Question 1

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) A

Question 2

Solve the problem.

-$$\cot ^ { 2 } \frac { u } { 2 } = \frac { \csc u + \cot u } { \csc u - \cot u }$$

Accepted Answer

$$\cot ^ { 2 } \frac { u } { 2 } = \frac { 1 } { \tan ^ { 2 } \frac { u } { 2 } } = \frac { 1 + \cos u } { 1 - \cos u } = \frac { \csc u + \cot u } { \csc u - \cot u }$$

Question 3

Write the following as an algebraic expression in u, u > 0.
-The figure shows a stationary spy satellite positioned 15,000 miles above the equator. What percel to the nearest tenth, of the equator can be seen from the satellite? The diameter of Earth is 7927 miles at the equator.

Accepted Answer

A)  $44.2 \%$ 
B)  $43.3 \%$ 
C)  $77.9 \%$ 
D)  $41.7 \%$ 
A)  $44.2 \%$ 
B)  $43.3 \%$ 
C)  $77.9 \%$ 
D)  $41.7 \%$ B

Question 4

Find the exact value by using a half-angle identity.
-$\cos 22.5 ^ { \circ }$

Accepted Answer

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

Question 5

Provide an appropriate response.
-Explain what is wrong with the following solution for the equation $\sin 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$.
$$\begin{array} { l } 
\sin 2 \theta = \sqrt { 3 } \
\sin \theta = \frac { \sqrt { 3 } } { 2 } \
\theta = \frac { \pi } { 3 } \text { or } \theta = \frac { 2 \pi } { 3 }
\end{array}$$

Accepted Answer

Answers will vary. @#LAT-DLM&

Question 6

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

Accepted Answer

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

Question 7

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$2 \sqrt { 3 } \sin 4 x = 3$

Accepted Answer

A)  $\{ 0 \}$ 
B)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$ 
A)  $\{ 0 \}$ 
B)  $\left\{ 0 , \frac { \pi } { 4 } , \pi \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } , \frac { 19 \pi } { 12 } \right\}$

Question 8

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$$\frac { 1 + \tan ^ { 2 } x } { \sec x }$$

Accepted Answer

A)  1 
B)  $- \sec x$ 
C)  $\sec x$ 
D)  $\csc x$ 
A)  1 
B)  $- \sec x$ 
C)  $\sec x$ 
D)  $\csc x$

Question 9

Solve the equation for solutions in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Round to the nearest degree.
-$\sin 2 \theta + \cos 2 \theta = 1$

Accepted Answer

A)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } \right\}$ 
A)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } \right\}$

Question 10

Give the degree measure of $$\theta$$
-$\theta = \arccos ( 0 )$

Accepted Answer

A)  $45 ^ { \circ }$ 
B)  $0 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $180 ^ { \circ }$ 
A)  $45 ^ { \circ }$ 
B)  $0 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $180 ^ { \circ }$

Question 11

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\cos 2 x = \sqrt { 2 } - \cos 2 x$

Accepted Answer

A)  $\varnothing$ 
B)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$ 
A)  $\varnothing$ 
B)  $\left\{ 0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 8 } , \frac { 9 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 15 \pi } { 8 } \right\}$

Question 12

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

Accepted Answer

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

Question 13

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

Question 14

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\sqrt { \frac { 1 - \cos 62 ^ { \circ } } { 2 } }$$

Accepted Answer

A)  $\cot 31 ^ { \circ }$ 
B)  $\cos 31 ^ { \circ }$ 
C)  $\sin 31 ^ { \circ }$ 
D)  $\tan 31 ^ { \circ }$ 
A)  $\cot 31 ^ { \circ }$ 
B)  $\cos 31 ^ { \circ }$ 
C)  $\sin 31 ^ { \circ }$ 
D)  $\tan 31 ^ { \circ }$

Question 15

Use identities to write the expression as a single function of x or $$\theta$$
-$$\cos \left( \theta + \frac { \pi } { 2 } \right)$$

Accepted Answer

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

Question 16

Solve the equation for solutions in the interval [0 $$\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$$ ). Round to the nearest degree.
-$$\cot \frac { \theta } { 3 } = 1$$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
D)  $\left\{ 135 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } , 315 ^ { \circ } \right\}$ 
C)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
D)  $\left\{ 135 ^ { \circ } \right\}$

Question 17

Use the fundamental identities to find the value of the trigonometric function.
-Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ is in quadrant IV.

