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What are the amplitude, period, and phase shift for the function?

amplitude: 8,000; period: 10; phase shift: 0

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Over this domain, when does the population reach 18,000? 13,000?

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What is the predicted population in 2007? 2010?

In 2007, the predicted population is 4,413. In 2010, the population will be 11,924.

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For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

Suppose the graph of the displacement function is shown in [link] , where the values on the x -axis represent the time in seconds and the y -axis represents the displacement in inches. Give the equation that models the vertical displacement of the weight on the spring.

A graph of a consine function over one period. Graphed on the domain of [0,10]. Range is [-5,5].
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At time = 0, what is the displacement of the weight?

5 in.

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At what time does the displacement from the equilibrium point equal zero?

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What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?

10 seconds

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Inverse Trigonometric Functions

For the following exercises, find the exact value without the aid of a calculator.

sin 1 ( cos ( π 6 ) )

π 3

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cos 1 ( tan ( 3 π 4 ) )

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sin ( sec 1 ( 3 5 ) )

No solution

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cot ( sin 1 ( 3 5 ) )

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tan ( cos 1 ( 5 13 ) )

12 5

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sin ( cos 1 ( x x + 1 ) )

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Graph f ( x ) = cos x and f ( x ) = sec x on the interval [ 0 , 2 π ) and explain any observations.

The graphs are not symmetrical with respect to the line y = x . They are symmetrical with respect to the y -axis.

A graph of cosine of x and secant of x. Cosine of x has maximums where secant has minimums and vice versa. Asymptotes at x=-3pi/2, -pi/2, pi/2, and 3pi/2.
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Graph f ( x ) = sin x and f ( x ) = csc x and explain any observations.

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Graph the function f ( x ) = x 1 x 3 3 ! + x 5 5 ! x 7 7 ! on the interval [ 1 , 1 ] and compare the graph to the graph of f ( x ) = sin x on the same interval. Describe any observations.

The graphs appear to be identical.

Two graphs of two identical functions on the interval [-1 to 1]. Both graphs appear sinusoidal.
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Chapter practice test

For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.

f ( x ) = 0.5 sin x

amplitude: 0.5; period: 2 π ; midline y = 0

A graph of two periods of a sinusoidal function, graphed over -2pi to 2pi. The range is [-0.5,0.5]. X-intercepts at multiples of pi.
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f ( x ) = 5 sin x

amplitude: 5; period: 2 π ; midline: y = 0

Two periods of a sine function, graphed over -2pi to 2pi. The range is [-5,5], amplitude of 5, period of 2pi.
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f ( x ) = cos ( x + π 3 ) + 1

amplitude: 1; period: 2 π ; midline: y = 1

A graph of two periods of a cosine function, graphed over -7pi/3 to 5pi/3. Range is [0,2], Period is 2pi, amplitude is1.
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f ( x ) = 5 sin ( 3 ( x π 6 ) ) + 4

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f ( x ) = 3 cos ( 1 3 x 5 π 6 )

amplitude: 3; period: 6 π ; midline: y = 0

A graph of two periods of a cosine function, over -7pi/2 to 17pi/2. The range is [-3,3], period is 6pi, and amplitude is 3.
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f ( x ) = 2 tan ( x 7 π 6 ) + 2

amplitude: none; period:   π ;   midline:   y = 0 , asymptotes:   x = 2 π 3 + π k , where   k   is an integer

A graph of two periods of a tangent function over -5pi/6 to 7pi/6. Period is pi, midline at y=0.
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f ( x ) = π cos ( 3 x + π )

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f ( x ) = 5 csc ( 3 x )

amplitude: none; period:   2 π 3 ;   midline:   y = 0 , asymptotes:   x = π 3 k , where   k   is an integer

A graph of two periods of a cosecant functinon, over -2pi/3 to 2pi/3. Vertical asymptotes at multiples of pi/3. Period of 2pi/3.
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f ( x ) = π sec ( π 2 x )

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f ( x ) = 2 csc ( x + π 4 ) 3

amplitude: none; period: 2 π ; midline: y = 3

A graph of two periods of a cosecant function, graphed from -9pi/4 to 7pi/4. Period is 2pi, midline at y=-3.
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For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.

Give in terms of a sine function.

A graph of two periods of a sine function, graphed from -2 to 2. Range is [-6,-2], period is 2, and amplitude is 2.
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Give in terms of a sine function.

