Question 1

Solve the equation for exact solutions over the interval [0, 2π).
-$\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$

Accepted Answer

A)  $\left\{ \frac { \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } \right\}$ 
D)  $\varnothing$ 
A)  $\left\{ \frac { \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } \right\}$ 
D)  $\varnothing$

Question 2

Write the following as an algebraic expression in u, u > 0.
-$\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)$

Accepted Answer

A)  $\frac { \sqrt { u ^ { 2 } + 3 } } { u ^ { 2 } + 3 }$ 
B)  $3 u$ 
C)  $\frac { 3 } { \mathrm { u } }$ 
D)  $\frac { u \sqrt { u ^ { 2 } + 9 } } { u ^ { 2 } + 9 }$ 
A)  $\frac { \sqrt { u ^ { 2 } + 3 } } { u ^ { 2 } + 3 }$ 
B)  $3 u$ 
C)  $\frac { 3 } { \mathrm { u } }$ 
D)  $\frac { u \sqrt { u ^ { 2 } + 9 } } { u ^ { 2 } + 9 }$

Question 3

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)

Question 4

Use identities to find the indicated value for each angle measure.
-$\tan \theta = \frac { 15 } { 8 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $- \frac { 161 } { 289 }$ 
B)  $\frac { 240 } { 289 }$ 
C)  $\frac { 161 } { 289 }$ 
D)  $- \frac { 240 } { 289 }$ 
A)  $- \frac { 161 } { 289 }$ 
B)  $\frac { 240 } { 289 }$ 
C)  $\frac { 161 } { 289 }$ 
D)  $- \frac { 240 } { 289 }$

Question 5

Factor the trigonometric expression and simplify.
-$$\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$$

Accepted Answer

A)  $\cot ^ { 2 } x + 1$ 
B)  1 
C)  $\sin ^ { 2 } x + 1$ 
D)  $\cot ^ { 2 } x - 1$ 
A)  $\cot ^ { 2 } x + 1$ 
B)  1 
C)  $\sin ^ { 2 } x + 1$ 
D)  $\cot ^ { 2 } x - 1$

Question 6

Use a calculator to give the value to the nearest degree.
-$\theta = \sin ^ { - 1 } ( 0.2079 )$

Accepted Answer

A)  $14 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $15 ^ { \circ }$ 
D)  $10 ^ { \circ }$ 
A)  $14 ^ { \circ }$ 
B)  $12 ^ { \circ }$ 
C)  $15 ^ { \circ }$ 
D)  $10 ^ { \circ }$

Question 7

Use the fundamental identities to simplify the expression.
-$\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Accepted Answer

A)  $- 1$ 
B)  1 
C)  $\tan ^ { 2 } x$ 
D)  $\cos ^ { 2 } x + 1$ 
A)  $- 1$ 
B)  1 
C)  $\tan ^ { 2 } x$ 
D)  $\cos ^ { 2 } x + 1$

Question 8

Find the exact value of the expression using the provided information.
-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)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$ 
A)  $\frac { \sqrt { 186 } - \sqrt { 15 } } { 18 }$ 
B)  $\frac { \sqrt { 55 } + \sqrt { 15 } } { 18 }$ 
C)  $\frac { \sqrt { 55 } - \sqrt { 15 } } { 18 }$ 
D)  $\frac { \sqrt { 186 } + \sqrt { 15 } } { 18 }$

Question 9

Solve the problem.
-The formula for the up-and-down motion of a weight on a spring is given by $S = a \sin \left( \frac { \sqrt { k } } { m } \right) t$
where $a$ is the radius of the circle, $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for $t$.

Accepted Answer

A)  $t = \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \arcsin \left( \frac { \mathrm { s } } { \mathrm { a } } \right)$ 
B)  $t = \frac { m } { \sqrt { k } } \arcsin \left( \frac { S } { a } \right)$ 
C)  $t = \frac { \sqrt { k } } { m } \sin \left( \frac { a } { S } \right)$ 
D)  $t = \frac { m } { \sqrt { k } } \sin \left( \frac { \mathrm { S } } { \mathrm { a } } \right)$ 
A)  $t = \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \arcsin \left( \frac { \mathrm { s } } { \mathrm { a } } \right)$ 
B)  $t = \frac { m } { \sqrt { k } } \arcsin \left( \frac { S } { a } \right)$ 
C)  $t = \frac { \sqrt { k } } { m } \sin \left( \frac { a } { S } \right)$ 
D)  $t = \frac { m } { \sqrt { k } } \sin \left( \frac { \mathrm { S } } { \mathrm { a } } \right)$

Question 10

Solve the problem.
-A vibrating wire has a vertical displacement given by $y = 4 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )$. Assume $b$ and $t$ are constants. Write $y$ as a cosine function of $x$.

