Question 1

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
-$$\cot x \cos x + \sin x = \csc x$$

Accepted Answer

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

Question 2

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi )$$
-$\sin ^ { 2 } 2 x = 1$

Accepted Answer

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

Question 3

Use a sum or difference identity to find the exact value.
-$$\frac { \tan 5 ^ { \circ } + \tan 25 ^ { \circ } } { 1 - \tan 5 ^ { \circ } \tan 25 ^ { \circ } }$$

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 4

Factor the trigonometric expression and simplify.
-$$1 - \sin ^ { 3 } x$$

Accepted Answer

A)  $( 1 - \sin x ) \left( 1 - 2 \sin x + \sin ^ { 2 } x \right)$ 
B)  $( 1 - \sin x ) ^ { 3 }$ 
C)  $( 1 - \sin x ) \left( \cos x + \sin ^ { 2 } x \right)$ 
D)  $( 1 - \sin x ) \left( 1 + \sin x + \sin ^ { 2 } x \right)$ 
A)  $( 1 - \sin x ) \left( 1 - 2 \sin x + \sin ^ { 2 } x \right)$ 
B)  $( 1 - \sin x ) ^ { 3 }$ 
C)  $( 1 - \sin x ) \left( \cos x + \sin ^ { 2 } x \right)$ 
D)  $( 1 - \sin x ) \left( 1 + \sin x + \sin ^ { 2 } x \right)$

Question 5

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 3 \sqrt { 13 } } { 13 }$, with $\mathrm { s }$ in quadrant $\mathrm { IV }$, and $\sin \mathrm { t } = - \frac { \sqrt { 10 } } { 10 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $\frac { 11 } { 3 }$ 
B)  $- \frac { 7 } { 9 }$ 
C)  $- \frac { 11 } { 9 }$ 
D)  $- \frac { 11 } { 3 }$ 
A)  $\frac { 11 } { 3 }$ 
B)  $- \frac { 7 } { 9 }$ 
C)  $- \frac { 11 } { 9 }$ 
D)  $- \frac { 11 } { 3 }$

Question 6

Find the exact value of the real number y.
-$$y = \arccos \left( - \frac { \sqrt { 2 } } { 2 } \right)$$

Accepted Answer

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

Question 7

Give the exact value of the expression.
-$$\cos \left( \arcsin \frac { 1 } { 4 } \right)$$

Accepted Answer

A)  $\frac { 4 \sqrt { 15 } } { 15 }$ 
B)  $\frac { 2 \sqrt { 15 } } { 15 }$ 
C)  $\frac { \sqrt { 15 } } { 4 }$ 
D)  $\frac { \sqrt { 15 } } { 2 }$ 
A)  $\frac { 4 \sqrt { 15 } } { 15 }$ 
B)  $\frac { 2 \sqrt { 15 } } { 15 }$ 
C)  $\frac { \sqrt { 15 } } { 4 }$ 
D)  $\frac { \sqrt { 15 } } { 2 }$

Question 8

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

Accepted Answer

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

Question 9

Verify that the equation is an identity.
-$$\csc ( s + t ) = \frac { \operatorname { cscscsc } t } { \cot t + \cot s }$$

Accepted Answer

\[\begin{array} { l } 
\csc ( s + t ) =

Question 10

Write the following as an algebraic expression in u, u > 0.
-The optimal angle of elevation $\theta$ that a shot-putter should aim for in order to throw the greatest distance depends on the velocity $\mathrm { v }$ of the throw and the initial height $\mathrm { h }$ of the shot. One model for $\theta$ that achieves this maximum distance is $\theta = \sin ^ { - 1 } \left( \sqrt { \frac { \mathrm { v } ^ { 2 } } { 2 \mathrm { v } ^ { 2 } + 64 \mathrm {~h} } } \right)$. Suppose a shot-putter can consistently release the steel ball with velocity of 49 feet per second from an initial height $h$ of $5.1$ feet. What angle, to the nearest degree, will maximize the distance?

