Question 1

Find the exact value of the composition.
-$\sin \left( \tan ^ { - 1 } ( 2 ) \right)$

Accepted Answer

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

Question 2

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = - \sin ( x + 6 ) - 5$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = - \sin ^ { - 1 } ( x - 5 ) + 6$ 
B)  $f ^ { - 1 } ( x ) = - \sin ^ { - 1 } ( x + 6 ) - 5$ 
C)  $f ^ { - 1 } ( x ) = - \sin ^ { - 1 } ( x + 5 ) - 6$ 
D)  $f ^ { - 1 } ( x ) = \sin ^ { - 1 } ( x + 5 ) - 6$ 
A)  $f ^ { - 1 } ( x ) = - \sin ^ { - 1 } ( x - 5 ) + 6$ 
B)  $f ^ { - 1 } ( x ) = - \sin ^ { - 1 } ( x + 6 ) - 5$ 
C)  $f ^ { - 1 } ( x ) = - \sin ^ { - 1 } ( x + 5 ) - 6$ 
D)  $f ^ { - 1 } ( x ) = \sin ^ { - 1 } ( x + 5 ) - 6$

Question 3

Express the product as a sum containing only sines or cosines.
-$\sin ( 4 \theta ) \cos ( 2 \theta )$

Accepted Answer

A)  $\sin \cos \left( 8 \theta ^ { 2 } \right)$ 
B)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \sin ( 2 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \cos ( 2 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 6 \theta ) - \cos ( 2 \theta ) ]$ 
A)  $\sin \cos \left( 8 \theta ^ { 2 } \right)$ 
B)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \sin ( 2 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \sin ( 6 \theta ) + \cos ( 2 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \cos ( 6 \theta ) - \cos ( 2 \theta ) ]$

Question 4

Establish the identity.
-$$\sec \left( \frac { \pi } { 2 } + u \right) = - \csc u$$

Accepted Answer

The answer of Establish the identity.
-\[\sec \left( \frac { \pi...

Question 5

Find the exact value of the expression.
-$\cos \left[ \tan ^ { - 1 } \left( \frac { \sqrt { 3 } } { 3 } \right) \right]$

Accepted Answer

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

Question 6

Establish the identity.
-$$\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)$$

Accepted Answer

The answer of Establish the identity.
-\[\csc ^ { 3 }...

Question 7

Establish the identity.
-$$\frac { \tan t } { 1 - \cot t } + \frac { 1 + \cot t } { \tan t } = \frac { \tan ^ { 2 } t + 2 - \csc ^ { 2 } t } { \tan t - 1 }$$

Accepted Answer

The answer of Establish the identity.
-\[\frac { \tan t }...

Question 8

Establish the identity.
-$$\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$$

Accepted Answer

The answer of Establish the identity.
-\[\tan ^ { 2 }...

Question 9

Find the exact solution of the equation.
-$$- 4 \tan ^ { - 1 } x = \pi$$

Accepted Answer

A)  $x = 0$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = - 1$ 
D)  $x = 1$ 
A)  $x = 0$ 
B)  $x = \frac { \pi } { 4 }$ 
C)  $x = - 1$ 
D)  $x = 1$

Question 10

Use a calculator to find the value of the expression rounded to two decimal places.
-$\cos ^ { - 1 } \left( - \frac { 4 } { 5 } \right)$

Accepted Answer

A)  -0.93 
B)  143.13 
C)  2.50 
D)  -53.13 
A)  -0.93 
B)  143.13 
C)  2.50 
D)  -53.13

Question 11

Find the exact value of the expression.
-$\sin 255 ^ { \circ }$

Accepted Answer

A)  $\frac { - \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$

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

В) $\frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$ 
C)  $\frac { - \sqrt { 2 } ( \sqrt { 3 } + 1 ) } { 4 }$ 
D)  $\frac { \sqrt { 2 } ( \sqrt { 3 } + 1 ) } { 4 }$

Question 12

Establish the identity.
-$$\frac { 1 - \sin t } { \cos t } = \frac { \cos t } { 1 + \sin t }$$

Accepted Answer

The answer of Establish the identity.
-\[\frac { 1 - \sin...

Question 13

Use a calculator to find the value of the expression in radian measure rounded to two decimal places.
-$\sec ^ { - 1 } \left( - \frac { 7 } { 3 } \right)$

Accepted Answer

A)  2.01 
B)  -2.01 
C)  0.50 
D)  1.13 
A)  2.01 
B)  -2.01 
C)  0.50 
D)  1.13

Question 14

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$$\sec \frac { \theta } { 2 } = \cos \frac { \theta } { 2 }$$

Accepted Answer

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

Question 15

Solve the problem.
-If $\sin \theta = - \frac { 4 } { 5 }$, and $\frac { 3 \pi } { 2 } < \theta < 2 \pi$, then find $\tan 2 \theta$.

