Question 1

Complete the identity.
-$\frac { \cos ( \alpha - \beta ) } { \sin \alpha \cos \beta } = ?$

Accepted Answer

A)  $\cos \alpha - \sin \beta$ 
B)  $\cot \alpha - 1$ 
C)  $\cot \alpha + \tan \beta$ 
D)  $\cos \alpha + \sin \beta$ 
A)  $\cos \alpha - \sin \beta$ 
B)  $\cot \alpha - 1$ 
C)  $\cot \alpha + \tan \beta$ 
D)  $\cos \alpha + \sin \beta$

Question 2

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

Accepted Answer

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

Question 3

Complete the identity.
-$$\frac { \sec \theta \sin \theta } { \tan \theta } - 1 = \text { ? }$$

Accepted Answer

A)  0 
B)  1 
C)  $- \sec ^ { 2 } \theta$ 
D)  $1 - \sin \theta$ 
A)  0 
B)  1 
C)  $- \sec ^ { 2 } \theta$ 
D)  $1 - \sin \theta$

Question 4

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places.
-$2 \csc \theta = 5$

Accepted Answer

A)  0.20 
B)  0.41 
C)  0.41, 2.73 
D)  0.20, 2.94 
A)  0.20 
B)  0.41 
C)  0.41, 2.73 
D)  0.20, 2.94

Question 5

Solve the equation. Give a general formula for all the solutions.
-$\cos \theta = 0$

Accepted Answer

A)  $\theta = \frac { \pi } { 2 } + k \pi$ 
B)  $\theta = 0 + \mathrm { k } \pi$ 
C)  $\theta = 0 + 2 \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 2 } + 2 k \pi$ 
A)  $\theta = \frac { \pi } { 2 } + k \pi$ 
B)  $\theta = 0 + \mathrm { k } \pi$ 
C)  $\theta = 0 + 2 \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 2 } + 2 k \pi$

Question 6

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$f ( x ) = 3 \sin x - 2$ A) $f ^ { - 1 } ( x ) = 3 \sin ^ { - 1 } x - 2$

Accepted Answer

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

Question 7

Solve the problem.
-$$\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$$

Accepted Answer

\[\begin{array} { l } 
\cos 3 x = \cos (

Question 8

Solve the problem.
-$$\sin 4 u = 2 \sin 2 u \cos 2 u$$

Accepted Answer

The answer of Solve the problem.
-\[\sin 4 u = 2...

Question 9

Write the trigonometric expression as an algebraic expression in u.
-$\tan \left( \csc ^ { - 1 } \mathrm { u } \right)$

Accepted Answer

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

Question 10

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.
-$7 \cot ^ { 2 } x - 5 = 0$

Accepted Answer

A)  103.2°, 145.2°, 283.2°, 325.2° 
B)  51.8°, 128.2° 
C)  70.5°, 109.5°, 180.0° 
D)  49.8°, 130.2°, 229.8°, 310.2° 
A)  103.2°, 145.2°, 283.2°, 325.2° 
B)  51.8°, 128.2° 
C)  70.5°, 109.5°, 180.0° 
D)  49.8°, 130.2°, 229.8°, 310.2°

Question 11

Solve the problem.
-Before exercising, an athlete measures her air flow and obtains $a = 0.65 \sin \left( \frac { 2 \pi } { 5 } t \right)$ where a is measured in liters per second and t is the time in seconds. If a > 0, the athlete is inhaling; if a < 0, the
athlete is exhaling. The time to complete one complete inhalation/exhalation sequence is a respiratory cycle.
Find the values of t for which the athlete's air flow is zero. Find all values of t for t < 20 seconds.

Accepted Answer

t = 0, 2.5

Question 12

Solve the problem.
-$$\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )$$

Accepted Answer

The answer of Solve the problem.
-\[\sin ^ { 3 }...

