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Academic Precalculus

Chapter 4

Sections 4.5-4.8Graphs, Inverses, and Applications

of Trigonometric Functions

24.5: Sine and Cosine Curves The graph of the sine function is called the sine curve. The graph of the cosine function is called the cosine

curve. One cycle of the repeating curve is called the period

of the function. Note that these curves repeat, representing multiple revolutions around the unit circle.

Using the Wolfram Demonstrations Project, determine the maximum points, zeros, and minimum points of the sine function and cosine function. Also determine the domain, range, and period.

34.5: Sketching Sine and Cosine by Key Points Sketch the graph of y = 3 cos x on the interval

[-2π,2π]

Sketch the graph of y = -2 sin x on the interval [-2π,2π]

Sketch the graph of y = ⅓ cos x on the interval [-π,5π]

44.5: Vertical Stretching/Shrinking On the Wolfram Demonstrations Project, click the

plus arrow at the end of each slider. Set the vertical stretch to 1 Set the phase shift to 0 Set the vertical shift to 0 Set the period to 6.28318

Adjust the slider for the vertical stretch. Describe the changes to the sine curve.

Set the vertical stretch back to 1 and the phase shift to -1.57079. What effect did that have on the curve?

Adjust the slider for the vertical stretch again. Does it have the same effect?

54.5: Amplitude of Sine and Cosine Curves The amplitude of

y = a sin x

and

y = a cos x

represents half the distance between the maximum and minimum values of the function and is given by

amplitude = |a|.

Note that if a is negative, then the graph is the reflection across the x-axis of the graph of the same graph with a positive.

64.5: Horizontal Stretching/Shrinking On the Wolfram Demonstrations Project, click the

plus arrow at the end of the frequency slider. Set the amplitude to 1 Set the frequency to 1 Set the horizontal shift to 0 Set the vertical shift to 0

Adjust the frequency slider or type in values to the field. Describe the changes to the sine curve.

Reset the frequency to 1, and change the graph to the cosine curve. What effect does adjusting the frequency have on the cosine curve?

74.5: Period of Sine and Cosine Functions Because y = a sin x completes one cycle in [0,2π),

y = a sin bx completes one cycle in [0,2π/b). The number b is called the frequency of the sine or

cosine function, and represents how many periods fit in the interval [0,2π).

Let b be a positive real number. The period of y = a sin bx and y = a cos bx is given by

period = 2π/b If 0 < b < 1 then the period is larger than 2π If b > 1 then the period is smaller than 2π If b is negative, then the even and odd identities are

used to rewrite the function.

84.5: Sketching Sine and Cosine by Key Points Find the period and then sketch the graph of

y = cos 3x on the interval [-2π,2π]

Find the period and then sketch the graph of y = -2 sin ½x on the interval [-2π,2π]

Find the period and then sketch the graph of y = ⅓ cos 2x on the interval [-π,5π]

94.5: Horizontal Translations On the Wolfram Demonstrations Project, click the

plus arrow at the end of the horizontal shift slider. Set the amplitude to 1 Set the frequency to 1 Set the horizontal shift to 0 Set the vertical shift to 0

Adjust the horizontal shift slider or type in values to the field. Describe the changes to the sine curve.

Reset the horizontal shift to 0, and change the graph to the cosine curve. What effect does adjusting the vertical shift have on the cosine curve?

104.5: Vertical Translations On the Wolfram Demonstrations Project, click the

plus arrow at the end of the vertical shift slider. Set the amplitude to 1 Set the frequency to 1 Set the horizontal shift to 0 Set the vertical shift to 0

Adjust the vertical shift slider or type in values to the field. Describe the changes to the sine curve.

Reset the vertical shift to 0, and change the graph to the cosine curve. What effect does adjusting the vertical shift have on the cosine curve?

114.5: Sine and Cosine Functions The graphs of

y = a sin (bx – c) + d

y = a cos (bx – c) + d

have the following characteristics (assume b > 0).

Amplitude = |a|Frequency = bPeriod = 2π/b

The left and right endpoints of a one-cycle interval can be determined by solving bx – c = 0 and bx – c = 2π

The vertical shift up is d.

124.5: Sketching Sine and Cosine by Key Points Find the amplitude, frequency, period, vertical shift,

and endpoints of a one-cycle interval, and then sketch the graph of each function on the interval [-3π,3π]: y = 2 cos x – 5 y = 2 cos (x – 5) y = 2 + 3 cos 2x y = -½ sin (πx + π) y = -3 sin (2πx + 4π) + 3

134.5: Mathematical Modeling The normal monthly temperatures in degrees

Fahrenheit in Albany, NY are given in the table:

Use a trigonometric function to model this data. Find the normal temperature in December. A painting company will accept exterior jobs only

when the normal temperature is 64° or higher. During what months will they accept exterior jobs?

