Question 1

Let $x = 2 \tan \theta$ , $0 \leq \theta < \pi / 2$) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

Accepted Answer

A)  $8 \tan ^ { 3 } \theta$ 
B)  $\frac { \cos ^ { 3 } \theta } { 8 \sin ^ { 2 } \theta }$ 
C)  $\sec ^ { 3 } \theta$ 
D)  $\frac { \cos ^ { 2 } \theta } { 4 \sin \theta }$ 
E)  $\frac { 8 \cos \theta } { \sin ^ { 2 } \theta }$ 
A)  $8 \tan ^ { 3 } \theta$ 
B)  $\frac { \cos ^ { 3 } \theta } { 8 \sin ^ { 2 } \theta }$ 
C)  $\sec ^ { 3 } \theta$ 
D)  $\frac { \cos ^ { 2 } \theta } { 4 \sin \theta }$ 
E)  $\frac { 8 \cos \theta } { \sin ^ { 2 } \theta }$

Question 2

Solve the given equation. $\tan \theta \sin \theta = - \sin \theta$

Accepted Answer

The answer of Solve the given equation. \(\tan \theta \sin...

Question 3

Solve the equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x + \sin x = 1$

Accepted Answer

A)  $\frac { \pi } { 6 } , \frac { \pi } { 2 }$ 
B)  $\frac { \pi } { 6 } , \frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 }$ 
D)  $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
E)  none 
A)  $\frac { \pi } { 6 } , \frac { \pi } { 2 }$ 
B)  $\frac { \pi } { 6 } , \frac { \pi } { 3 }$ 
C)  $\frac { \pi } { 2 } , \frac { 7 \pi } { 6 }$ 
D)  $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
E)  none

Question 4

Find $\sin 2 x$ , given that $\sin x = \frac { 3 } { 5 }$
And $x$ Is in quadrant II

Accepted Answer

A)  $- \frac { 24 } { 25 }$ 
B)  $\frac { 24 } { 7 }$ 
C)  $ \frac{4 \sqrt{6}}{7} $ 
D)  $\frac { \sqrt { 6 } } { 25 }$ 
E)  $- \frac { 4 } { 7 }$ 
A)  $- \frac { 24 } { 25 }$ 
B)  $\frac { 24 } { 7 }$ 
C)  $ \frac{4 \sqrt{6}}{7} $ 
D)  $\frac { \sqrt { 6 } } { 25 }$ 
E)  $- \frac { 4 } { 7 }$

Question 5

Solve the equation in the interval $[ 0,2 \pi )$ 
. $2 \cos ^ { 2 } x + \sin x = 1$

Accepted Answer

The answer of Solve the equation in the interval \([...

Question 6

Find the exact value of $\sin 2 x$ given that $\sec x = \frac { 3 } { 2 } , \quad \csc y = 3$
, and x and y are in quadrant I

Accepted Answer

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

Question 7

Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$

Accepted Answer

The answer of Find the exact value. \(\sin \left( \sin...

Question 8

Find the exact value. $\operatorname { COS } \frac { 5 \pi } { 12 }$

Accepted Answer

A)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
B)  $\frac { 5 \pi } { 12 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
E)  none 
A)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
B)  $\frac { 5 \pi } { 12 }$ 
C)  $\frac { \sqrt { 6 } } { 4 }$ 
D)  $\frac { \sqrt { 3 } } { 2 }$ 
E)  none

Question 9

Find the exact value. $\left( \cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 } \right) ^ { 2 }$

Accepted Answer

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

Question 10

Find the exact value. $\cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 }$

Accepted Answer

The answer of Find the exact value. \(\cos \frac {...

Question 11

Let $x = 2 \tan \theta$ , $0 \leq \theta < \frac { \pi } { 2 }$) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

Accepted Answer

A)  $8 \tan ^ { 3 } \theta$ 
B)  $\frac { \cos ^ { 3 } \theta } { 8 \sin ^ { 2 } \theta }$ 
C)  $\sec ^ { 3 } \theta$ 
D)  $\frac { \cos ^ { 2 } \theta } { 4 \sin \theta }$ 
E)  $\frac { 8 \cos \theta } { \sin ^ { 2 } \theta }$ 
A)  $8 \tan ^ { 3 } \theta$ 
B)  $\frac { \cos ^ { 3 } \theta } { 8 \sin ^ { 2 } \theta }$ 
C)  $\sec ^ { 3 } \theta$ 
D)  $\frac { \cos ^ { 2 } \theta } { 4 \sin \theta }$ 
E)  $\frac { 8 \cos \theta } { \sin ^ { 2 } \theta }$

Question 12

Find the exact value. $\cos \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$

Accepted Answer

A)  $13 / 6$ 
B)  $5 / 6$ 
C)  $13 / 119$ 
D)  $169 / 119$ 
E)  $119 / 169$ 
A)  $13 / 6$ 
B)  $5 / 6$ 
C)  $13 / 119$ 
D)  $169 / 119$ 
E)  $119 / 169$

Question 13

Write $ \frac{sec ~x - \cos ~x}{\tan~ x}$in terms of $\sin x$ and $\cos x$then simplify