Question 13

Find the exact value of the expression.
-$\cot ^ { - 1 } ( - 1 )$

Accepted Answer

A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 3 \pi } { 4 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 3 }$ 
A)  $\frac { 2 \pi } { 3 }$ 
B)  $\frac { 3 \pi } { 4 }$ 
C)  $\frac { \pi } { 4 }$ 
D)  $\frac { \pi } { 3 }$

Question 14

Find the exact value of the expression.
-$\tan \left( \tan ^ { - 1 } \frac { 3 } { 4 } + \sin ^ { - 1 } \frac { 1 } { 2 } \right)$

Accepted Answer

A)  $\frac { 2 \sqrt { 3 } + 2 \sqrt { 10 } } { 9 }$ 
B)  $\frac { 2 \sqrt { 3 } } { 5 }$

$\frac { 2 \sqrt { 6 } } { 5 }$ 
D)  $\frac { 9 + 4 \sqrt { 3 } } { 12 - 3 \sqrt { 3 } }$ 
A)  $\frac { 2 \sqrt { 3 } + 2 \sqrt { 10 } } { 9 }$ 
B)  $\frac { 2 \sqrt { 3 } } { 5 }$

$\frac { 2 \sqrt { 6 } } { 5 }$ 
D)  $\frac { 9 + 4 \sqrt { 3 } } { 12 - 3 \sqrt { 3 } }$

Question 15

Solve the problem.
-Find $\cos \frac { \theta } { 2 }$, given that $\sec \theta = 4$ and $\theta$ terminates in $0 < \theta < \pi / 2$.

Accepted Answer

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

Question 16

Find the exact value of the expression.
-$$\frac { \tan 70 ^ { \circ } - \tan \left( - 50 ^ { \circ } \right) } { 1 + \tan 70 ^ { \circ } \tan \left( - 50 ^ { \circ } \right) }$$

Accepted Answer

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

Question 17

Solve the problem.
-If $\cos \theta = - \frac { 5 } { 13 }$, and $\theta$ terminates in quadrant II, then find $\cos 2 \theta$.

Accepted Answer

A)  $- \frac { 120 } { 169 }$ 
B)  $\frac { 120 } { 169 }$ 
C)  $\frac { 119 } { 169 }$ 
D)  $- \frac { 119 } { 169 }$ 
A)  $- \frac { 120 } { 169 }$ 
B)  $\frac { 120 } { 169 }$ 
C)  $\frac { 119 } { 169 }$ 
D)  $- \frac { 119 } { 169 }$

Question 18

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.
-tan2 x + 5 tan x + 3 = 0

Accepted Answer

A)  103.1°, 145.1°, 283.1°, 325.1° 
B)  70.5°, 109.5°, 180.0° 
C)  51.8°, 128.2° 
D)  49.8°, 130.2°, 229.8°, 310.2° 
A)  103.1°, 145.1°, 283.1°, 325.1° 
B)  70.5°, 109.5°, 180.0° 
C)  51.8°, 128.2° 
D)  49.8°, 130.2°, 229.8°, 310.2°

Question 19

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$$\sin ( 4 \theta ) = \frac { \sqrt { 3 } } { 2 }$$

Accepted Answer

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

Question 20

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

Accepted Answer

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

Complete the identity. - $\frac { \cos ( \alpha - \beta ) } { \sin \alpha \cos \beta } = ?$

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

Complete the identity. - $\frac { \sec \theta \sin \theta } { \tan \theta } - 1 = \text { ? }$

Use a calculator to solve the equation on the interval 0 $0 \leq \theta < 2 \pi$ . Round the answer to two decimal places. - $2 \csc \theta = 5$

Solve the equation. Give a general formula for all the solutions. - $\cos \theta = 0$

Solve the problem. - $\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos x$

Solve the problem. - $\sin 4 u = 2 \sin 2 u \cos 2 u$

Write the trigonometric expression as an algebraic expression in u. - $\tan \left( \csc ^ { - 1 } \mathrm { u } \right)$

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. - $7 \cot ^ { 2 } x - 5 = 0$

Solve the problem. - $\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )$

Find the exact value of the expression. - $\cot ^ { - 1 } ( - 1 )$

Find the exact value of the expression. - $\tan \left( \tan ^ { - 1 } \frac { 3 } { 4 } + \sin ^ { - 1 } \frac { 1 } { 2 } \right)$

Solve the problem. -Find $\cos \frac { \theta } { 2 }$ , given that $\sec \theta = 4$ and $\theta$ terminates in $0 < \theta < \pi / 2$ .

