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(a) To define the inverse sine function, we restrict the domain of sine to the interval __________. On this interval the sine function is one-to-one, and its inverse function sin −1 is defined by sin −1 x = y ⇔ sin _____ = _____. For example, sin − 1 1 2 = _ _ _ _ _ because sin _____ = _____. (b) To define the inverse cosine function, we restrict the domain of cosine to the interval __________. On this interval the cosine function is one-to-one and its inverse function cos −1 is defined by cos −1 x = y ⇔ cos _____ = _____. For example, cos − 1 1 2 = _ _ _ _ _ because cos _____ = _____.

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Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759

Solutions

Chapter
Section
BuyFindarrow_forward

Precalculus: Mathematics for Calcu...

7th Edition
James Stewart + 2 others
Publisher: Brooks Cole
ISBN: 9781305071759
Chapter 5.5, Problem 1E
Textbook Problem
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  1. (a) To define the inverse sine function, we restrict the domain of sine to the interval __________. On this interval the sine function is one-to-one, and its inverse function sin−1 is defined by sin−1 x = y ⇔ sin _____ = _____. For example, sin 1 1 2 = _ _ _ _ _ because sin _____ = _____.
  2. (b) To define the inverse cosine function, we restrict the domain of cosine to the interval __________. On this interval the cosine function is one-to-one and its inverse function cos−1 is defined by cos−1 x = y ⇔ cos _____ = _____. For example, cos 1 1 2 = _ _ _ _ _ because cos _____ = _____.

(a)

To determine

To fill: The statement “To define the inverse function , we restrict the domain of sine to the interval _______.On this interval the sine function is one to one, and its inverse function sin1x=ysin _______=_______. For example, sin112= _______.because sin _______=_______.”

Explanation of Solution

To prove sin1x=ysiny=x as shown below,

Theorem used:

“If a function f is continuous as well as one to one on the interval, then its inverse is also one to one and continuous on that interval.”

Calculation:

To obtain the domain of the sine function, take the help of the graph as shown in below.

From the Figure 1, it is observed that the domain of the sine function is [π2,π2] .

To prove the if and only if statement, first assume that y=sin1x is given.

Then, the claim to prove is x=siny .

That is,

Use the above theorem mentioned above to conclude about the inverse function y=sin1x is continuous

(b)

To determine

To fill: The statement “To define the inverse function , we restrict the domain of cosine to the interval                         . On this interval the cosine function is one to one, and its inverse function cos1x=ycos              =              . For example, cos112=            because cos              =              .”

