Mathematics
Inverse Trigonometric Functions
Inverse trigonometric functions are functions that "undo" the actions of trigonometric functions. They are used to find the angle when the value of a trigonometric function is known. Common inverse trigonometric functions include arcsine, arccosine, and arctangent, denoted as sin⁻¹, cos⁻¹, and tan⁻¹ respectively. These functions are essential in solving trigonometric equations and in various applications of mathematics and physics.
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10 Key excerpts on "Inverse Trigonometric Functions"
eBook - PDFPrecalculus
A Prelude to Calculus
- Sheldon Axler(Author)
- 2016(Publication Date)
- Wiley(Publisher)
This chapter concludes with an investigation into transformations of trigono- metric functions, which are used to model periodic events. Redoing function transformations in the context of trigonometric functions will also help us review the results from Chapter 1 on how function transformations change graphs. 351 352 Chapter 5 Trigonometric Algebra and Geometry 5.1 Inverse Trigonometric Functions Learning Objectives By the end of this section you should be able to • compute values of cos -1 , sin -1 , and tan -1 ; • sketch the radius of the unit circle corresponding to the arccosine, arcsine, and arctangent of a number; • use the Inverse Trigonometric Functions to find angles in a right triangle, given the lengths of two sides; • find the angles in an isosceles triangle, given the lengths of the sides; • use tan -1 to find the angle a line with given slope makes with the horizontal axis. Several of the most important functions in mathematics are defined as the inverse functions of familiar functions. For example, the cube root is defined as the inverse function of x 3 , and the logarithm base 3 is defined as the inverse function of 3 x . In this section, we will define the inverses of the cosine, sine, and tangent The Inverse Trigonometric Functions provide remarkably useful tools for solving many problems. functions. These inverse functions are called the arccosine, the arcsine, and the arctangent. Neither cosine nor sine nor tangent is one-to-one when defined on its usual domain. Thus we will need to restrict the domains of these functions to obtain one-to-one functions that have inverses. The Arccosine Function Recall that a function is called one-to-one if it assigns distinct values to distinct numbers in its domain. The cosine function, whose domain is the entire real line, is As usual, we will assume throughout this section that all angles are measured in radians unless explicitly stated otherwise. not one-to-one because, for example, cos 0 = cos 2π.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 5 Inverse Trigonometric Functions and Trigonometric Functions Inverse Trigonometric Functions In mathematics, the Inverse Trigonometric Functions or cyclometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions. Since none of the six trigonometric functions are one-to-one (by failing the horizontal line test), they must be restricted in order to have inverse functions. For example, just as the square root function is defined such that y 2 = x , the function y = arcsin( x ) is defined so that sin( y ) = x . There are multiple numbers y such that sin( y ) = x ; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also a rcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin( x ) will evaluate only to a single value, called its principal value. These properties apply to all the Inverse Trigonometric Functions. The principal inverses are listed in the following table. Name Usual notation Definition Domain of x for real result Range of usual principal value (radians) Range of usual principal value (degrees) arcsine y = arcsin x x = sin y −1 ≤ x ≤ 1 −π/2 ≤ y ≤ π/2 −90° ≤ y ≤ 90° arccosine y = arccos x x = cos y −1 ≤ x ≤ 1 0 ≤ y ≤ π 0° ≤ y ≤ 180° arctangent y = arctan x x = tan y all real numbers −π/2 < y < π/2 −90° < y < 90° arccotangent y = arccot x x = cot y all real numbers 0 < y < π 0° < y < 180°
eBook - PDF- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
4 Equations and Applications IN THIS PART . . . Become acquainted with inverse trig functions. Identify the domains and ranges of the inverse trig functions. Recognize the pairings of the quadrants used by each inverse function. Solve trig equations using identities and inverse functions. Write expressions to include infinitely many answers. Find the areas of triangles using trig functions in the formulas. CHAPTER 14 Investigating Inverse Trig Functions 233 Chapter 14 Investigating Inverse Trig Functions A s thrilling and fulfilling as the original six trig functions are, they just aren’t complete without their inverses. An inverse trig function behaves like the inverse of any other type of function — it undoes what the original function did. In mathematics, functions