Question 1

Determine two coterminal angles (one positive and one negative) for each angle. Give the answers in radians.

Accepted Answer

A)  positive: $\frac { 19 \pi } { 9 }$

negative: $\frac { 37 \pi } { 19 }$ 
B)  positive: $\quad - \frac { 35 \pi } { 18 }$

negative: $\frac { 37 \pi } { 18 }$ 
C)  positive: $\quad - \frac { 37 \pi } { 18 }$

negative: $\frac { 35 \pi } { 18 }$ 
D)  positive: $\frac { 37 \pi } { 18 }$

negative: $\quad - \frac { 35 \pi } { 18 }$ 
E)  positive: $\frac { 37 \pi } { 19 }$

negative: $\quad - \frac { 35 \pi } { 17 }$ 
A)  positive: $\frac { 19 \pi } { 9 }$

negative: $\frac { 37 \pi } { 19 }$ 
B)  positive: $\quad - \frac { 35 \pi } { 18 }$

negative: $\frac { 37 \pi } { 18 }$ 
C)  positive: $\quad - \frac { 37 \pi } { 18 }$

negative: $\frac { 35 \pi } { 18 }$ 
D)  positive: $\frac { 37 \pi } { 18 }$

negative: $\quad - \frac { 35 \pi } { 18 }$ 
E)  positive: $\frac { 37 \pi } { 19 }$

negative: $\quad - \frac { 35 \pi } { 17 }$

Question 2

Find the cosine of $\theta$ .

Accepted Answer

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

Question 3

For a person at rest, the velocity v (in liters per second) of air flow into and out of the lungs during a respiratory cycle is given by $0.9 \sin \frac { \pi t } { 7 }$ , where t is the time in seconds. Inhalation occurs when $v > 0$ and exhalation occurs when $v < 0$ . Find the time for one full respiratory cycle.

Accepted Answer

A)  $14 \pi$ seconds 
B)  $\pi$ seconds 
C) 14 seconds 
D)  $2 \pi$ seconds 
E) 7 seconds 
A)  $14 \pi$ seconds 
B)  $\pi$ seconds 
C) 14 seconds 
D)  $2 \pi$ seconds 
E) 7 seconds

Question 4

The normal average daily temperature in degrees Fahrenheit for a city is given by $51 - 23 \cos \frac { 5 \pi ( t - 33 ) } { 365 }$ where t is the time in days, with $t = 1$ corresponding to January 1. Find the warmest day.

Accepted Answer

A) March 17 
B) March 16 
C) April 17 
D) April 16 
E) April 15 
A) March 17 
B) March 16 
C) April 17 
D) April 16 
E) April 15

Question 5

Sketch the graph of the function $y = 3 \tan \pi x$ .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 6

Find the period and amplitude of the function $y = \frac { 7 } { 2 } \sin \left( \frac { \pi x } { 2 } \right)$ .

Accepted Answer

A) Period: $2 \pi$ ; Amplitude:7 
B) Period: $2 \pi$ ; Amplitude: $\frac { 7 } { 2 }$ 
C) Period: $2$ ; Amplitude: $\frac { 7 } { 2 }$ 
D) Period: $4$ ; Amplitude: 7 
E) Period: $4$ ; Amplitude: $\frac { 7 } { 2 }$ 
A) Period: $2 \pi$ ; Amplitude:7 
B) Period: $2 \pi$ ; Amplitude: $\frac { 7 } { 2 }$ 
C) Period: $2$ ; Amplitude: $\frac { 7 } { 2 }$ 
D) Period: $4$ ; Amplitude: 7 
E) Period: $4$ ; Amplitude: $\frac { 7 } { 2 }$

Question 7

Find an equation of the tangent line to the graph of the function at the given point. $y = \cot x$ $\left( \frac { 3 \pi } { 4 } , - 1 \right)$