Accepted Answer

A)  $\frac { 7 } { 24 }$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $- \frac { 7 } { 24 }$ 
D)  $\frac { 24 } { 7 }$ 
A)  $\frac { 7 } { 24 }$ 
B)  $- \frac { 24 } { 7 }$ 
C)  $- \frac { 7 } { 24 }$ 
D)  $\frac { 24 } { 7 }$

Question 16

Find the exact value of the expression.
-$\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

Accepted Answer

A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 13 } { 4 }$ 
D)  $- \frac { 1 } { 2 }$ 
A)  $\frac { \sqrt { 3 } } { 2 }$ 
B)  $- \frac { \sqrt { 3 } } { 2 }$ 
C)  $- \frac { 13 } { 4 }$ 
D)  $- \frac { 1 } { 2 }$

Question 17

Establish the identity.
-$$\sin \left( x - \frac { \pi } { 4 } \right) = \frac { \sqrt { 2 } } { 2 } ( \sin x - \cos x )$$

Accepted Answer

The answer of Establish the identity.
-\[\sin \left( x - \frac...

Question 18

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = 4 \tan ( 9 x )$$

Accepted Answer

A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 } \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 9 } \right)$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 \tan ( 9 \mathrm { x } ) }$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 4 \tan ^ { - 1 } ( 9 \mathrm { x } )$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 9 } \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right)$ 
A)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 } \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 9 } \right)$ 
B)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 4 \tan ( 9 \mathrm { x } ) }$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = 4 \tan ^ { - 1 } ( 9 \mathrm { x } )$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 9 } \tan ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right)$

Question 19

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal
place.
-$\sin ^ { 2 } x + 8 \sin x - 4 = 0$

Accepted Answer

A)  208.2°, 331.8° 
B)  28.2°, 151.8° 
C)  28.2°, 151.8°, 208.2°, 331.8° 
D)  No solution 
A)  208.2°, 331.8° 
B)  28.2°, 151.8° 
C)  28.2°, 151.8°, 208.2°, 331.8° 
D)  No solution

Question 20

Use a calculator to find the value of the expression in radian measure rounded to two decimal places.
-$\sec ^ { - 1 } \left( - \frac { 5 } { 4 } \right)$

Accepted Answer

A)  143.13 
B)  -0.93 
C)  2.50 
D)  -53.13 
A)  143.13 
B)  -0.93 
C)  2.50 
D)  -53.13

Find the exact value of the composition. - $\sin \left( \tan ^ { - 1 } ( 2 ) \right)$

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f. - $f ( x ) = - \sin ( x + 6 ) - 5$

Express the product as a sum containing only sines or cosines. - $\sin ( 4 \theta ) \cos ( 2 \theta )$

Establish the identity. - $\sec \left( \frac { \pi } { 2 } + u \right) = - \csc u$

Find the exact value of the expression. - $\cos \left[ \tan ^ { - 1 } \left( \frac { \sqrt { 3 } } { 3 } \right) \right]$

Establish the identity. - $\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)$

Establish the identity. - $\frac { \tan t } { 1 - \cot t } + \frac { 1 + \cot t } { \tan t } = \frac { \tan ^ { 2 } t + 2 - \csc ^ { 2 } t } { \tan t - 1 }$

Establish the identity. - $\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$

Find the exact solution of the equation. - $- 4 \tan ^ { - 1 } x = \pi$

Use a calculator to find the value of the expression rounded to two decimal places. - $\cos ^ { - 1 } \left( - \frac { 4 } { 5 } \right)$

Find the exact value of the expression. - $\sin 255 ^ { \circ }$

Establish the identity. - $\frac { 1 - \sin t } { \cos t } = \frac { \cos t } { 1 + \sin t }$

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - $\sec ^ { - 1 } \left( - \frac { 7 } { 3 } \right)$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\sec \frac { \theta } { 2 } = \cos \frac { \theta } { 2 }$

Solve the problem. -If $\sin \theta = - \frac { 4 } { 5 }$ , and $\frac { 3 \pi } { 2 } < \theta < 2 \pi$ , then find $\tan 2 \theta$ .