can have inverses if they’re one-to-one, meaning each output value occurs only once. This whole inverse idea is going to take some fast talking when it comes to trig functions, because they keep repeat- ing values over and over as angles are formed with every full rotation of the circle — so you’re going to wonder how these functions and inverses can be one- to-one. If you need a refresher on basic inverse functions, just refer to the section on inverses in the Appendix for the lowdown on them and how you determine one. Writing It Right You use inverse trig functions when you want to know what angle is involved in equations such as sin x 1 2 or sec 2 x , or tan 2 1 x . In typical algebra equations, you can solve for the value of x by dividing each side of the equation by the coef- ficient or by adding the same thing to each side, and so on. But you can’t do that with the function sin x 1 2 . IN THIS CHAPTER » Acquainting yourself with inverse notation » Setting limits on inverse trig functions » Determining domain and range of inverse trig functions
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
Several of the most important functions in mathematics are defined as the inverse functions of familiar functions. For example, the cube root is defined as the inverse function of x 3 , and the logarithm base 3 is defined as the inverse function of 3 x . In this section, we will define the inverses of the cosine, sine, and tangent The inverse trigono- metric functions pro- vide a remarkably useful tool for solv- ing many problems. functions. These inverse functions are called the arccosine, the arcsine, and the arctangent. Neither cosine nor sine nor tangent is one-to-one when defined on its usual domain. Thus we will need to restrict the domains of these functions to obtain one-to-one functions that have inverses. The Arccosine Function Consider the cosine function, whose domain is the entire real line. The cosine As usual, we will as- sume throughout this section that all an- gles are measured in radians unless explic- itly stated otherwise. function is not one-to-one because, for example, cos 0 = cos 2π . 2 Π Π Π 2 Π 1 1 The graph of cosine on the interval [-2π, 2π]. For example, suppose we are told that x is a number such that cos x = 0, The graph above fails the horizontal line test—there are hori- zontal lines that in- tersect the graph in more than one point. Thus the cosine func- tion is not one-to-one. and we are asked to find the value of x. Of course cos π 2 = 0, but also cos 3π 2 = 0; we also have cos(- π 2 ) = 0 and cos(- 3π 2 ) = 0 and so on. Thus with the information given we have no way to determine a unique value of x such that cos x = 0. Hence the cosine function does not have an inverse. We faced a similar dilemma when we wanted to define the square root function as the inverse of the function x 2 . The domain of the function x 2 is the entire real line. This function is not one-to-one; thus it does not have an inverse. For example, if we are told that x 2 = 16, then we cannot determine whether x = 4 or x = -4.
eBook - PDF- Cynthia Y. Young(Author)
- 2017(Publication Date)
- Wiley(Publisher)
■ ■ Understand that the domain of the cosine function is restricted 30, p4 in order for the inverse cosine function to exist. ■ ■ Understand that the domain of the tangent function is restricted A 2 p 2 , p 2 B in order for the inverse tangent function to exist. ■ ■ Understand that the cotangent, cosecant, and secant inverse functions are not found from the reciprocal of the tangent, sine, and cosine functions respectively, but rather from the inverse secant, inverse cosecant, and inverse cotangent identities. ■ ■ Visualize the quadrants in order to find exact values of trigonometric expressions involving Inverse Trigonometric Functions. 6.1 Inverse Trigonometric Functions In Appendix A.6, one-to-one functions and inverse functions are discussed. Here we present a summary of that section. A function is one-to-one if it passes the horizontal line test: No two x-values map to the same y-value. Notice that the sine function does not pass the horizontal line test. 2π π –π –2π x y = sin x –1 1 y y = 1 2 However, if we restrict the domain to 2 p 2 # x # p 2 , then the restricted function is one-to-one. 2π π –π –2π x –1 1 y Recall that if y 5 ƒ 1 x 2 , then x 5 ƒ 21 1 y 2 . The following are the properties of inverse functions: 1. If ƒ is a one-to-one function, then the inverse function ƒ 21 exists. 2. The domain of ƒ 21 5 the range of ƒ. The range of ƒ 21 5 the domain of ƒ. 6.1 Inverse Trigonometric Functions 293 3. ƒ 21 1 ƒ 1 x 22 5 x for all x in the domain of ƒ. ƒ 1 ƒ 21 1 x 22 5 x for all x in the domain of ƒ 21 . 4. The graph of ƒ 21 is the reflection of the graph of ƒ about the line y 5 x. If the point 1 a, b 2 lies on the graph of a function, then the point 1 b, a 2 lies on the graph of its inverse.