Accepted Answer

A)  $y = - 2 x + \frac { 3 } { 2 } \pi - 1$ 
B)  $y = 2 x + \frac { 1 } { 2 } \pi - 1$ 
C)  $y = - 2 x - \frac { 1 } { 2 } \pi$ 
D)  $y = - x - \frac { 5 } { 2 } \pi - 1$ 
E)  $y = 2 x - \frac { 3 } { 2 } \pi + 1$ 
A)  $y = - 2 x + \frac { 3 } { 2 } \pi - 1$ 
B)  $y = 2 x + \frac { 1 } { 2 } \pi - 1$ 
C)  $y = - 2 x - \frac { 1 } { 2 } \pi$ 
D)  $y = - x - \frac { 5 } { 2 } \pi - 1$ 
E)  $y = 2 x - \frac { 3 } { 2 } \pi + 1$

Question 8

Determine the relative extrema of the function $e ^ { 5 x } \cos x$ on the interval $( 0,2 \pi )$ .

Accepted Answer

A) relative minimum: $\left( - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi } { 4 } } , \frac { 5 \pi } { 4 } \right)$ relative maximum: $\left( \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi } { 4 } } , \frac { \pi } { 4 } \right)$ 
B) relative minimum: $\left( \frac { \pi } { 4 } , - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi } { 4 } } \right)$ relative maximum: $\left( \frac { 5 \pi } { 4 } , \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi } { 4 } } \right)$ 
C) relative minimum: $\left( \frac { 5 \pi } { 4 } , - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi } { 4 } } \right)$ relative maximum: $\left( \frac { \pi } { 4 } , \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi  } { 4 } } \right)$ 
D) relative minimum: $\left( \frac { 5 \pi } { 4 } , \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi  } { 4 } } \right)$ relative maximum: $\left( \frac { \pi } { 4 } , - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi  } { 4 } } \right)$ 
E) relative minimum: $\left( \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi } { 4 } } , \frac { 5 \pi } { 4 } \right)$ relative maximum: $\left( - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi  } { 4 } } , \frac { \pi } { 4 } \right)$ 
A) relative minimum: $\left( - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi } { 4 } } , \frac { 5 \pi } { 4 } \right)$ relative maximum: $\left( \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi } { 4 } } , \frac { \pi } { 4 } \right)$ 
B) relative minimum: $\left( \frac { \pi } { 4 } , - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi } { 4 } } \right)$ relative maximum: $\left( \frac { 5 \pi } { 4 } , \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi } { 4 } } \right)$ 
C) relative minimum: $\left( \frac { 5 \pi } { 4 } , - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi } { 4 } } \right)$ relative maximum: $\left( \frac { \pi } { 4 } , \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi  } { 4 } } \right)$ 
D) relative minimum: $\left( \frac { 5 \pi } { 4 } , \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi  } { 4 } } \right)$ relative maximum: $\left( \frac { \pi } { 4 } , - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi  } { 4 } } \right)$ 
E) relative minimum: $\left( \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 5 \pi } { 4 } } , \frac { 5 \pi } { 4 } \right)$ relative maximum: $\left( - \frac { \sqrt { 2 } } { 2 } e ^ { \frac { 25 \pi  } { 4 } } , \frac { \pi } { 4 } \right)$

Question 9

Sketch the graph of the function $y = \csc 2 x$ .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 10

Find $\csc \theta$ from the given graph.

Accepted Answer

A)  $\frac { 15 } { 17 }$ 
B)  $- \frac { 8 } { 17 }$ 
C)  $- \frac { 17 } { 8 }$ 
D)  $\frac { 17 } { 8 }$ 
E)  $- \frac { 17 } { 15 }$ 
A)  $\frac { 15 } { 17 }$ 
B)  $- \frac { 8 } { 17 }$ 
C)  $- \frac { 17 } { 8 }$ 
D)  $\frac { 17 } { 8 }$ 
E)  $- \frac { 17 } { 15 }$

Question 11

Find the derivative of the trigonometric function. $y = \cos 3 x + \sin ^ { 2 } x$