Accepted Answer

A)  $74 ^ { \circ }$ 
B)  $70 ^ { \circ }$ 
C)  $47 ^ { \circ }$ 
D)  $43 ^ { \circ }$ 
A)  $74 ^ { \circ }$ 
B)  $70 ^ { \circ }$ 
C)  $47 ^ { \circ }$ 
D)  $43 ^ { \circ }$

Question 11

Verify that each equation is an identity.
-$$\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )$$

Accepted Answer

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

Question 12

Find the exact value of the expression using the provided information.
-Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { \sqrt { 3 } } { 3 }$, with $\mathrm { s }$ in quadrant IV, and $\sin \mathrm { t } = - \frac { \sqrt { 5 } } { 6 }$, with $\mathrm { t }$ 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 13

Give the degree measure of $$\theta$$
-$$\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )$$

Accepted Answer

A)  $120 ^ { \circ }$ 
B)  $135 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $315 ^ { \circ }$ 
A)  $120 ^ { \circ }$ 
B)  $135 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $315 ^ { \circ }$

Question 14

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$2 \cos ^ { 2 } 4 x - 1$$

Accepted Answer

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

Question 15

Find the exact value of the expression using the provided information.
-Find $\tan ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$, with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$, with $t$ in quadrant II.

Accepted Answer

A)  $- \frac { 220 } { 221 }$ 
B)  $\frac { 220 } { 21 }$ 
C)  $\frac { 20 } { 3 }$ 
D)  $- \frac { 220 } { 171 }$ 
A)  $- \frac { 220 } { 221 }$ 
B)  $\frac { 220 } { 21 }$ 
C)  $\frac { 20 } { 3 }$ 
D)  $- \frac { 220 } { 171 }$

Question 16

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

Question 17

Use a sum or difference identity to find the exact value.
-$$\sin \frac { 11 \pi } { 12 }$$

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 18

Rewrite the following as a product of trigonometric functions.
-$\sin 1 ^ { \circ } - \sin 37 ^ { \circ }$

Accepted Answer

A)  $2 \sin 19 ^ { \circ } \sin \left( - 18 ^ { \circ } \right)$ 
B)  $2 \cos 19 ^ { \circ } \cos \left( - 18 ^ { \circ } \right)$ 
C)  $2 \cos 19 ^ { \circ } \sin \left( - 18 ^ { \circ } \right)$ 
D)  $2 \sin 19 ^ { \circ } \cos \left( - 18 ^ { \circ } \right)$ 
A)  $2 \sin 19 ^ { \circ } \sin \left( - 18 ^ { \circ } \right)$ 
B)  $2 \cos 19 ^ { \circ } \cos \left( - 18 ^ { \circ } \right)$ 
C)  $2 \cos 19 ^ { \circ } \sin \left( - 18 ^ { \circ } \right)$ 
D)  $2 \sin 19 ^ { \circ } \cos \left( - 18 ^ { \circ } \right)$

Question 19

Use the fundamental identities to find the value of the trigonometric function.
-Find $\sec \theta$ if $\cos \theta = \frac { 1 } { 4 }$ and $\sin \theta > 0$.

Accepted Answer

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

Question 20

Decide whether the expression is or is not an identity.
-$$\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0$$

Accepted Answer

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

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\cot x \cos x + \sin x = \csc x$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi )$ - $\sin ^ { 2 } 2 x = 1$

Use a sum or difference identity to find the exact value. - $\frac { \tan 5 ^ { \circ } + \tan 25 ^ { \circ } } { 1 - \tan 5 ^ { \circ } \tan 25 ^ { \circ } }$

Factor the trigonometric expression and simplify. - $1 - \sin ^ { 3 } x$

Find the exact value of the real number y. - $y = \arccos \left( - \frac { \sqrt { 2 } } { 2 } \right)$

Give the exact value of the expression. - $\cos \left( \arcsin \frac { 1 } { 4 } \right)$

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

Verify that the equation is an identity. - $\csc ( s + t ) = \frac { \operatorname { cscscsc } t } { \cot t + \cot s }$

Verify that each equation is an identity. - $\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )$

Give the degree measure of $\theta$ - $\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )$

Use an identity to write the expression as a single trigonometric function or as a single number. - $2 \cos ^ { 2 } 4 x - 1$

Find the exact value of the expression using the provided information. -Find $\tan ( s + t )$ given that $\cos s = - \frac { 5 } { 13 }$ , with $s$ in quadrant II, and $\sin t = \frac { 8 } { 17 }$ , with $t$ in quadrant II.