Accepted Answer

A)  $y = 4 \cos ( 2 \pi x - b t )$ 
B)  $y = 4 \cos ( 2 \pi x + b t )$ 
C)  $y = 8 \cos ( 2 \pi x + b t )$ 
D)  $y = \cos ( 2 \pi x - b t )$ 
A)  $y = 4 \cos ( 2 \pi x - b t )$ 
B)  $y = 4 \cos ( 2 \pi x + b t )$ 
C)  $y = 8 \cos ( 2 \pi x + b t )$ 
D)  $y = \cos ( 2 \pi x - b t )$

Question 11

Verify that each equation is an identity.
-$$\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$$

Accepted Answer

\[\begin{array} { l } 
\cot x \sec ^ { 4

Question 12

Complete the sentence so the result is an identity. Let x be any real number.
-$$ـــــــ + \sin ^ { 2 } x = 1$$

Accepted Answer

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

Question 13

Use an identity to write the expression as a single trigonometric function or as a single number.
-$$\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 1 - \cos 22 ^ { \circ } } }$$

Accepted Answer

A)  $\sin 11 ^ { \circ }$ 
B)  $\tan 11 ^ { \circ }$ 
C)  $\cot 11 ^ { \circ }$ 
D)  $\cos 11 ^ { \circ }$ 
A)  $\sin 11 ^ { \circ }$ 
B)  $\tan 11 ^ { \circ }$ 
C)  $\cot 11 ^ { \circ }$ 
D)  $\cos 11 ^ { \circ }$

Question 14

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

Accepted Answer

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

Question 15

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-$\frac { \csc x \cot x } { \sec x }$

Accepted Answer

A)  $\sec ^ { 2 } x$ 
B)  $\csc ^ { 2 } x$ 
C)  $\cot ^ { 2 } x$ 
D)  1 
A)  $\sec ^ { 2 } x$ 
B)  $\csc ^ { 2 } x$ 
C)  $\cot ^ { 2 } x$ 
D)  1

Question 16

Verify that each 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 each equation is an identity.
-\[\cot...

Question 17

Solve the problem.
-A wooden block sitting at rest on a flat, level board will just begin to slide when the board is tilted up to a critical angle of inclination, $\theta _ { \mathrm { S } }$. An experiment that determines this critical angle can be used to find the coefficient of static friction, $\mu _ { \mathrm { S } }$, for the block on the board by using the relation $\mu _ { \mathrm { S } } = \frac { \sin \theta _ { \mathrm { S } } } { \cos \theta _ { \mathrm { S } } }$. Rewrite this relation using a quotient identity.

Accepted Answer

A)  $\mu _ { \mathrm { S } } = \cot \theta _ { \mathrm { S } }$ 
B)  $\mu _ { \mathrm { S } } = \sec \theta _ { \mathrm { S } }$ 
C)  $\mu _ { \mathrm { S } } = \tan ^ { 2 } \theta _ { \mathrm { S } }$ 
D)  $\mu _ { \mathrm { S } } = \tan \theta _ { \mathrm { S } }$ 
A)  $\mu _ { \mathrm { S } } = \cot \theta _ { \mathrm { S } }$ 
B)  $\mu _ { \mathrm { S } } = \sec \theta _ { \mathrm { S } }$ 
C)  $\mu _ { \mathrm { S } } = \tan ^ { 2 } \theta _ { \mathrm { S } }$ 
D)  $\mu _ { \mathrm { S } } = \tan \theta _ { \mathrm { S } }$

Question 18

Use the fundamental identities to simplify the expression.
-$\frac { \tan \theta } { \sec \theta }$

Accepted Answer

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

Question 19

Solve the equation for x, where x is restricted to the given interval.
-$y = - 5 \sin 8 x$, for $x$ in $\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right]$

Accepted Answer

A)  $x = \frac { 1 } { 8 } \arcsin \frac { y } { 5 }$ 
B)  $x = - \frac { 1 } { 5 } \arcsin \frac { y } { 8 }$ 
C)  $x = \frac { 1 } { 8 } \arcsin \left( - \frac { y } { 5 } \right)$ 
D)  $x = \frac { 1 } { 5 } \arcsin \left( - \frac { y } { 8 } \right)$ 
A)  $x = \frac { 1 } { 8 } \arcsin \frac { y } { 5 }$ 
B)  $x = - \frac { 1 } { 5 } \arcsin \frac { y } { 8 }$ 
C)  $x = \frac { 1 } { 8 } \arcsin \left( - \frac { y } { 5 } \right)$ 
D)  $x = \frac { 1 } { 5 } \arcsin \left( - \frac { y } { 8 } \right)$

