 
D)  $\frac { 3 - 4 \sqrt { } } { 10 }$

Question 6

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

Accepted Answer

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

Question 7

Use a sum or difference identity to find the exact value.
-$\sin ( x + 2 \pi )$

Accepted Answer

A)  $- \cos x$ 
B)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
C)  $\sin x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
A)  $- \cos x$ 
B)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
C)  $\sin x$ 
D)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$

Question 8

Use the fundamental identities to find the value of the trigonometric function.
-$$\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$$

Accepted Answer

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

Question 9

Write the product as a sum or difference of trigonometric functions.
-$\cos \left( - \frac { 7 \pi } { 12 } \right)$

Accepted Answer

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

Question 10

Use a sum or difference identity to find the exact value.
-$\cos 34 ^ { \circ } \sin 25 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 } \left( \cos 59 ^ { \circ } + \cos 9 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \sin 59 ^ { \circ } - \sin 9 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \sin 59 ^ { \circ } + \sin 9 ^ { \circ } \right)$ 
D)  $\frac { 1 } { 2 } \left( \cos 59 ^ { \circ } - \cos 9 ^ { \circ } \right)$ 
A)  $\frac { 1 } { 2 } \left( \cos 59 ^ { \circ } + \cos 9 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \sin 59 ^ { \circ } - \sin 9 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \sin 59 ^ { \circ } + \sin 9 ^ { \circ } \right)$ 
D)  $\frac { 1 } { 2 } \left( \cos 59 ^ { \circ } - \cos 9 ^ { \circ } \right)$

Question 11

Verify that each equation is an identity.
-$$\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$$

Accepted Answer

\[\begin{array} { l } 
\cos ( 4 \theta )

Question 12

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

Accepted Answer

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

Question 13

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-10 cos 26° cos 13°

Accepted Answer

A) 5(sin 39° - sin 13°) 
B) 10(sin 39° + sin 13°) 
C) 5(cos 39° - cos 13°) 
D) 5(cos 39° + cos 13°) 
A) 5(sin 39° - sin 13°) 
B) 10(sin 39° + sin 13°) 
C) 5(cos 39° - cos 13°) 
D) 5(cos 39° + cos 13°)

Question 14

Express the function as a trigonometric function of x.
-$$1 + \tan ^ { 2 } x = \frac { \cos ^ { 2 } x } { 1 - \cos ^ { 2 } x }$$

Accepted Answer

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

Question 15

Perform the transformation.
-$$\frac { \tan 170 ^ { \circ } - \tan 50 ^ { \circ } } { 1 + \tan 170 ^ { \circ } \tan 50 ^ { \circ } }$$

Accepted Answer

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

Question 16

Use the fundamental identities to find the value of the trigonometric function.
-Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.

Accepted Answer

A)  $- \frac { \sqrt { 11 } } { 6 }$ 
B)  $\frac { \sqrt { 33 } } { 4 }$ 
C)  $- \frac { 5 \sqrt { 29 } } { 29 }$ 
D)  $\frac { 4 } { 5 }$ 
A)  $- \frac { \sqrt { 11 } } { 6 }$ 
B)  $\frac { \sqrt { 33 } } { 4 }$ 
C)  $- \frac { 5 \sqrt { 29 } } { 29 }$ 
D)  $\frac { 4 } { 5 }$

Question 17

Verify that each equation is an identity.
-$$\frac { 1 - \sec \theta } { \tan \theta } + \frac { \tan \theta } { 1 - \sec \theta } = - 2 \csc \theta$$

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Using a sum or difference identity, write the following as an expression involving functions of x.
-$\sin \left( \frac { \pi } { 4 } - x \right)$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\sin x$ 
C)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
D)  $- \cos x$ 
A)  $\frac { \sqrt { 2 } } { 2 } \cos x - \frac { \sqrt { 2 } } { 2 } \sin x$ 
B)  $\sin x$ 
C)  $\frac { \sqrt { 2 } } { 2 } \cos x + \frac { \sqrt { 2 } } { 2 } \sin x$ 
D)  $- \cos x$

Question 20

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\sqrt { \frac { 1 + \cos 26 ^ { \circ } } { 2 } }$$

Accepted Answer

A)  $\sin 13 ^ { \circ }$ 
B)  $\cot 13 ^ { \circ }$ 
C)  $\tan 13 ^ { \circ }$ 
D)  $\cos 13$ 
A)  $\sin 13 ^ { \circ }$ 
B)  $\cot 13 ^ { \circ }$ 
C)  $\tan 13 ^ { \circ }$ 
D)  $\cos 13$

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\sin \theta = \frac { 3 } { 4 }$ and $\theta$ is in quadrant II.