Accepted Answer

A)  $1$ 
B)  $\sin ^ { 2 } x$ 
C)  $\sin x$ 
D)  $\cos x$ 
E)  $- \cos ^ { 2 } x$ 
A)  $1$ 
B)  $\sin ^ { 2 } x$ 
C)  $\sin x$ 
D)  $\cos x$ 
E)  $- \cos ^ { 2 } x$

Question 14

$\cos \left( x + \frac { \pi } { 2 } \right) + \sin \left( x + \frac { \pi } { 2 } \right) = \cos x - \sin x$

Accepted Answer

A) True 
 B)False

Question 15

Solve the equation in the interval $[ 0,2 \pi )$ 
. $\csc ^ { 2 } x - 4 = 0$

Accepted Answer

The answer of Solve the equation in the interval \([...

Question 16

Find the exact value. $\cos 15 ^ { \circ }$

Accepted Answer

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

Question 17

Write $\left( \sin ^ { 2 } x \right) \left( 1 + \cot ^ { 2 } x \right)$ 
in terms of $\sin x$ 
and $\cos x$ 
then simplify.

Accepted Answer

The answer of Write \(\left( \sin ^ { 2 }...

Question 18

Solve the equation in the interval $[ 0,2 \pi )$ . $\sin x = \cos x$

Accepted Answer

A)  $x = \frac { \pi } { 4 }$ , $x = \frac { 5 \pi } { 4 }$ 
B)  $x = \frac { 3 \pi } { 4 }$, $x = \frac { 5 \pi } { 4 }$ 
C)  $x = \frac { \pi } { 4 }$, $x = \frac { 7 \pi } { 4 }$ 
D)  $x = \frac { 5 \pi } { 4 }$, $x = \frac { 7 \pi } { 4 }$ 
E)  none 
A)  $x = \frac { \pi } { 4 }$ , $x = \frac { 5 \pi } { 4 }$ 
B)  $x = \frac { 3 \pi } { 4 }$, $x = \frac { 5 \pi } { 4 }$ 
C)  $x = \frac { \pi } { 4 }$, $x = \frac { 7 \pi } { 4 }$ 
D)  $x = \frac { 5 \pi } { 4 }$, $x = \frac { 7 \pi } { 4 }$ 
E)  none

Question 19

Find $\tan 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ and $x$ is in quadrant II

Accepted Answer

A)  $- 24$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $\frac { 4 \sqrt { 6 } } { 7 }$ 
D)  $\frac { \sqrt { 6 } } { 7 }$ 
E)  $- \frac { 4 } { 7 }$ 
A)  $- 24$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $\frac { 4 \sqrt { 6 } } { 7 }$ 
D)  $\frac { \sqrt { 6 } } { 7 }$ 
E)  $- \frac { 4 } { 7 }$

Question 20

Find the exact value. $\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$

Accepted Answer

The answer of Find the exact value. \(\sin \left( 2...

Let $x = 2 \tan \theta$ , $0 \leq \theta < \pi / 2$ ) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

Solve the given equation. $\tan \theta \sin \theta = - \sin \theta$

Solve the equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x + \sin x = 1$

Find $\sin 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ And $x$ Is in quadrant II

Solve the equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x + \sin x = 1$

Find the exact value of $\sin 2 x$ given that $\sec x = \frac { 3 } { 2 } , \quad \csc y = 3$ , and x and y are in quadrant I

Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$

Find the exact value. $\operatorname { COS } \frac { 5 \pi } { 12 }$

Find the exact value. $\left( \cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 } \right) ^ { 2 }$

Find the exact value. $\cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 }$

Let $x = 2 \tan \theta$ , $0 \leq \theta < \frac { \pi } { 2 }$ ) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

Find the exact value. $\cos \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$

Write $\frac{sec ~x - \cos ~x}{\tan~ x}$ in terms of $\sin x$ and $\cos x$ then simplify

$\cos \left( x + \frac { \pi } { 2 } \right) + \sin \left( x + \frac { \pi } { 2 } \right) = \cos x - \sin x$

Solve the equation in the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } x - 4 = 0$

Find the exact value. $\cos 15 ^ { \circ }$

Write $\left( \sin ^ { 2 } x \right) \left( 1 + \cot ^ { 2 } x \right)$ in terms of $\sin x$ and $\cos x$ then simplify.

Solve the equation in the interval $[ 0,2 \pi )$ . $\sin x = \cos x$

Find $\tan 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ and $x$ is in quadrant II

Find the exact value. $\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$

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Exam 8: Polar Coordinates and Parametric Equations

Let x=2tan⁡θx = 2 \tan \thetax=2tanθ , 0≤θ<π/20 \leq \theta < \pi / 20≤θ<π/2 ) Simplify the expression. 1x24+x2\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }x24+x2​1​

Solve the given equation. tan⁡θsin⁡θ=−sin⁡θ\tan \theta \sin \theta = - \sin \thetatanθsinθ=−sinθ

Solve the equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2cos⁡2x+sin⁡x=12 \cos ^ { 2 } x + \sin x = 12cos2x+sinx=1