Find the exact value of the expression. - $\frac { \tan 70 ^ { \circ } - \tan \left( - 50 ^ { \circ } \right) } { 1 + \tan 70 ^ { \circ } \tan \left( - 50 ^ { \circ } \right) }$

Solve the problem. -If $\cos \theta = - \frac { 5 } { 13 }$ , and $\theta$ terminates in quadrant II, then find $\cos 2 \theta$ .

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. -tan2 x + 5 tan x + 3 = 0

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\sin ( 4 \theta ) = \frac { \sqrt { 3 } } { 2 }$

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

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

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Exam 6: Analytic Trigonometry

Complete the identity. - cos⁡(α−β)sin⁡αcos⁡β=?\frac { \cos ( \alpha - \beta ) } { \sin \alpha \cos \beta } = ?sinαcosβcos(α−β)​=?

Find the exact value of the expression. - tan⁡(sin⁡−122)\tan \left( \sin ^ { - 1 } \frac { \sqrt { 2 } } { 2 } \right)tan(sin−122​​)

Complete the identity. - sec⁡θsin⁡θtan⁡θ−1= ? \frac { \sec \theta \sin \theta } { \tan \theta } - 1 = \text { ? }tanθsecθsinθ​−1= ?

Use a calculator to solve the equation on the interval 0 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . Round the answer to two decimal places. - 2csc⁡θ=52 \csc \theta = 52cscθ=5

Solve the equation. Give a general formula for all the solutions. - cos⁡θ=0\cos \theta = 0cosθ=0

Solve the problem. - cos⁡3x=cos⁡3x−3sin⁡2xcos⁡x\cos 3 x = \cos ^ { 3 } x - 3 \sin ^ { 2 } x \cos xcos3x=cos3x−3sin2xcosx

Solve the problem. - sin⁡4u=2sin⁡2ucos⁡2u\sin 4 u = 2 \sin 2 u \cos 2 usin4u=2sin2ucos2u

Write the trigonometric expression as an algebraic expression in u. - tan⁡(csc⁡−1u)\tan \left( \csc ^ { - 1 } \mathrm { u } \right)tan(csc−1u)

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. - 7cot⁡2x−5=07 \cot ^ { 2 } x - 5 = 07cot2x−5=0

Solve the problem. - sin⁡33x=12(sin⁡3x)(1−cos⁡6x)\sin ^ { 3 } 3 x = \frac { 1 } { 2 } ( \sin 3 x ) ( 1 - \cos 6 x )sin33x=21​(sin3x)(1−cos6x)

Find the exact value of the expression. - cot⁡−1(−1)\cot ^ { - 1 } ( - 1 )cot−1(−1)

Find the exact value of the expression. - tan⁡(tan⁡−134+sin⁡−112)\tan \left( \tan ^ { - 1 } \frac { 3 } { 4 } + \sin ^ { - 1 } \frac { 1 } { 2 } \right)tan(tan−143​+sin−121​)

Solve the problem. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that sec⁡θ=4\sec \theta = 4secθ=4 and θ\thetaθ terminates in 0<θ<π/20 < \theta < \pi / 20<θ<π/2 .

Find the exact value of the expression. - tan⁡70∘−tan⁡(−50∘)1+tan⁡70∘tan⁡(−50∘)\frac { \tan 70 ^ { \circ } - \tan \left( - 50 ^ { \circ } \right) } { 1 + \tan 70 ^ { \circ } \tan \left( - 50 ^ { \circ } \right) }1+tan70∘tan(−50∘)tan70∘−tan(−50∘)​

Solve the problem. -If cos⁡θ=−513\cos \theta = - \frac { 5 } { 13 }cosθ=−135​ , and θ\thetaθ terminates in quadrant II, then find cos⁡2θ\cos 2 \thetacos2θ .

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. -tan2 x + 5 tan x + 3 = 0

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - sin⁡(4θ)=32\sin ( 4 \theta ) = \frac { \sqrt { 3 } } { 2 }sin(4θ)=23​​

Establish the identity. - −tan⁡2x+sec⁡2x=sec⁡2xcos⁡2x- \tan ^ { 2 } x + \sec ^ { 2 } x = \sec ^ { 2 } x \cos ^ { 2 } x−tan2x+sec2x=sec2xcos2x