A graph of two periods of a sine function, graphed over -2 to 2. Range is [-2,2], period is 2, and amplitude is 2.

amplitude: 2; period: 2; midline: y = 0 ; f ( x ) = 2 sin ( π ( x 1 ) )

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Give in terms of a tangent function.

A graph of two periods of a tangent function, graphed over -3pi/4 to 5pi/4. Vertical asymptotes at x=-pi/4, 3pi/4. Period is pi.
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For the following exercises, find the amplitude, period, phase shift, and midline.

y = sin ( π 6 x + π ) 3

amplitude: 1; period: 12; phase shift: −6 ; midline y = −3

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y = 8 sin ( 7 π 6 x + 7 π 2 ) + 6

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The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D , in terms of t .

D ( t ) = 68 12 sin ( π 12 x )

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Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

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For the following exercises, find the period and horizontal shift of each function.

g ( x ) = 3 tan ( 6 x + 42 )

period: π 6 ; horizontal shift: −7

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n ( x ) = 4 csc ( 5 π 3 x 20 π 3 )

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Write the equation for the graph in [link] in terms of the secant function and give the period and phase shift.

A graph of 2 periods of a secant function, graphed over -2 to 2. The period is 2 and there is no phase shift.

f ( x ) = sec ( π x ) ; period: 2; phase shift: 0

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If tan x = 3 , find tan ( x ) .

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If sec x = 4 , find sec ( x ) .

4

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For the following exercises, graph the functions on the specified window and answer the questions.

Graph m ( x ) = sin ( 2 x ) + cos ( 3 x ) on the viewing window [ 10 , 10 ] by [ 3 , 3 ] . Approximate the graph’s period.

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Graph n ( x ) = 0.02 sin ( 50 π x ) on the following domains in x : [ 0 , 1 ] and [ 0 , 3 ] . Suppose this function models sound waves. Why would these views look so different?

The views are different because the period of the wave is 1 25 . Over a bigger domain, there will be more cycles of the graph.

Two side-by-side graphs of a sinusodial function. The first graph is graphed over 0 to 1, the second graph is graphed over 0 to 3. There are many periods for each.
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Graph f ( x ) = sin x x on [ 0.5 , 0.5 ] and explain any observations.

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For the following exercises, let f ( x ) = 3 5 cos ( 6 x ) .

What is the largest possible value for f ( x ) ?

3 5

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What is the smallest possible value for f ( x ) ?

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Where is the function increasing on the interval [ 0 , 2 π ] ?

On the approximate intervals ( 0.5 , 1 ) , ( 1.6 , 2.1 ) , ( 2.6 , 3.1 ) , ( 3.7 , 4.2 ) , ( 4.7 , 5.2 ) , ( 5.6 , 6.28 )

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For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.

Sine curve with amplitude 3, period π 3 , and phase shift ( h , k ) = ( π 4 , 2 )

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Cosine curve with amplitude 2, period π 6 , and phase shift ( h , k ) = ( π 4 , 3 )

f ( x ) = 2 cos ( 12 ( x + π 4 ) ) + 3

A graph of one period of a cosine function, graphed over -pi/4 to 0. Range is [1,5], period is pi/6.
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For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.

f ( x ) = 5 cos ( 3 x ) + 4 sin ( 2 x )

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f ( x ) = e sin t

This graph is periodic with a period of 2 π .

A graph of two periods of a sinusoidal function, The graph has a period of 2pi.
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For the following exercises, find the exact value.

cos 1 ( sin ( π ) )

π 2

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cos 1 ( tan ( 7 π 4 ) )

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cos ( sin 1 ( 1 2 x ) )

1 ( 1 2 x ) 2

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cos ( tan 1 ( x 2 ) )

1 1 + x 4

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For the following exercises, suppose sin t = x x + 1 .

Given [link] , find the measure of angle θ to three decimal places. Answer in radians.

An illustration of a right triangle with angle theta. Opposite the angle theta is a side with length 12, adjacent to the angle theta is a side with length 19.
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For the following exercises, determine whether the equation is true or false.

arcsin ( sin ( 5 π 6 ) ) = 5 π 6

False

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arccos ( cos ( 5 π 6 ) ) = 5 π 6

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The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.

approximately 0.07 radians

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Practice Key Terms 6

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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