Month Temp Month Temp

Jan 21 Jul 72

Mar 34 Sep 61

May 58 Nov 40

144.6: Graph of the Tangent Function Using the Wolfram Demonstrations Project,

determine the zeros and asymptotes of the tangent function. Also determine the domain, period, and in which quadrants the tangent function is increasing or decreasing.

Sketching the graph of y = a tan (bx – c) + d is similar to what was done for the sine and cosine curves in that you locate key points. Two consecutive asymptotes can be found by

solving bx – c = -π/2 and bx – c = π/2. The midpoint between two consecutive asymptotes

will have y-coordinate d and be a point of inflection where the curve changes direction.

154.6: Sketching Tangent by Key Points Find the frequency, period, vertical shift, and

endpoints of a one-cycle interval, and then sketch the graph of each function on the interval [-3π,3π]: y = tan x y = 2 tan x – 1 y = 2 tan (x – 1) y = 2 + 3 tan 2x y = -½ tan (πx + π) y = -3 tan (2πx + 4π) + 3

164.6: Graph of the Cotangent Function Using the Wolfram Demonstrations Project,

determine the zeros and asymptotes of the cotangent function. Also determine the domain, period, and in which quadrants the tangent function is increasing or decreasing.

Sketching the graph of y = a cot (bx – c) + d is similar to what was done for the tangent curve in that you locate key points. Two consecutive asymptotes can be found by

solving bx – c = 0 and bx – c = π. The midpoint between two consecutive asymptotes

will have y-coordinate d and be a point of inflection where the curve changes direction.

174.6: Sketching Tangent by Key Points Find the frequency, period, vertical shift, and

endpoints of a one-cycle interval, and then sketch the graph of each function on the interval [-3π,3π]: y = cot x y = 2 cot x – 1 y = 2 cot (x – 1) y = 2 + 3 cot 2x y = -½ cot (πx + π) y = -3 cot (2πx + 4π) + 3

184.6: Graphs of the Secant and Cosecant Functions Using the WDP, explore the graphs of the secant and

cosecant functions to determine the maximums, minimums, and asymptotes. Determine the domain and period of each.

Sketching the graph of y = a sec (bx – c) + d or y = a csc (bx – c) + d is similar to what was done for the other curves in that you locate key points. Minimums of the secant or cosecant occur at the

maxiumus of cosine or sine, respectively. Maximums of the secant or cosecant occur at the

minumus of cosine or sine, respectively. Asymptotes of the secant or cosecant occur at the

zeros of the cosine or sine, respectively.

194.6: Sketching Secant and Cosecant by Key Points Find the frequency, period, vertical shift, and

endpoints of a one-cycle interval, and then sketch the graph of each function on the interval [-3π,3π]: y = sec x y = 2 csc x – 1 y = 2 csc (x – 1) y = 2 + 3 sec 2x y = -½ sec (πx + π) y = -3 csc (2πx + 4π) + 3

204.6: Dance of the Trig Functions

214.6: Damped Trigonometric Graphs The product of two functions can be graphed using

properties of the individual functions. Graph x sin x

Because |sin x| ≤ 1, then 0 ≤ |x| |sin x| ≤ |x| and thus -|x| ≤ x sin x ≤ |x|. So the graph of f(x) = x sin x lies between the lines y = x and y = -x.

Further, the graph will touch the bounding lines at π/2 + nπ and will have x-intercepts at nπ.

224.6: Analyzing a Damped Sine Curve In situations like the previous example, the factor that

creates the bounding curves is said to be the damping factor.

Analyze the graph of f(x) = ex sin 4x. Find the bounding curves. Find where f(x) touches the bounding curves. Find the zeros. Sketch.

234.7: Inverse Functions Recall that for a function to have an inverse function,

it must be one-to-one, that is, it must pass the horizontal line test. None of the trig functions pass the horizontal line

test over their entire domain, although restricted domains can be found where the trig function takes on its full range of values and is still one-to-one.

Graph sine, cosine, and tangent and identify a restricted domain near zero where the function is one-to-one and takes on the full range of values.

244.7: Inverse Sine Function The inverse sine function is denoted y = sin-1 x and

denotes the angle whose sine is x. It is also sometimes denoted y = arcsin x where the

arcsine comes from the association of a central angle with its intercepted arc length on a unit circle. So arcsin x means the central angle (or arc measure) whose sine is x.

Graph y = sin-1 x by hand and find its domain and range with the WDP

Domain: [-1, 1]Range: [-π/2, π/2]

254.7: Evaluating the Inverse Sine Function The inverse sine of x is the angle whose sine is x If possible, find the exact value:

y=arcsin(−12 ) y=arcsin (−1)

y=sin−1(√32 ) y=sin−1(−1

2 )y=sin−1 (√3) y=sin−1 (2)

264.7: Inverse Cosine Function The inverse cosine function is denoted y = cos-1 x and

denotes the angle whose cosine is x. It is also sometimes denoted y = arccos x where the

arccosine comes from the association of a central angle with its intercepted arc length on a unit circle. So arccos x means the central angle (or arc measure) whose cosine is x.