Find sin⁡2x\sin 2 xsin2x , given that sin⁡x=35\sin x = \frac { 3 } { 5 }sinx=53​ And xxx Is in quadrant II

Solve the equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . 2cos⁡2x+sin⁡x=12 \cos ^ { 2 } x + \sin x = 12cos2x+sinx=1

Find the exact value of sin⁡2x\sin 2 xsin2x given that sec⁡x=32,csc⁡y=3\sec x = \frac { 3 } { 2 } , \quad \csc y = 3secx=23​,cscy=3 , and x and y are in quadrant I

Find the exact value. sin⁡(sin⁡−135−cos⁡−112)\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)sin(sin−153​−cos−121​)

Find the exact value. COS⁡5π12\operatorname { COS } \frac { 5 \pi } { 12 }COS125π​

Find the exact value. (cos⁡3π8+cos⁡π8)2\left( \cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 } \right) ^ { 2 }(cos83π​+cos8π​)2

Find the exact value. cos⁡3π8+cos⁡π8\cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 }cos83π​+cos8π​

Let x=2tan⁡θx = 2 \tan \thetax=2tanθ , 0≤θ<π20 \leq \theta < \frac { \pi } { 2 }0≤θ<2π​ ) Simplify the expression. 1x24+x2\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }x24+x2​1​

Find the exact value. cos⁡(2tan⁡−1512)\cos \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)cos(2tan−1125​)

Write sec x−cos⁡ xtan⁡ x \frac{sec ~x - \cos ~x}{\tan~ x}tan xsec x−cos x​ in terms of sin⁡x\sin xsinx and cos⁡x\cos xcosx then simplify

cos⁡(x+π2)+sin⁡(x+π2)=cos⁡x−sin⁡x\cos \left( x + \frac { \pi } { 2 } \right) + \sin \left( x + \frac { \pi } { 2 } \right) = \cos x - \sin xcos(x+2π​)+sin(x+2π​)=cosx−sinx

Solve the equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . csc⁡2x−4=0\csc ^ { 2 } x - 4 = 0csc2x−4=0

Find the exact value. cos⁡15∘\cos 15 ^ { \circ }cos15∘

Write (sin⁡2x)(1+cot⁡2x)\left( \sin ^ { 2 } x \right) \left( 1 + \cot ^ { 2 } x \right)(sin2x)(1+cot2x) in terms of sin⁡x\sin xsinx and cos⁡x\cos xcosx then simplify.

Solve the equation in the interval [0,2π)[ 0,2 \pi )[0,2π) . sin⁡x=cos⁡x\sin x = \cos xsinx=cosx

Find tan⁡2x\tan 2 xtan2x , given that sin⁡x=35\sin x = \frac { 3 } { 5 }sinx=53​ and xxx is in quadrant II

Find the exact value. sin⁡(2tan⁡−1512)\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)sin(2tan−1125​)

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

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Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Sequences and Series

Counting and Probability

Filters

Let $x = 2 \tan \theta$ , $0 \leq \theta < \pi / 2$ ) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

Solve the given equation. $\tan \theta \sin \theta = - \sin \theta$

Solve the equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x + \sin x = 1$

Find $\sin 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ And $x$ Is in quadrant II

Solve the equation in the interval $[ 0,2 \pi )$ . $2 \cos ^ { 2 } x + \sin x = 1$

Find the exact value of $\sin 2 x$ given that $\sec x = \frac { 3 } { 2 } , \quad \csc y = 3$ , and x and y are in quadrant I

Find the exact value. $\sin \left( \sin ^ { - 1 } \frac { 3 } { 5 } - \cos ^ { - 1 } \frac { 1 } { 2 } \right)$

Find the exact value. $\operatorname { COS } \frac { 5 \pi } { 12 }$

Find the exact value. $\left( \cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 } \right) ^ { 2 }$

Find the exact value. $\cos \frac { 3 \pi } { 8 } + \cos \frac { \pi } { 8 }$

Let $x = 2 \tan \theta$ , $0 \leq \theta < \frac { \pi } { 2 }$ ) Simplify the expression. $\frac { 1 } { x ^ { 2 } \sqrt { 4 + x ^ { 2 } } }$

Find the exact value. $\cos \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$

Write $\frac{sec ~x - \cos ~x}{\tan~ x}$ in terms of $\sin x$ and $\cos x$ then simplify

$\cos \left( x + \frac { \pi } { 2 } \right) + \sin \left( x + \frac { \pi } { 2 } \right) = \cos x - \sin x$

Solve the equation in the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } x - 4 = 0$

Find the exact value. $\cos 15 ^ { \circ }$

Write $\left( \sin ^ { 2 } x \right) \left( 1 + \cot ^ { 2 } x \right)$ in terms of $\sin x$ and $\cos x$ then simplify.

Solve the equation in the interval $[ 0,2 \pi )$ . $\sin x = \cos x$

Find $\tan 2 x$ , given that $\sin x = \frac { 3 } { 5 }$ and $x$ is in quadrant II

Find the exact value. $\sin \left( 2 \tan ^ { - 1 } \frac { 5 } { 12 } \right)$