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 a sum or difference identity to find the exact value. - $\sin \frac { 11 \pi } { 12 }$

Rewrite the following as a product of trigonometric functions. - $\sin 1 ^ { \circ } - \sin 37 ^ { \circ }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sec \theta$ if $\cos \theta = \frac { 1 } { 4 }$ and $\sin \theta > 0$ .

Decide whether the expression is or is not an identity. - $\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0$

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

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - cot⁡xcos⁡x+sin⁡x=csc⁡x\cot x \cos x + \sin x = \csc xcotxcosx+sinx=cscx

Solve the equation for solutions in the interval [0, 2 [0,2π)[ 0,2 \pi )[0,2π) - sin⁡22x=1\sin ^ { 2 } 2 x = 1sin22x=1

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

Factor the trigonometric expression and simplify. - 1−sin⁡3x1 - \sin ^ { 3 } x1−sin3x

Find the exact value of the real number y. - y=arccos⁡(−22)y = \arccos \left( - \frac { \sqrt { 2 } } { 2 } \right)y=arccos(−22​​)

Give the exact value of the expression. - cos⁡(arcsin⁡14)\cos \left( \arcsin \frac { 1 } { 4 } \right)cos(arcsin41​)

Determine all solutions of the equation in radians. -Find tan⁡x2\tan \frac { x } { 2 }tan2x​ , given that tan⁡x=3\tan x = 3tanx=3 and xxx terminates in π<x<3π2\pi < x < \frac { 3 \pi } { 2 }π<x<23π​

Verify that the equation is an identity. - csc⁡(s+t)=cscscsc⁡tcot⁡t+cot⁡s\csc ( s + t ) = \frac { \operatorname { cscscsc } t } { \cot t + \cot s }csc(s+t)=cott+cotscscscsct​

Verify that each equation is an identity. - sin⁡(4u)=2sin⁡(2u)cos⁡(2u)\sin ( 4 u ) = 2 \sin ( 2 u ) \cos ( 2 u )sin(4u)=2sin(2u)cos(2u)

Give the degree measure of θ\thetaθ - θ=sec⁡−1(−2)\theta = \sec ^ { - 1 } ( - \sqrt { 2 } )θ=sec−1(−2​)

Use an identity to write the expression as a single trigonometric function or as a single number. - 2cos⁡24x−12 \cos ^ { 2 } 4 x - 12cos24x−1

Find the exact value of the expression using the provided information. -Find tan⁡(s+t)\tan ( s + t )tan(s+t) given that cos⁡s=−513\cos s = - \frac { 5 } { 13 }coss=−135​ , with sss in quadrant II, and sin⁡t=817\sin t = \frac { 8 } { 17 }sint=178​ , with ttt in quadrant II.

Determine all solutions of the equation in radians. -Find cos⁡x2\cos \frac { x } { 2 }cos2x​ , given that cos⁡x=14\cos x = \frac { 1 } { 4 }cosx=41​ and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​

Use a sum or difference identity to find the exact value. - sin⁡11π12\sin \frac { 11 \pi } { 12 }sin1211π​

Rewrite the following as a product of trigonometric functions. - sin⁡1∘−sin⁡37∘\sin 1 ^ { \circ } - \sin 37 ^ { \circ }sin1∘−sin37∘

Use the fundamental identities to find the value of the trigonometric function. -Find sec⁡θ\sec \thetasecθ if cos⁡θ=14\cos \theta = \frac { 1 } { 4 }cosθ=41​ and sin⁡θ>0\sin \theta > 0sinθ>0 .

Decide whether the expression is or is not an identity. - sec⁡2x−tan⁡2x=0\sqrt { \sec ^ { 2 } x - \tan ^ { 2 } x } = 0sec2x−tan2x​=0