Graph y = cos-1 x by hand and find its domain and range with the WDP

Domain: [-1, 1]Range: [0, π]

274.7: Evaluating the Inverse Cosine Function The inverse cosine of x is the angle whose cosine is x If possible, find the exact value:

y=arccos(−12 ) y=arccos (−1)

y=cos−1(√32 ) y=cos−1(−1

2 )y=cos−1 (√3) y=cos−1 (2)

284.7: Inverse Tangent Function The inverse tangent function is denoted y = tan-1 x and

denotes the angle whose tangent is x. It is also sometimes denoted y = arctan x where the

arctangent comes from the association of a central angle with its intercepted arc length on a unit circle. So arctan x means the central angle (or arc measure) whose tangent is x.

Graph y = tan-1 x by hand and find its domain and range with the WDP

Domain: RRange: [-π/2, π/2]

294.7: Evaluating the Inverse Tangent Function The inverse sine of x is the angle whose sine is x If possible, find the exact value:

y=arctan(−12 ) y=arctan (−1)

y=tan−1(√32 ) y=tan−1(−1

2 )y=tan−1 (√3) y=tan−1 (2)

304.7: Composition of Functions Recall that composition of functions implies that for

all x in the domains of f and f-1,

f(f-1(x)) = x and f-1(f(x)) = x If -1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2

sin(sin-1(x)) = x and sin-1(sin(y)) = y If -1 ≤ x ≤ 1 and 0 ≤ y ≤ π

cos(cos-1(x)) = x and cos-1(cos(y)) = y If x is a real number and -π/2 ≤ y ≤ π/2

tan(tan-1(x)) = x and tan-1(tan(y)) = y Note that if x or y are outside these ranges, then these

rules do not apply!

314.7: Evaluating Composition of Functions If possible, find the exact value:

tan( tan−1(−14)) cos−1(cos(5π4 ))

sin(sin−1(π)) cos(tan−1(−34 ))

sin−1(sin (π)) sin(cos−1(23))

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324.7: Evaluating Composition of Functions Find an algebraic value in terms of x:

sin(cos−1(3x)) ,0≤x< 1/3

cot (cos−1(4x)) ,0≤x< 1/4

sec(tan−1(x))

tan(sin(2x)) ,0≤x< 1 /2

334.8: Applications Involving Right Triangles Unless an application suggests otherwise, we

normally let the angles of a right triangle be A, B, and C, where C is the right angle, and let the sides be a, b, and c, where side a is opposite angle A, etc.

Solve the right triangle for all unknown sides and angles: mA = 34.2° and b = 19.4 m = 25° and c = 16

344.8: Ladder Problems / Angle of Elevation A ladder 16 feet long leans against the side of a

house. Find the height from the top of the ladder to the ground if the angle of elevation of the ladder is 74°.

354.8: Angle of Depression From the time a small airplane is 100 feet high and

1600 ground feet from the runway, the plane descends in a straight line to the runway. Determine the plane's angle of descent.

364.8: Bearings In surveying and navigation, directions are normally

given in terms of bearings, which is the acute angle a path or line of sight makes with a fixed north-south line. S 35° E is the angle 35° east of south

A sailboat leaves a pier and heads due west at 8 knots (nautical miles per hour). After 15 minutes, the sailboat tacks, changing course to N 16° W at 10 knots. Find the sailboat's bearing and distance from the pier after 12 minutes on this course.

374.8: Harmonic Motion Motions that repeat in a regular cycle are periodic

motions, ie: springs. There is one position where the net force on the object

is zero. At that position, the object is at equilibrium. Whenever the object is pulled away, the net force on

the system becomes nonzero and pulls the object back toward equilibrium.

In order for an object to be in simple harmonic motion, the restoring force (the force that tries to return the object to its equilibrium position) must be proportional to the displacement from equilibrium.

F spring=−k⋅x (Hooke's Law)

384.8: Harmonic Motion The distance from the equilibrium position for an

object in simple harmonic motion at time t is given by

d = a sin ωt or d = a cos ωt The amplitude, A = |a|, of the motion is the

maximum distance that the object moves away from equilibrium.

The period, T = 2π /ω, is the time needed for an object to repeat one complete cycle of the motion.

Related to the period of the simple harmonic motion is its frequency of vibration, f = ω/2π, which is the number of cycles that repeat in one time period.

394.8: Harmonic Motion For the simple harmonic motion described by

d = ¼ sin 16π t

find the amplitude, the period, the frequency, the displacement at 10.25 seconds, and the least positive time for which d = 0.

Find a model for simple harmonic motion for a spring whose period is 6 seconds and whose amplitude is 4 cm.

40Chapter 4 Review Trigonometry Formulas

Definitions Laws and Identities

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