Derivatives of Inverse Trigonometric Functions

8 Key excerpts on "Derivatives of Inverse Trigonometric Functions"

• 

  eBook - PDF

  Precalculus

  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.

  Sign up to read

  Learn more about book
• 

  No longer available |Learn more

  Elementary Special Functions in Mathematics

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used for instance in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations. ________________________ WORLD TECHNOLOGIES ________________________ Right-angled triangle definitions A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle. ________________________ WORLD TECHNOLOGIES ________________________ (Top): Trigonometric function sin θ for selected angles θ , π − θ , π + θ , and 2 π − θ in the four quadrants. (Bottom) Graph of sine function versus angle. Angles from the top panel are identified. The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Trigonometry For Dummies

  • 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

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Each definition has advantages and disadvantages, but we will use the current definition to conform with the conventions used by the CAS programs Mathematica, Maple, and Sage. 390 Chapter 6 / Exponential, Logarithmic, and Inverse Trigonometric Functions continuous at all points where the denominator is nonzero and either x ≤ −1 or x ≥ 1. Note that π + 4 tan −1 x = 0 if x = tan (−π / 4) = −1. Thus, f is continuous on the intervals (−∞, −1) and [1, +∞). EVALUATING INVERSE TRIGONOMETRIC FUNCTIONS A common problem in trigonometry is to find an angle whose sine is known. For example, you might want to find an angle x in radian measure such that sin x = 1 2 (4) and, more generally, for a given value of y in the interval −1 ≤ y ≤ 1 you might want to solve the equation sin x = y (5) Because sin x repeats periodically, this equation has infinitely many solutions for x; however, if we solve this equation as x = sin −1 y then we isolate the specific solution that lies in the interval [−π / 2, π / 2], since this is the range of the inverse sine. For example, Figure 6.7.2 shows four solutions of Equation (4), namely, −11π / 6, −7π / 6, π / 6, and 5π / 6. Of these, π / 6 is the solution in the interval [−π / 2, π / 2], so sin −1 ( 1 2 ) = π / 6 (6) Figure 6.7.2 In general, if we view x = sin −1 y as an angle in radian measure whose sine is y, then the restriction −π / 2 ≤ x ≤ π / 2 imposes the geometric requirement that the angle x in standard position terminate in either the first or fourth quadrant or on an axis adjacent to those quadrants. TECHNOLOGY MASTERY Refer to the documentation for your calculating utility to determine how to calculate inverse sines, inverse cosines, and inverse tangents; and then con- firm Equation (6) numerically by show- ing that sin −1 (0.5) ≈ 0.523598775598 . . . ≈ π/6 Example 3 Find exact values of (a) sin −1 (1 / √ 2 ) (b) sin −1 (−1) by inspection, and confirm your results numerically using a calculating utility.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Calculus Lifesaver

  All the Tools You Need to Excel at Calculus

  • Adrian Banner(Author)
  • 2009(Publication Date)
  • Princeton University Press

    (Publisher)

  C h a p te r 10 Inverse Functions and Inverse Trig Functions In the previous chapter, we looked at exponentials and logarithms. We got a lot of mileage out of the fact that e x and ln( x ) are inverses of each other. In this chapter, we’ll look at some more general properties of inverse functions, then examine inverse trig functions (and their hyperbolic cousins) in greater detail. Here’s the game plan: • using the derivative to show that a function has an inverse; • finding the derivative of inverse functions; • inverse trig functions, one by one; and • inverse hyperbolic functions. 10.1 The Derivative and Inverse Functions In Section 1.2 of Chapter 1, we reviewed the basics of inverse functions. I strongly suggest you take a quick look over that section before reading further, familiarizing yourself with the general idea. Now that we know some calculus, we can say more. In particular, we’re going to explore two connections between derivatives and inverse functions. 10.1.1 Using the derivative to show that an inverse exists Suppose that you have a differentiable function f whose derivative is always positive. What do you think the graph of this function looks like? Well, the slope of the tangent has to be positive everywhere, so the function can’t dip up and down: it has to go upward as we look from left to right. In other words, the function must be increasing . We’ll prove this fact in the next chapter (see Section 11.3.1 and also Sec-tion 11.2), but it at least seems clear that it should be true. In any case, if our function f is always increasing, then it must satisfy the horizontal line test. No horizontal line could possibly hit the graph of y = f ( x ) twice. Since the horizontal line test is satisfied by f , we know that f has an inverse. This has given us a nice strategy for showing that a function has an inverse: show that its derivative is always positive on its domain.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Resequenced for Students in STEM