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Chapter 5 Solutions

Precalculus: Mathematics for Calculus (Standalone Book)
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Phase and Phase Difference A pair of sine curves...Ch. 5.6 - Phase and Phase Difference A pair of sine curves...Ch. 5.6 - Phase and Phase Difference A pair of sine curves...Ch. 5.6 - Phase and Phase Difference A pair of sine curves...Ch. 5.6 - A Bobbing Cork A cork floating in a lake is...Ch. 5.6 - FM Radio Signals The carrier wave for an FM radio...Ch. 5.6 - Blood Pressure Each time your heart beats, your...Ch. 5.6 - Predator Population Model In a predator/prey...Ch. 5.6 - Mass-Spring System A mass attached to a spring is...Ch. 5.6 - Tides The graph shows the variation of the water...Ch. 5.6 - Tides The Bay of Fundy in Nova Scotia has the...Ch. 5.6 - Mass-Spring System A mass suspended from a spring...Ch. 5.6 - Mass-Spring System A mass is suspended on a...Ch. 5.6 - Mass-Spring System The frequency of oscillation of...Ch. 5.6 - Ferris Wheel A Ferris wheel has a radius of 10 m,...Ch. 5.6 - Cock Pendulum The pendulum in a grandfather clock...Ch. 5.6 - Variable Stars The variable star Zeta Gemini has a...Ch. 5.6 - Variable Stars Astronomers believe that the radius...Ch. 5.6 - Biological Clocks Circadian rhythms are biological...Ch. 5.6 - Electric Generator The armature in an electric...Ch. 5.6 - Electric Generator The graph shows an oscilloscope...Ch. 5.6 - Doppler Effect When a car with its horn blowing...Ch. 5.6 - Motion of a Building A strong gust of wind strikes...Ch. 5.6 - Shock Absorber When a car hits a certain bump on...Ch. 5.6 - Tuning Fork A tuning fork is struck and oscillates...Ch. 5.6 - Guitar String A guitar string is pulled at point P...Ch. 5.6 - Two Fans Electric fans A and B have radius 1 ft...Ch. 5.6 - Alternating Current Alternating current is...Ch. 5.6 - DISCUSS: Phases of Sine The phase of a sine curve...Ch. 5.6 - DISCUSS: Phases of the Moon During the course of a...Ch. 5 - (a) What is the unit circle, and what is the...Ch. 5 - Let t be a real number, and let P(x, y) be the...Ch. 5 - (a) Describe the steps we use to find the value of...Ch. 5 - (a) What is a periodic function? (b) What are the...Ch. 5 - (a) What is an even function, and what is an odd...Ch. 5 - (a) State the reciprocal identities. (b) Slate the...Ch. 5 - (a) Graph the sine and cosine functions. (b) What...Ch. 5 - (a) Graph the tangent and cotangent functions. (b)...Ch. 5 - (a) Graph the cosecant and secant functions. (b)...Ch. 5 - (a) Define the inverse sine function, the inverse...Ch. 5 - (a) What is simple harmonic motion? (b) What is...Ch. 5 - Suppose that an object is in simple harmonic...Ch. 5 - Consider the following models of harmonic motion....Ch. 5 - Terminal Points A point P(x, y) is given. (a) Show...Ch. 5 - Terminal Points A point P(x, y) is given. (a) Show...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Reference Number and Terminal Point A real number...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Values of Trigonometric Functions Find the value...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Fundamental Identities Use the fundamental...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Values of Trigonometric Functions Find the values...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Horizontal Shifts A trigonometric function is...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Functions from a Graph The graph of one period of...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Graphing Trigonometric Functions Find the period,...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Evaluating Expressions Involving Inverse...Ch. 5 - Amplitude, Period, Phase, and Horizontal Shift For...Ch. 5 - Amplitude, Period, Phase, and Horizontal Shift For...Ch. 5 - Phase and Phase Difference A pair of sine curves...Ch. 5 - Phase and Phase Difference A pair of sine curves...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Even and Odd Functions A function is given. (a)...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Sine and Cosine Curves with Variable Amplitude...Ch. 5 - Maxima and Minima Find the maximum and minimum...Ch. 5 - Maxima and Minima Find the maximum and minimum...Ch. 5 - Solving Trigonometric Equations Graphically Find...Ch. 5 - Solving Trigonometric Equations Graphically Find...Ch. 5 - Discover the Period of a Trigonometric Function...Ch. 5 - Simple Harmonic Motion A point P moving in simple...Ch. 5 - Simple Harmonic Motion A mass suspended from a...Ch. 5 - Damped Harmonic Motion The top floor of a building...Ch. 5 - The point P(x, y) is on the unit circle in...Ch. 5 - The point P in the figure at the left has...Ch. 5 - Find the exact value. (a) sin76 (b) cos134 (c)...Ch. 5 - Express tan t in terms of sin t, if the terminal...Ch. 5 - If cost=817 and if the terminal point determined...Ch. 5 - A trigonometric function is given. (a) Find the...Ch. 5 - A trigonometric function is given. (a) Find the...Ch. 5 - Find the period, and graph the function. 8. y =...Ch. 5 - Find the period, and graph the function. 9....Ch. 5 - Find the exact value of each expression, if it is...Ch. 5 - The graph shown at left is one period of a...Ch. 5 - The sine curves y1=30sin(6t2) and y2=30sin(6t3)...Ch. 5 - Let f(x)=cosx1+x2. (a) Use a graphing device to...Ch. 5 - A mass suspended from a spring oscillates in...Ch. 5 - An object is moving up and down in damped harmonic...

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