Question 1

Verify that each equation is an identity.
-$$\tan \theta \cdot \csc \theta = \sec \theta$$

Accepted Answer

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

Question 2

Solve the problem.
-A vibrating wire has a vertical displacement given by $y = 9 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )$. Assume $b$ and $t$ are constants. Write $y$ as a cosine function of $x$.

Accepted Answer

A)  $\mathrm { y } = \cos ( 2 \pi \mathrm { x } - \mathrm { bt } )$ 
B)  $y = 9 \cos ( 2 \pi x + b t )$ 
C)  $y = 9 \cos ( 2 \pi x - b t )$ 
D)  $y = 18 \cos ( 2 \pi x + b t )$ 
A)  $\mathrm { y } = \cos ( 2 \pi \mathrm { x } - \mathrm { bt } )$ 
B)  $y = 9 \cos ( 2 \pi x + b t )$ 
C)  $y = 9 \cos ( 2 \pi x - b t )$ 
D)  $y = 18 \cos ( 2 \pi x + b t )$

Question 3

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

Accepted Answer

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

Question 4

Use a sum or difference identity to find the exact value.
-$\cos ( \theta - \pi )$

Accepted Answer

A)  $\sin \theta$ 
B)  $\cos \theta$ 
C)  $- \cos \theta$ 
D)  $- \sin \theta$ 
A)  $\sin \theta$ 
B)  $\cos \theta$ 
C)  $- \cos \theta$ 
D)  $- \sin \theta$

Question 5

Perform the indicated operations and simplify the result so there are no quotients.
-$\sec \left( 6 \theta + 17 ^ { \circ } \right) = \csc \left( 2 \theta - 7 ^ { \circ } \right)$

Accepted Answer

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

Question 6

Use identities to fill in the blank with the appropriate trigonometric function name.
-Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 1 } { 2 }$, with $\mathrm { s }$ in quadrant $\mathrm { IV }$, and $\sin \mathrm { t } = \frac { 1 } { 4 }$, with $\mathrm { t }$ 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 7

Use identities to find the indicated value for each angle measure.
-$\sin \left( 360 ^ { \circ } + x \right)$

Accepted Answer

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

Question 8

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 9

Use a sum or difference identity to find the exact value.
-$$\sin \frac { 7 \pi } { 12 }$$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } + 1 } { 2 }$ 
D)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
C)  $\frac { \sqrt { 3 } + 1 } { 2 }$ 
D)  $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$

Question 10

Decide whether the expression is or is not an identity.
-Find $\cos \frac { \theta } { 2 }$, given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 10 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$

Question 11

Use the fundamental identities to simplify the expression.
-Find $\tan ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$, with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 8 } { 17 }$, with $\mathrm { t }$ in quadrant II.

Accepted Answer

A)  $- \frac { 220 } { 171 }$ 
B)  $- \frac { 140 } { 171 }$ 
C)  $\frac { 20 } { 3 }$ 
D)  $- \frac { 20 } { 3 }$ 
A)  $- \frac { 220 } { 171 }$ 
B)  $- \frac { 140 } { 171 }$ 
C)  $\frac { 20 } { 3 }$ 
D)  $- \frac { 20 } { 3 }$

Question 12

Using a sum or difference identity, write the following as an expression involving functions of x.
-$\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  $\sqrt { 3 }$ 
C)  $\frac { \sqrt { 2 } } { 4 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$ 
A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  $\sqrt { 3 }$ 
C)  $\frac { \sqrt { 2 } } { 4 }$ 
D)  $\frac { \sqrt { 3 } } { 3 }$

Question 13

Tell whether the statement is true or false.
-cos (ϴ + 90°)

Accepted Answer

A) -cos ϴ 
B) -sin ϴ 
C) cos ϴ 
D) sin ϴ 
A) -cos ϴ 
B) -sin ϴ 
C) cos ϴ 
D) sin ϴ

Question 14

Write the product as a sum or difference of trigonometric functions.
-Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 10 } } { 3 }$ and $\theta$ is in quadrant $\mathrm { I }$.

Accepted Answer

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

Question 15

Determine all solutions of the equation in radians.
-Find $\cos \frac { x } { 2 }$, given that $\cos x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
A)  $\frac { \sqrt { 6 } } { 4 }$ 
B)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 10 } } { 4 }$ 
D)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$

Question 16

Use identities to find the indicated value for each angle measure.
-Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 2 }$ and $\theta$ is in quadrant IV.