Question 20

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 $\mathrm { IV }$, and $\sin \mathrm { t } = - \frac { \sqrt { 5 } } { 6 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

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

Solve the equation for exact solutions over the interval [0, 2π). - $\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$

Write the following as an algebraic expression in u, u > 0. - $\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)$

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

Use identities to find the indicated value for each angle measure. - $\tan \theta = \frac { 15 } { 8 } , \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

Factor the trigonometric expression and simplify. - $\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } x$

Use a calculator to give the value to the nearest degree. - $\theta = \sin ^ { - 1 } ( 0.2079 )$

Use the fundamental identities to simplify the expression. - $\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)$

Find the exact value of the expression using the provided information. -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.

Solve the problem. -A vibrating wire has a vertical displacement given by $y = 4 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )$ . Assume $b$ and $t$ are constants. Write $y$ as a cosine function of $x$ .

Verify that each equation is an identity. - $\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$

Complete the sentence so the result is an identity. Let x be any real number. - $ـــــــ + \sin ^ { 2 } x = 1$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 1 - \cos 22 ^ { \circ } } }$

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

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\frac { \csc x \cot x } { \sec x }$

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

Use the fundamental identities to simplify the expression. - $\frac { \tan \theta } { \sec \theta }$

Solve the equation for x, where x is restricted to the given interval. - $y = - 5 \sin 8 x$ , for $x$ in $\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right]$

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

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Exam 7: Trigonometric Identities and Equations

Solve the equation for exact solutions over the interval [0, 2π). - sec⁡2x−2=tan⁡2x\sec ^ { 2 } x - 2 = \tan ^ { 2 } xsec2x−2=tan2x

Write the following as an algebraic expression in u, u > 0. - tan⁡(arcsec⁡u2+9u)\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)tan(arcsecuu2+9​​)

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

Use identities to find the indicated value for each angle measure. - tan⁡θ=158,π<θ<3π2\tan \theta = \frac { 15 } { 8 } , \pi < \theta < \frac { 3 \pi } { 2 } \quadtanθ=815​,π<θ<23π​ Find sin⁡(2θ)\sin ( 2 \theta )sin(2θ)

Factor the trigonometric expression and simplify. - sin⁡2x+sin⁡2xcot⁡2x\sin ^ { 2 } x + \sin ^ { 2 } x \cot ^ { 2 } xsin2x+sin2xcot2x

Use a calculator to give the value to the nearest degree. - θ=sin⁡−1(0.2079)\theta = \sin ^ { - 1 } ( 0.2079 )θ=sin−1(0.2079)

Use the fundamental identities to simplify the expression. - sin⁡2x(cot⁡2x+1)\sin ^ { 2 } x \left( \cot ^ { 2 } x + 1 \right)sin2x(cot2x+1)

Solve the problem. -A vibrating wire has a vertical displacement given by y=4(cos⁡2πxcos⁡bt+sin⁡2πxsin⁡bt)y = 4 ( \cos 2 \pi x \cos b t + \sin 2 \pi x \sin b t )y=4(cos2πxcosbt+sin2πxsinbt) . Assume bbb and ttt are constants. Write yyy as a cosine function of xxx .

Verify that each equation is an identity. - cot⁡xsec⁡4x=cot⁡x+2tan⁡x+tan⁡3x\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } xcotxsec4x=cotx+2tanx+tan3x

Complete the sentence so the result is an identity. Let x be any real number. - ـــــــ+sin⁡2x=1ـــــــ + \sin ^ { 2 } x = 1ـــــــ+sin2x=1

Use an identity to write the expression as a single trigonometric function or as a single number. - 1+cos⁡22∘1−cos⁡22∘\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 1 - \cos 22 ^ { \circ } } }1−cos22∘1+cos22∘​​

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - 1+tan⁡2x=cos⁡2x1−cos⁡2x1 + \tan ^ { 2 } x = \frac { \cos ^ { 2 } x } { 1 - \cos ^ { 2 } x }1+tan2x=1−cos2xcos2x​

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - csc⁡xcot⁡xsec⁡x\frac { \csc x \cot x } { \sec x }secxcscxcotx​

Verify that each 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​

Use the fundamental identities to simplify the expression. - tan⁡θsec⁡θ\frac { \tan \theta } { \sec \theta }secθtanθ​

Solve the equation for x, where x is restricted to the given interval. - y=−5sin⁡8xy = - 5 \sin 8 xy=−5sin8x , for xxx in [−π16,π16]\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right][−16π​,16π​]