Find the exact value of the expression. - $\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }$

Establish the identity. - $\sin \left( x - \frac { \pi } { 4 } \right) = \frac { \sqrt { 2 } } { 2 } ( \sin x - \cos x )$

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f. - $f ( x ) = 4 \tan ( 9 x )$

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal place. - $\sin ^ { 2 } x + 8 \sin x - 4 = 0$

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - $\sec ^ { - 1 } \left( - \frac { 5 } { 4 } \right)$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Review

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Exam 6: Analytic Trigonometry

Find the exact value of the composition. - sin⁡(tan⁡−1(2))\sin \left( \tan ^ { - 1 } ( 2 ) \right)sin(tan−1(2))

Find the inverse function f−1\mathrm { f } ^ { - 1 }f−1 of the function f. - f(x)=−sin⁡(x+6)−5f ( x ) = - \sin ( x + 6 ) - 5f(x)=−sin(x+6)−5

Express the product as a sum containing only sines or cosines. - sin⁡(4θ)cos⁡(2θ)\sin ( 4 \theta ) \cos ( 2 \theta )sin(4θ)cos(2θ)

Establish the identity. - sec⁡(π2+u)=−csc⁡u\sec \left( \frac { \pi } { 2 } + u \right) = - \csc usec(2π​+u)=−cscu

Find the exact value of the expression. - cos⁡[tan⁡−1(33)]\cos \left[ \tan ^ { - 1 } \left( \frac { \sqrt { 3 } } { 3 } \right) \right]cos[tan−1(33​​)]

Establish the identity. - csc⁡3xtan⁡2x=csc⁡x(1+tan⁡2x)\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)csc3xtan2x=cscx(1+tan2x)

Establish the identity. - tan⁡t1−cot⁡t+1+cot⁡ttan⁡t=tan⁡2t+2−csc⁡2ttan⁡t−1\frac { \tan t } { 1 - \cot t } + \frac { 1 + \cot t } { \tan t } = \frac { \tan ^ { 2 } t + 2 - \csc ^ { 2 } t } { \tan t - 1 }1−cotttant​+tant1+cott​=tant−1tan2t+2−csc2t​

Establish the identity. - tan⁡2x=sec⁡2x−sin⁡2x−cos⁡2x\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } xtan2x=sec2x−sin2x−cos2x

Find the exact solution of the equation. - −4tan⁡−1x=π- 4 \tan ^ { - 1 } x = \pi−4tan−1x=π

Use a calculator to find the value of the expression rounded to two decimal places. - cos⁡−1(−45)\cos ^ { - 1 } \left( - \frac { 4 } { 5 } \right)cos−1(−54​)

Find the exact value of the expression. - sin⁡255∘\sin 255 ^ { \circ }sin255∘

Establish the identity. - 1−sin⁡tcos⁡t=cos⁡t1+sin⁡t\frac { 1 - \sin t } { \cos t } = \frac { \cos t } { 1 + \sin t }cost1−sint​=1+sintcost​

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - sec⁡−1(−73)\sec ^ { - 1 } \left( - \frac { 7 } { 3 } \right)sec−1(−37​)

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - sec⁡θ2=cos⁡θ2\sec \frac { \theta } { 2 } = \cos \frac { \theta } { 2 }sec2θ​=cos2θ​

Solve the problem. -If sin⁡θ=−45\sin \theta = - \frac { 4 } { 5 }sinθ=−54​ , and 3π2<θ<2π\frac { 3 \pi } { 2 } < \theta < 2 \pi23π​<θ<2π , then find tan⁡2θ\tan 2 \thetatan2θ .

Find the exact value of the expression. - sin⁡195∘cos⁡75∘−cos⁡195∘sin⁡75∘\sin 195 ^ { \circ } \cos 75 ^ { \circ } - \cos 195 ^ { \circ } \sin 75 ^ { \circ }sin195∘cos75∘−cos195∘sin75∘

Establish the identity. - sin⁡(x−π4)=22(sin⁡x−cos⁡x)\sin \left( x - \frac { \pi } { 4 } \right) = \frac { \sqrt { 2 } } { 2 } ( \sin x - \cos x )sin(x−4π​)=22​​(sinx−cosx)

Find the inverse function f−1\mathrm { f } ^ { - 1 }f−1 of the function f. - f(x)=4tan⁡(9x)f ( x ) = 4 \tan ( 9 x )f(x)=4tan(9x)

Use a graphing utility to solve the equation on the interval 0 0∘≤x<360∘0 ^ { \circ } \leq x < 360 ^ { \circ }0∘≤x<360∘ . Express the solution(s) rounded to one decimal place. - sin⁡2x+8sin⁡x−4=0\sin ^ { 2 } x + 8 \sin x - 4 = 0sin2x+8sinx−4=0

Use a calculator to find the value of the expression in radian measure rounded to two decimal places. - sec⁡−1(−54)\sec ^ { - 1 } \left( - \frac { 5 } { 4 } \right)sec−1(−45​)