Accepted Answer

A)  $- 3 \sin 3 x + 2 \sin x \cos x$ 
B)  $3 \sin 3 x - 2 \sin x \cos x$ 
C)  $3 \sin 3 x + 2 \sin ^ { 2 } x \cos x$ 
D)  $- \sin 3 x + 2 \sin ^ { 2 } x \cos ^ { 2 } x$ 
E)  $- 3 \sin ^ { 2 } x + \sin ^ { 2 } x \cos ^ { 2 } x$ 
A)  $- 3 \sin 3 x + 2 \sin x \cos x$ 
B)  $3 \sin 3 x - 2 \sin x \cos x$ 
C)  $3 \sin 3 x + 2 \sin ^ { 2 } x \cos x$ 
D)  $- \sin 3 x + 2 \sin ^ { 2 } x \cos ^ { 2 } x$ 
E)  $- 3 \sin ^ { 2 } x + \sin ^ { 2 } x \cos ^ { 2 } x$

Question 12

Find the indefinite integral. $\int \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } d x$

Accepted Answer

A)  $\frac { 2 } { 3 } \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } + C$ 
B)  $\frac { 2 } { 3 } \sec ^ { 6 } \frac { x } { 4 } + C$ 
C)  $\frac { 1 } { 6 } \sec ^ { 7 } \frac { x } { 4 } + C$ 
D)  $\frac { 1 } { 4 } \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } + C$ 
E)  $\frac { 1 } { 3 } \tan \frac { x } { 4 } + C$ 
A)  $\frac { 2 } { 3 } \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } + C$ 
B)  $\frac { 2 } { 3 } \sec ^ { 6 } \frac { x } { 4 } + C$ 
C)  $\frac { 1 } { 6 } \sec ^ { 7 } \frac { x } { 4 } + C$ 
D)  $\frac { 1 } { 4 } \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } + C$ 
E)  $\frac { 1 } { 3 } \tan \frac { x } { 4 } + C$

Question 13

Find the area of the equilateral triangle with sides of length $s = 18$ in. Round your answer to two decimal places.

Accepted Answer

A) 121.50 square inches 
B) 561.18 square inches 
C) 140.30 square inches 
D) 162.00 square inches 
E) 486.00 square inches 
A) 121.50 square inches 
B) 561.18 square inches 
C) 140.30 square inches 
D) 162.00 square inches 
E) 486.00 square inches

Question 14

Find the indefinite integral of $\int e ^ { x } \sin e ^ { x } d x$ .

Accepted Answer

A)  $e ^ { x } \cos e ^ { x } + C$ 
B)  $e ^ { x } - \sin e ^ { x } + C$ 
C)  $- \cos e ^ { x } + C$ 
D)  $\cos e ^ { x } + C$ 
E)  $\sin e ^ { x } + C$ 
A)  $e ^ { x } \cos e ^ { x } + C$ 
B)  $e ^ { x } - \sin e ^ { x } + C$ 
C)  $- \cos e ^ { x } + C$ 
D)  $\cos e ^ { x } + C$ 
E)  $\sin e ^ { x } + C$

Question 15

Evaluate the definite integral $\int _ { \frac { \pi  } { 36 } } ^ { \frac { \pi  } { 22 } } \csc 6 x \cot 6 x d x$ .

Accepted Answer

A) 6 
B) -6 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  $\infty$ 
A) 6 
B) -6 
C)  $- \frac { 1 } { 6 }$ 
D)  $\frac { 1 } { 6 }$ 
E)  $\infty$

Question 16

Sketch the graph of the function $y = 2 \sec 3 x$ .

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 17

In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is 4.5 $^\circ$ . After you drive 13 miles closer to the mountain, the angle of elevation is 11 $^\circ$ . Approximate the height of the mountain. Round your answer to two decimal places.

Accepted Answer

A) 45.50 miles 
B) 8.84 miles 
C) 1.94 miles 
D) 1.72 miles 
E) 17.69 miles 
A) 45.50 miles 
B) 8.84 miles 
C) 1.94 miles 
D) 1.72 miles 
E) 17.69 miles

Question 18

Find the indefinite integral of $\int \frac { \sec x \tan x } { \sec x - 2 } d x$ .