Accepted Answer

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

Question 17

Tell whether the statement is true or false.
-

Accepted Answer

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

Question 18

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 19

Use an identity to write the expression as a single trigonometric function or as a single number.
-$\cos \frac { 5 \pi } { 12 }$

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 20

Use the fundamental identities to simplify the expression.
-$$---+ \sin ^ { 2 } x = 1$$

Accepted Answer

A)  $\sin ^ { 2 } x$ 
B)  $\csc ^ { 2 } x$ 
C)  $\tan ^ { 2 } x$ 
D)  $\cos ^ { 2 } x$ 
A)  $\sin ^ { 2 } x$ 
B)  $\csc ^ { 2 } x$ 
C)  $\tan ^ { 2 } x$ 
D)  $\cos ^ { 2 } x$

Verify that each equation is an identity. - $\tan \theta \cdot \csc \theta = \sec \theta$

Solve the problem. -A vibrating wire has a vertical displacement given by $y = 9 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )$ . Assume $b$ and $t$ are constants. Write $y$ as a cosine function of $x$ .

Use a sum or difference identity to find the exact value. - $\cos ( \theta - \pi )$

Perform the indicated operations and simplify the result so there are no quotients. - $\sec \left( 6 \theta + 17 ^ { \circ } \right) = \csc \left( 2 \theta - 7 ^ { \circ } \right)$

Use identities to find the indicated value for each angle measure. - $\sin \left( 360 ^ { \circ } + x \right)$

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

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 12 }$

Decide whether the expression is or is not an identity. -Find $\cos \frac { \theta } { 2 }$ , given that $\sin \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$ .

Use the fundamental identities to simplify the expression. -Find $\tan ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 5 } { 13 }$ , with $\mathrm { s }$ in quadrant II, and $\sin \mathrm { t } = \frac { 8 } { 17 }$ , with $\mathrm { t }$ in quadrant II.

Using a sum or difference identity, write the following as an expression involving functions of x. - $\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }$

Tell whether the statement is true or false. -cos (ϴ + 90°)

Write the product as a sum or difference of trigonometric functions. -Find $\cot \theta$ if $\csc \theta = \frac { \sqrt { 10 } } { 3 }$ and $\theta$ is in quadrant $\mathrm { I }$ .

Determine all solutions of the equation in radians. -Find $\cos \frac { x } { 2 }$ , given that $\cos x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Use identities to find the indicated value for each angle measure. -Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 2 }$ and $\theta$ is in quadrant IV.

Tell whether the statement is true or false. -

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

Use an identity to write the expression as a single trigonometric function or as a single number. - $\cos \frac { 5 \pi } { 12 }$

Use the fundamental identities to simplify the expression. - $---+ \sin ^ { 2 } x = 1$

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. - tan⁡θ⋅csc⁡θ=sec⁡θ\tan \theta \cdot \csc \theta = \sec \thetatanθ⋅cscθ=secθ

Solve the problem. -A vibrating wire has a vertical displacement given by y=9(cos⁡2πxcos⁡bt+sin⁡2πxsin⁡bt)y = 9 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )y=9(cos2πxcosbt+sin2πxsinbt) . Assume bbb and ttt are constants. Write yyy as a cosine function of xxx .

Use a sum or difference identity to find the exact value. - cos⁡(θ−π)\cos ( \theta - \pi )cos(θ−π)

Perform the indicated operations and simplify the result so there are no quotients. - sec⁡(6θ+17∘)=csc⁡(2θ−7∘)\sec \left( 6 \theta + 17 ^ { \circ } \right) = \csc \left( 2 \theta - 7 ^ { \circ } \right)sec(6θ+17∘)=csc(2θ−7∘)

Use identities to find the indicated value for each angle measure. - sin⁡(360∘+x)\sin \left( 360 ^ { \circ } + x \right)sin(360∘+x)

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

Use a sum or difference identity to find the exact value. - sin⁡7π12\sin \frac { 7 \pi } { 12 }sin127π​

Decide whether the expression is or is not an identity. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that sin⁡θ=14\sin \theta = \frac { 1 } { 4 }sinθ=41​ and θ\thetaθ terminates in 0<θ<90∘0 < \theta < 90 ^ { \circ }0<θ<90∘ .

Using a sum or difference identity, write the following as an expression involving functions of x. - sin⁡22.5∘cos⁡22.5∘\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }sin22.5∘cos22.5∘