Use an identity to write the expression as a single trigonometric function or as a single number. -sin (3ϴ - 17°)= cos (ϴ + 43°)

Find the exact value of the expression using the provided information. - $\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }$

Perform the transformation. - $\sin \theta \cos \theta \sec \theta \csc \theta$

Find the exact value of the expression using the provided information. -Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 1 } { 2 }$ , with s in quadrant III, and $\cos t = - \frac { 3 } { 5 }$ , with $\mathrm { t }$ in quadrant III.

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

Use a sum or difference identity to find the exact value. - $\sin ( x + 2 \pi )$

Use the fundamental identities to find the value of the trigonometric function. - $\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta$

Write the product as a sum or difference of trigonometric functions. - $\cos \left( - \frac { 7 \pi } { 12 } \right)$

Use a sum or difference identity to find the exact value. - $\cos 34 ^ { \circ } \sin 25 ^ { \circ }$

Verify that each equation is an identity. - $\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \theta$

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

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. -10 cos 26° cos 13°

Express the function as a trigonometric function of x. - $1 + \tan ^ { 2 } x = \frac { \cos ^ { 2 } x } { 1 - \cos ^ { 2 } x }$

Perform the transformation. - $\frac { \tan 170 ^ { \circ } - \tan 50 ^ { \circ } } { 1 + \tan 170 ^ { \circ } \tan 50 ^ { \circ } }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\tan \theta = \frac { 2 } { 5 }$ and $\theta$ is in quadrant III.

Verify that each equation is an identity. - $\frac { 1 - \sec \theta } { \tan \theta } + \frac { \tan \theta } { 1 - \sec \theta } = - 2 \csc \theta$

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

Using a sum or difference identity, write the following as an expression involving functions of x. - $\sin \left( \frac { \pi } { 4 } - x \right)$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\sqrt { \frac { 1 + \cos 26 ^ { \circ } } { 2 } }$

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 the fundamental identities to find the value of the trigonometric function. -Find tan⁡θ\tan \thetatanθ if sin⁡θ=34\sin \theta = \frac { 3 } { 4 }sinθ=43​ and θ\thetaθ is in quadrant II.

Use an identity to write the expression as a single trigonometric function or as a single number. -sin (3ϴ - 17°)= cos (ϴ + 43°)

Find the exact value of the expression using the provided information. - sin⁡25∘cos⁡35∘+cos⁡25∘sin⁡35∘\sin 25 ^ { \circ } \cos 35 ^ { \circ } + \cos 25 ^ { \circ } \sin 35 ^ { \circ }sin25∘cos35∘+cos25∘sin35∘

Perform the transformation. - sin⁡θcos⁡θsec⁡θcsc⁡θ\sin \theta \cos \theta \sec \theta \csc \thetasinθcosθsecθcscθ

Verify that the equation is an identity. - sin⁡(x+π2)=cos⁡x\sin \left( x + \frac { \pi } { 2 } \right) = \cos xsin(x+2π​)=cosx

Use a sum or difference identity to find the exact value. - sin⁡(x+2π)\sin ( x + 2 \pi )sin(x+2π)

Use the fundamental identities to find the value of the trigonometric function. - 1cot⁡2θ+sec⁡θcos⁡θ\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \thetacot2θ1​+secθcosθ

Write the product as a sum or difference of trigonometric functions. - cos⁡(−7π12)\cos \left( - \frac { 7 \pi } { 12 } \right)cos(−127π​)

Use a sum or difference identity to find the exact value. - cos⁡34∘sin⁡25∘\cos 34 ^ { \circ } \sin 25 ^ { \circ }cos34∘sin25∘

Verify that each equation is an identity. - cos⁡(4θ)=cos⁡4θ−6sin⁡2θcos⁡2θ+sin⁡4θ\cos ( 4 \theta ) = \cos ^ { 4 } \theta - 6 \sin ^ { 2 } \theta \cos ^ { 2 } \theta + \sin ^ { 4 } \thetacos(4θ)=cos4θ−6sin2θcos2θ+sin4θ

Verify that the equation is an identity. - cot⁡2u2=csc⁡u+cot⁡ucsc⁡u−cot⁡u\cot ^ { 2 } \frac { u } { 2 } = \frac { \csc u + \cot u } { \csc u - \cot u }cot22u​=cscu−cotucscu+cotu​