Accepted Answer

Question 19

Find the period of the trigonometric function. $y = 3 \sec 5 x$

Accepted Answer

A)  $\frac { \pi } { 5 }$ 
B)  $\frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 2 \pi } { 5 }$ 
E)  $\frac { 3 \pi } { 5 }$ 
A)  $\frac { \pi } { 5 }$ 
B)  $\frac { 2 \pi } { 3 }$ 
C)  $\frac { \pi } { 3 }$ 
D)  $\frac { 2 \pi } { 5 }$ 
E)  $\frac { 3 \pi } { 5 }$

Question 20

Find the cosine of $$\theta$$ .

Accepted Answer

A)  $- \frac { 5 } { 12 }$ 
B)  $- \frac { 13 } { 12 }$ 
C)  $\frac { 13 } { 5 }$ 
D)  $- \frac { 12 } { 13 }$ 
E)  $\frac { 5 } { 13 }$ 
A)  $- \frac { 5 } { 12 }$ 
B)  $- \frac { 13 } { 12 }$ 
C)  $\frac { 13 } { 5 }$ 
D)  $- \frac { 12 } { 13 }$ 
E)  $\frac { 5 } { 13 }$

Determine two coterminal angles (one positive and one negative) for each angle. Give the answers in radians.

Find the cosine of $\theta$ . $Find the cosine of \theta .$

The normal average daily temperature in degrees Fahrenheit for a city is given by $51 - 23 \cos \frac { 5 \pi ( t - 33 ) } { 365 }$ where t is the time in days, with $t = 1$ corresponding to January 1. Find the warmest day.

Sketch the graph of the function $y = 3 \tan \pi x$ .

Find the period and amplitude of the function $y = \frac { 7 } { 2 } \sin \left( \frac { \pi x } { 2 } \right)$ . $Find the period and amplitude of the function y = \frac { 7 } { 2 } \sin \left( \frac { \pi x } { 2 } \right) .$

Find an equation of the tangent line to the graph of the function at the given point. $y = \cot x$ $\left( \frac { 3 \pi } { 4 } , - 1 \right)$

Determine the relative extrema of the function $e ^ { 5 x } \cos x$ on the interval $( 0,2 \pi )$ .

Sketch the graph of the function $y = \csc 2 x$ .

Find $\csc \theta$ from the given graph. $Find \csc \theta from the given graph.$

Find the derivative of the trigonometric function. $y = \cos 3 x + \sin ^ { 2 } x$

Find the indefinite integral. $\int \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } d x$

Find the area of the equilateral triangle with sides of length $s = 18$ in. Round your answer to two decimal places.

Find the indefinite integral of $\int e ^ { x } \sin e ^ { x } d x$ .

Evaluate the definite integral $\int _ { \frac { \pi } { 36 } } ^ { \frac { \pi } { 22 } } \csc 6 x \cot 6 x d x$ .

Sketch the graph of the function $y = 2 \sec 3 x$ .

Find the indefinite integral of $\int \frac { \sec x \tan x } { \sec x - 2 } d x$ .

Find the period of the trigonometric function. $y = 3 \sec 5 x$

Find the cosine of $\theta$ . $Find the cosine of \theta .$

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Exam 15: Trigonometric Functions Web

Determine two coterminal angles (one positive and one negative) for each angle. Give the answers in radians.

Find the cosine of θ\thetaθ .

The normal average daily temperature in degrees Fahrenheit for a city is given by 51−23cos⁡5π(t−33)36551 - 23 \cos \frac { 5 \pi ( t - 33 ) } { 365 }51−23cos3655π(t−33)​ where t is the time in days, with t=1t = 1t=1 corresponding to January 1. Find the warmest day.

Sketch the graph of the function y=3tan⁡πxy = 3 \tan \pi xy=3tanπx .

Find the period and amplitude of the function y=72sin⁡(πx2)y = \frac { 7 } { 2 } \sin \left( \frac { \pi x } { 2 } \right)y=27​sin(2πx​) .