Accepted Answer

A)  4 
B)  $- \sqrt { 15 }$ 
C)  $- \sqrt { 17 }$ 
D)  $- \frac { \sqrt { 15 } } { 15 }$ 
A)  4 
B)  $- \sqrt { 15 }$ 
C)  $- \sqrt { 17 }$ 
D)  $- \frac { \sqrt { 15 } } { 15 }$

Question 18

Find the exact value of the real number y.
-$y = \tan ^ { - 1 } ( 1 )$

Accepted Answer

A)  $\frac { 5 \pi } { 4 }$ 
B)  $\frac { 3 \pi } { 4 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { 7 \pi } { 4 }$ 
A)  $\frac { 5 \pi } { 4 }$ 
B)  $\frac { 3 \pi } { 4 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { 7 \pi } { 4 }$

Question 19

Determine all solutions of the equation in radians.
-Find $\sin \frac { x } { 2 }$, given that $\sin x = \frac { 1 } { 4 }$ 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 20

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$$( \sin x + \cos x ) ^ { 2 } = 1$$

Accepted Answer

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

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

Solve the problem. - $\cot ^ { 2 } \frac { u } { 2 } = \frac { \csc u + \cot u } { \csc u - \cot u }$

Write the following as an algebraic expression in u, u > 0. -The figure shows a stationary spy satellite positioned 15,000 miles above the equator. What percel to the nearest tenth, of the equator can be seen from the satellite? The diameter of Earth is 7927 miles at the equator.

Find the exact value by using a half-angle identity. - $\cos 22.5 ^ { \circ }$

Provide an appropriate response. -Explain what is wrong with the following solution for the equation $\sin 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$ . 2\theta= \theta= \theta= or \theta=

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

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $2 \sqrt { 3 } \sin 4 x = 3$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\frac { 1 + \tan ^ { 2 } x } { \sec x }$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\sin 2 \theta + \cos 2 \theta = 1$

Give the degree measure of $\theta$ - $\theta = \arccos ( 0 )$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\cos 2 x = \sqrt { 2 } - \cos 2 x$

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

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 an identity to write the expression as a single trigonometric function or as a single number. - $\sqrt { \frac { 1 - \cos 62 ^ { \circ } } { 2 } }$

Use identities to write the expression as a single function of x or $\theta$ - $\cos \left( \theta + \frac { \pi } { 2 } \right)$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\cot \frac { \theta } { 3 } = 1$

Use the fundamental identities to find the value of the trigonometric function. -Find $\tan \theta$ if $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ is in quadrant IV.

Find the exact value of the real number y. - $y = \tan ^ { - 1 } ( 1 )$

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

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $( \sin x + \cos x ) ^ { 2 } = 1$

Review of Basic Concepts

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

Filters

Exam 8: Trigonometric Identities and Equations

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

Solve the problem. - 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​

Write the following as an algebraic expression in u, u > 0. -The figure shows a stationary spy satellite positioned 15,000 miles above the equator. What percel to the nearest tenth, of the equator can be seen from the satellite? The diameter of Earth is 7927 miles at the equator.

Find the exact value by using a half-angle identity. - cos⁡22.5∘\cos 22.5 ^ { \circ }cos22.5∘

Provide an appropriate response. -Explain what is wrong with the following solution for the equation sin⁡2θ=3\sin 2 \theta = \sqrt { 3 }sin2θ=3​ in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2\theta= \theta= \theta= or \theta=

Use a sum or difference identity to find the exact value. - tan⁡7π12\tan \frac { 7 \pi } { 12 }tan127π​

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - 23sin⁡4x=32 \sqrt { 3 } \sin 4 x = 323​sin4x=3

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - 1+tan⁡2xsec⁡x\frac { 1 + \tan ^ { 2 } x } { \sec x }secx1+tan2x​

Solve the equation for solutions in the interval [0 [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Round to the nearest degree. - sin⁡2θ+cos⁡2θ=1\sin 2 \theta + \cos 2 \theta = 1sin2θ+cos2θ=1

Give the degree measure of θ\thetaθ - θ=arccos⁡(0)\theta = \arccos ( 0 )θ=arccos(0)

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - cos⁡2x=2−cos⁡2x\cos 2 x = \sqrt { 2 } - \cos 2 xcos2x=2​−cos2x

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 an identity to write the expression as a single trigonometric function or as a single number. - 1−cos⁡62∘2\sqrt { \frac { 1 - \cos 62 ^ { \circ } } { 2 } }21−cos62∘​​

Use identities to write the expression as a single function of x or θ\thetaθ - cos⁡(θ+π2)\cos \left( \theta + \frac { \pi } { 2 } \right)cos(θ+2π​)

Solve the equation for solutions in the interval [0 [0∘,360∘)\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)[0∘,360∘) ). Round to the nearest degree. - cot⁡θ3=1\cot \frac { \theta } { 3 } = 1cot3θ​=1

Use the fundamental identities to find the value of the trigonometric function. -Find tan⁡θ\tan \thetatanθ if cos⁡θ=14\cos \theta = \frac { 1 } { 4 }cosθ=41​ and θ\thetaθ is in quadrant IV.

Find the exact value of the real number y. - y=tan⁡−1(1)y = \tan ^ { - 1 } ( 1 )y=tan−1(1)

Determine all solutions of the equation in radians. -Find sin⁡x2\sin \frac { x } { 2 }sin2x​ , given that sin⁡x=14\sin x = \frac { 1 } { 4 }sinx=41​ and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ .

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - (sin⁡x+cos⁡x)2=1( \sin x + \cos x ) ^ { 2 } = 1(sinx+cosx)2=1