  • David Dwyer, Mark Gruenwald(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  By the Pythagorean Theorem, the side adjacent to θ must then be √ 1 - x 2 . Thus, cos θ = √ 1 - x 2 1 = p 1 - x 2 (3.14) Using the fact that cos(-θ) = cos θ, and considering special cases for θ = 0 and θ = π 2 , we can conclude that Equation 3.14 holds for all angles - π 2 ≤ θ ≤ π 2 and all -1 ≤ x ≤ 1. Thus, cos(sin -1 x) = √ 1 - x 2 for all -1 ≤ x ≤ 1. Substituting this result into Equation 3.13, we have d dx [sin -1 x] = 1 √ 1 - x 2 The differentiation formulas for the other inverse trigonometric functions can be derived in a similar fashion. The formulas are summarized as follows: Derivatives of Inverse Trigonometric Functions d dx [sin -1 x] = 1 √ 1 - x 2 d dx [cos -1 x] = - 1 √ 1 - x 2 d dx [tan -1 x] = 1 1 + x 2 Example 5 Differentiating inverse trigonometric functions Differentiate. a. y = sin -1 (2x 3 ) b. f (x) = e 2x arccos x Solution a. In this case we must combine the Chain Rule with the rule for the derivative of the inverse sine function. dy dx = 1 p 1 - (2x 3 ) 2 d dx  2x 3  = 1 √ 1 - 4x 6 · 6x 2 = 6x 2 √ 1 - 4x 6 3.9. INVERSE FUNCTIONS AND THEIR DERIVATIVES 193 b. Here we apply the Product Rule. f 0 (x) = e 2x d dx [arccos x] + arccos x d dx [e 2x ] = e 2x -1 √ 1 - x 2 + arccos x · 2e 2x = e 2x  2 arccos x - 1 √ 1 - x 2  Example 6 Finding angular rate of change A space shuttle launch is being observed from a location 5 miles away from the launch pad. In Figure 3.55, the height of the shuttle is denoted by h and the angle of elevation by θ. a. Express θ as a function of h. b. Assuming θ and h are functions of time t, find the rate of change of θ with respect to t (sometimes called the angular velocity) when h = 1 mile, at which time the velocity of the shuttle is dh dt = 508 miles per hour. 5 θ h Figure 3.55 Solution a. Since tan θ = h 5 and θ is acute, we can use the inverse tangent function to write θ = tan -1 h 5 b.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  33–1 Derivatives of the Sine and Cosine Functions Derivative of sin u Approximated Graphically Before deriving a formula for the derivative of the sine function, let us use a sketch to get an indication of its shape. ◆◆◆ Example 1: Graph y = sin x, with x in radians. Use the slopes of the tangents at points on that graph to sketch the graph of the derivative. Solution: We graph y = sin x as shown in Fig. 33-1(a) and the slopes as shown in Fig. 33-1(b). Note that the slope of the sine curve is zero at points A, B, C, and D, so the derivative curve must cross the x axis at points A′, B′, C′, and D′. We estimate the slope to be 1 at points E and F, which gives us points E′ and F′ on the derivative curve. Similarly, the slope is -1 at G, giving us point G′ on the derivative curve. We then note that the sine curve is rising from A to B and from C to D, so the derivative curve must be positive in those intervals. Similarly, the sine curve is falling from B to C, so the derivative curve is negative in this interval. Using all of this information, we sketch in the derivative curve. 33 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Find the derivatives of the sine and cosine functions. • Solve applied problems requiring derivatives of the trigonometric functions. • Find derivatives of the inverse trigonometric functions. • Find derivatives of logarithmic functions. • Find derivatives of exponential functions. In this chapter we extend our ability to take derivatives to include the trigonometric, logarith- mic, and exponential functions. This will enable us to solve a larger range of problems than was possible before. After learning the rules for taking derivatives of these functions, we apply them to problems quite similar to those in chapters 28 and 29; that is, tangents, related rates, maximum–minimum, and the rest.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  382 CHAPTER 6 Exponential, Logarithmic, and Inverse Trigonometric Functions 53. Writing A student objects that it is circular reasoning to make the definition ln x =  x 1 1 t dt since to evaluate the integral we need to know the value of ln x. Write a short paragraph that answers this student’s objection. 54. Writing Write a short paragraph that compares Definition 6.6.1 with the definition of the natural logarithm function given in Section 6.1. Be sure to discuss the issues surrounding continu- ity and differentiability. 6.6 | Quick Check Answers 1. −1 2. a. 5 6 b. 7 12 3. e 4. y = 2 +  x 0 cos t 3 dt 5. − e −x 1 + e −4x 6.7 Derivatives and Integrals Involving Inverse Trigonometric Functions A common problem in trigonometry is to find an angle x using a known value of sin x, cos x, or some other trigonometric function. Problems of this type involve the computation of inverse trigonometric functions. In this section we will study these functions from the viewpoint of general inverse functions, with the goal of developing derivative formulas for the inverse trigonometric functions. We will also derive some related integration formulas that involve inverse trigonometric functions. Inverse Trigonometric Functions The six basic trigonometric functions do not have inverses because their graphs repeat periodi- cally and hence do not pass the horizontal line test. To circumvent this problem we will restrict the domains of the trigonometric functions to produce one-to-one functions and then define the “inverse trigonometric functions” to be the inverses of these restricted functions. The top part of Figure 6.7.1 shows geometrically how these restrictions are made for sin x, cos x, tan x, and sec x, and the bottom part of the figure shows the graphs of the corresponding inverse functions sin −1 x, cos −1 x, tan −1 x, sec −1 x (also denoted by arcsin x, arccos x, arctan x, and arcsec x).

  Sign up to read

  Learn more about book