eBook - PDFThe Calculus Lifesaver
All the Tools You Need to Excel at Calculus
- Adrian Banner(Author)
- 2009(Publication Date)
- Princeton University Press(Publisher)
208 • Inverse Functions and Inverse Trig Functions 10.2 Inverse Trig Functions Now it’s time to investigate the inverse trig functions. We’ll see how to define them, what their graphs look like, and how to differentiate them. Let’s look at them one at a time, beginning with inverse sine. 10.2.1 Inverse sine Let’s start by looking at the graph of y = sin( x ) once again: 1 0 -1 -3 π -5 π 2 -2 π -3 π 2 -π -π 2 3 π 5 π 2 2 π 3 π 2 π π 2 y = sin( x ) Does the sine function have an inverse? You can see from the above graph that the horizontal line test fails pretty miserably. In fact, every horizontal line of height between -1 and 1 intersects the graph infinitely many times, which is a lot more than the zero or one time we can tolerate. So, using the tactic described in Section 1.2.3 in Chapter 1, we throw away as little of the domain as possible in order to pass the horizontal line test. There are many options, but the sensible one is to restrict the domain to the interval [ -π/ 2 , π/ 2]. Here’s the effect of this: 1 0 -1 -3 π -5 π 2 -2 π -3 π 2 -π -π 2 3 π 5 π 2 2 π 3 π 2 π π 2 y = sin( x ) , -π 2 ≤ x ≤ π 2 The solid portion of the curve is all we have left after we restrict the domain. Clearly we can’t go to the right of π/ 2 or else we’ll start repeating the values immediately to the left of π/ 2 as the curve dips back down. A similar thing happens at -π/ 2. So, we’re stuck with our interval. OK, if f ( x ) = sin( x ) with domain [ -π/ 2 , π/ 2], then it satisfies the hor-izontal line test, so it has an inverse f -1 . We’ll write f -1 ( x ) as sin -1 ( x ) or arcsin( x ). (Beware: the first of these notations is a little confusing at first, since sin -1 ( x ) does not mean the same thing as (sin( x )) -1 , even though sin 2 ( x ) = (sin( x )) 2 and sin 3 ( x ) = (sin( x )) 3 .) So, what is the domain of the inverse sine function? Well, since the range of f ( x ) = sin( x ) is [ -1 , 1], the domain of the inverse function is [ -1 , 1].
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
2. State the definitions of the inverse cosine and inverse tangent functions (page 480). For examples of evaluating Inverse Trigonometric Functions, see Examples 3 and 4. 3. State the inverse properties of trigonometric functions (page 482). For examples of finding composite functions involving Inverse Trigonometric Functions, see Examples 5–7. Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 484 Chapter 6 Trigonometry 6.6 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. GO DIGITAL Vocabulary and Concept Check In Exercises 1–4, fill in the blanks. Function Alternative Notation Domain Range 1. y = arcsin x __________ __________ - π 2 ≤ y ≤ π 2 2. __________ y = cos -1 x -1 ≤ x ≤ 1 __________ 3. y = arctan x __________ __________ __________ 4. A trigonometric function has an __________ function only when its domain is restricted. 5. What notation can you use to represent the inverse cosecant function? 6. Does arccos x = 1cos x? Skills and Applications Evaluating an Inverse Trigonometric Function In Exercises 7–20, find the exact value of the expression, if possible. 7. arcsin 1 2 8. arcsin 0 9. arccos 0 10. arccos 1 2 11. arctan √3 3 12. arctan 1 13. arcsin 3 14. arcsin √3 15. tan -1 (-√3 ) 16. cos -1 (-2) 17. arccos ( - 1 2 ) 18. arcsin √2 2 19. sin -1 ( - √3 2 ) 20. tan -1 ( - √3 3 ) Graphing an Inverse Trigonometric Function In Exercises 21 and 22, use a graphing utility to graph f, g, and y = x in the same viewing window to verify geometrically that g is the inverse function of f.
- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
Chapter 5 Inverse Trigonometric Functions; Trigonometric Equations 199 Chapter 5 Inverse Trigonometric Functions; Trigonometric Equations EXERCISE 5.1 Inverse Sine, Cosine, and Tangent Functions 1. For a function to be one-to-one, each element of the range must correspond to a single element of the domain. Since there are, for example, an infinite number of domain elements (x = n, n any integer) corresponding to range element 0, y = sin x is not a one-to-one function. 3. No. For example, sin 6 = 1 2 and sin 5 6 = 1 2 , so more than one element of the domain [0, ] corresponds to the element 1 2 of the range. 5. Make a careful drawing. Measure 53°. 7. Make a careful drawing. Measure 66°. 9. Make a careful drawing. Measure 31°. Exercise 5.1 Inverse Sine, Cosine, and Tangent Functions 200 11. y = sin –1 0 is equivalent to sin y = 0. No reference triangle can be drawn, but the only y between – 2 and 2 which has sine equal to 0 is y = 0. Thus, sin –1 0 = 0. (1, 0) 13. y = arccos 3 2 is equivalent to cos y = 3 2 . What y between 0 and has cosine equal to 3 2 ? y must be associated with a reference triangle in the first quadrant. Reference triangle is a special 30°–60° triangle. y = 6 , arccos 3 2 = 6 y 2 1 3 15. y = tan –1 1 is equivalent to tan y = 1. What y between – 2 and 2 has tangent equal to 1? y must be associated with a reference triangle in the first quadrant. Reference triangle is a special 45° triangle. y = 4 , tan –1 1 = 4 y 1 2 1 17. y = cos –1 1 2 is equivalent to cos y = 1 2 . What y between 0 and has cosine equal to 1 2 ? y must be associated with a reference triangle in the first quadrant. Reference triangle is a special 30°–60° triangle. y = 3 , cos –1 1 2 = 3 2 y 1 3 19. Calculator in radian mode: cos –1 (–0.9999) = 3.127 21. Calculator in radian mode: tan –1 4.056 = 1.329 23. 3.142 is not in the domain of the inverse sine function. –1 3.142 1 is false. arcsin 3.142 is not defined.
eBook - PDFPrecalculus
Functions and Graphs
- Earl Swokowski, Jeffery Cole(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The following general relationships involving f and f 2 1 were discussed in Section 4.1. The Inverse Trigonometric Functions 6.6 460 CHAPTER 6 Analytic Trigonometry Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. We shall use relationship 1 to define each of the Inverse Trigonometric Functions. The sine function is not one-to-one, since different numbers, such as p y 6, 5 p y 6 , and 2 7 p y 6 , yield the same function value s 1 2 d . If we restrict the domain to f 2 p y 2, p y 2 g , then, as illustrated by the blue portion of the graph of y 5 sin x in Figure 1, we obtain a one-to-one (increasing) function that takes on every value of the sine function once and only once. We use this new function with domain f 2 p y 2, p y 2 g and range f 2 1, 1 g to define the inverse sine function. The domain of the inverse sine function is f 2 1, 1 g , and the range is f 2 p y 2, p y 2 g . The notation y 5 sin 2 1 x is sometimes read “ y is the inverse sine of x .” The equation x 5 sin y in the definition allows us to regard y as an angle, so y 5 sin 2 1 x may also be read “ y is the angle whose sine is x ” (with 2 p y 2 # y # p y 2 ). The inverse sine function is also called the arcsine function, and arcsin x may be used in place of sin 2 1 x . If t 5 arcsin x , then sin t 5 x , and t may be interpreted as an arc length on the unit circle U with center at the origin. We will use both notations— sin 2 1 and arcsin—throughout our work. Several values of the inverse sine function are listed in the next chart.
eBook - PDF- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 1.7 Inverse Trigonometric Functions 189 Exploration True or False? In Exercises 106–109, determine whether the statement is true or false. Justify your answer. 106. sin 5π 6 = 1 2 arcsin 1 2 = 5π 6 107. tan parenleft.alt4 - π 4 parenright.alt4 = -1 arctan(-1) = - π 4 108. arctan x = arcsin x arccos x 109. sin -1 x = 1 sin x 110. HOW DO YOU SEE IT? Use the figure below to determine the value(s) of x for which each statement is true. π 2 π y x - 1 1 - 4 2 2 ( ) , π y = arcsin x y = arccos x (a) arcsin x < arccos x (b) arcsin x = arccos x (c) arcsin x > arccos x 111. Inverse Cotangent Function Define the inverse cotangent function by restricting the domain of the cotangent function to the interval (0, π ), and sketch the graph of the inverse trigonometric function. 112. Inverse Secant Function Define the inverse secant function by restricting the domain of the secant function to the intervals [0, π H208622) and (π H208622, π ], and sketch the graph of the inverse trigonometric function. 113. Inverse Cosecant Function Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals [-π H208622, 0) and (0, π H208622], and sketch the graph of the inverse trigonometric function. 114. Writing Use the results of Exercises 111 –113 to explain how to graph (a) the inverse cotangent function, (b) the inverse secant function, and (c) the inverse cosecant function on a graphing utility. Evaluating an Inverse Trigonometric Function In Exercises 115–120, use the results of Exercises 111–113 to find the exact value of the expression. 115. arcsec radical.alt22 116. arcsec 1 117. arccot(-1) 118.
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