Find an equation of the tangent line to the graph of the function at the given point. y=cot⁡xy = \cot xy=cotx (3π4,−1)\left( \frac { 3 \pi } { 4 } , - 1 \right)(43π​,−1)

Determine the relative extrema of the function e5xcos⁡xe ^ { 5 x } \cos xe5xcosx on the interval (0,2π)( 0,2 \pi )(0,2π) .

Sketch the graph of the function y=csc⁡2xy = \csc 2 xy=csc2x .

Find csc⁡θ\csc \thetacscθ from the given graph.

Find the derivative of the trigonometric function. y=cos⁡3x+sin⁡2xy = \cos 3 x + \sin ^ { 2 } xy=cos3x+sin2x

Find the indefinite integral. ∫sec⁡6x4tan⁡x4dx\int \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } d x∫sec64x​tan4x​dx

Find the area of the equilateral triangle with sides of length s=18s = 18s=18 in. Round your answer to two decimal places.

Find the indefinite integral of ∫exsin⁡exdx\int e ^ { x } \sin e ^ { x } d x∫exsinexdx .

Evaluate the definite integral ∫π36π22csc⁡6xcot⁡6xdx\int _ { \frac { \pi } { 36 } } ^ { \frac { \pi } { 22 } } \csc 6 x \cot 6 x d x∫36π​22π​​csc6xcot6xdx .

Sketch the graph of the function y=2sec⁡3xy = 2 \sec 3 xy=2sec3x .

Find the indefinite integral of ∫sec⁡xtan⁡xsec⁡x−2dx\int \frac { \sec x \tan x } { \sec x - 2 } d x∫secx−2secxtanx​dx .

Find the period of the trigonometric function. y=3sec⁡5xy = 3 \sec 5 xy=3sec5x

Find the cosine of θ\thetaθ .

Fundamental Concepts of Algebra

Equations and Inequalities

Functions and Graphs

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Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants

Limits and Derivatives

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Find the cosine of $\theta$ . $Find the cosine of \theta .$

The normal average daily temperature in degrees Fahrenheit for a city is given by $51 - 23 \cos \frac { 5 \pi ( t - 33 ) } { 365 }$ where t is the time in days, with $t = 1$ corresponding to January 1. Find the warmest day.

Sketch the graph of the function $y = 3 \tan \pi x$ .

Find the period and amplitude of the function $y = \frac { 7 } { 2 } \sin \left( \frac { \pi x } { 2 } \right)$ . $Find the period and amplitude of the function y = \frac { 7 } { 2 } \sin \left( \frac { \pi x } { 2 } \right) .$

Find an equation of the tangent line to the graph of the function at the given point. $y = \cot x$ $\left( \frac { 3 \pi } { 4 } , - 1 \right)$

Determine the relative extrema of the function $e ^ { 5 x } \cos x$ on the interval $( 0,2 \pi )$ .

Sketch the graph of the function $y = \csc 2 x$ .

Find $\csc \theta$ from the given graph. $Find \csc \theta from the given graph.$

Find the derivative of the trigonometric function. $y = \cos 3 x + \sin ^ { 2 } x$

Find the indefinite integral. $\int \sec ^ { 6 } \frac { x } { 4 } \tan \frac { x } { 4 } d x$

Find the area of the equilateral triangle with sides of length $s = 18$ in. Round your answer to two decimal places.

Find the indefinite integral of $\int e ^ { x } \sin e ^ { x } d x$ .

Evaluate the definite integral $\int _ { \frac { \pi } { 36 } } ^ { \frac { \pi } { 22 } } \csc 6 x \cot 6 x d x$ .

Sketch the graph of the function $y = 2 \sec 3 x$ .

Find the indefinite integral of $\int \frac { \sec x \tan x } { \sec x - 2 } d x$ .

Find the period of the trigonometric function. $y = 3 \sec 5 x$

Find the cosine of $\theta$ . $Find the cosine of \theta .$