7 INVERSE FUNCTIONS

7.6 Inverse Trigonometric Functions In this section, we will learn about: Inverse trigonometric functions and their derivatives. INVERSE FUNCTIONS

Here, we apply the ideas of Section 7.1 to find the derivatives of the so-called inverse trigonometric functions. INVERSE TRIGONOMETRIC FUNCTIONS

However, we have a slight difficulty in this task.  As the trigonometric functions are not one-to-one, they don’t have inverse functions.  The difficulty is overcome by restricting the domains of these functions so that they become one-to-one. INVERSE TRIGONOMETRIC FUNCTIONS

Here, you can see that the sine function y = sin x is not one-to-one.  Use the Horizontal Line Test. INVERSE TRIGONOMETRIC FUNCTIONS

However, here, you can see that the function f(x) = sin x,, is one-to-one. INVERSE TRIGONOMETRIC FUNCTIONS

The inverse function of this restricted sine function f exists and is denoted by sin -1 or arcsin.  It is called the inverse sine function or the arcsine function. INVERSE SINE FUNCTION / ARCSINE FUNCTION

As the definition of an inverse function states that we have:  Thus, if -1 ≤ x ≤ 1, sin -1 x is the number between and whose sine is x. INVERSE SINE FUNCTIONSEquation 1

Evaluate: a. b. INVERSE SINE FUNCTIONSExample 1

We have:  This is because, and lies between and. Example 1 aINVERSE SINE FUNCTIONS

Let, so.  Then, we can draw a right triangle with angle θ.  So, we deduce from the Pythagorean Theorem that the third side has length. Example 1 bINVERSE SINE FUNCTIONS

 This enables us to read from the triangle that: INVERSE SINE FUNCTIONSExample 1b

In this case, the cancellation equations for inverse functions become: INVERSE SINE FUNCTIONSEquations 2

The inverse sine function, sin -1, has domain [-1, 1] and range. INVERSE SINE FUNCTIONS

The graph is obtained from that of the restricted sine function by reflection about the line y = x. INVERSE SINE FUNCTIONS

We know that:  The sine function f is continuous, so the inverse sine function is also continuous.  The sine function is differentiable, so the inverse sine function is also differentiable (from Section 3.4). INVERSE SINE FUNCTIONS

We could calculate the derivative of sin -1 by the formula in Theorem 7 in Section 7.1. However, since we know that is sin -1 differentiable, we can just as easily calculate it by implicit differentiation as follows. INVERSE SINE FUNCTIONS

Let y = sin -1 x.  Then, sin y = x and –π/2 ≤ y ≤ π/2.  Differentiating sin y = x implicitly with respect to x, we obtain: INVERSE SINE FUNCTIONS

Now, cos y ≥ 0 since –π/2 ≤ y ≤ π/2, so INVERSE SINE FUNCTIONSFormula 3

If f(x) = sin -1 (x 2 – 1), find: (a) the domain of f. (b) f ’(x). (c) the domain of f ’. INVERSE SINE FUNCTIONSExample 2

Since the domain of the inverse sine function is [-1, 1], the domain of f is: INVERSE SINE FUNCTIONSExample 2 a

Combining Formula 3 with the Chain Rule, we have: Example 2 bINVERSE SINE FUNCTIONS

The domain of f ’ is: Example 2 cINVERSE SINE FUNCTIONS

The inverse cosine function is handled similarly.  The restricted cosine function f(x) = cos x, 0 ≤ x ≤ π, is one-to-one.  So, it has an inverse function denoted by cos -1 or arccos. INVERSE COSINE FUNCTIONSEquation 4

The cancellation equations are: INVERSE COSINE FUNCTIONSEquation 5

The inverse cosine function,cos -1, has domain [-1, 1] and range, and is a continuous function. INVERSE COSINE FUNCTIONS

Its derivative is given by:  The formula can be proved by the same method as for Formula 3.  It is left as Exercise 17. INVERSE COSINE FUNCTIONSFormula 6

The tangent function can be made one-to-one by restricting it to the interval. INVERSE TANGENT FUNCTIONS

Thus, the inverse tangent function is defined as the inverse of the function f(x) = tan x,.  It is denoted by tan -1 or arctan. INVERSE TANGENT FUNCTIONSEquation 7

Simplify the expression cos(tan -1 x)  Let y = tan -1 x.  Then, tan y = x and. E. g. 3—Solution 1INVERSE TANGENT FUNCTIONS

We want to find cos y.  However, since tan y is known, it is easier to find sec y first.  Therefore, INVERSE TANGENT FUNCTIONSE. g. 3—Solution 1

Thus, INVERSE TANGENT FUNCTIONSE. g. 3—Solution 1

Instead of using trigonometric identities, it is perhaps easier to use a diagram.  If y = tan -1 x, then tan y = x.  We can read from the figure (which illustrates the case y > 0) that: INVERSE TANGENT FUNCTIONSE. g. 3—Solution 2

The inverse tangent function, tan -1 = arctan, has domain and range. INVERSE TANGENT FUNCTIONS

We know that:  So, the lines are vertical asymptotes of the graph of tan. INVERSE TANGENT FUNCTIONS

The graph of tan -1 is obtained by reflecting the graph of the restricted tangent function about the line y = x.  It follows that the lines y = π/2 and y = -π/2 are horizontal asymptotes of the graph of tan -1. INVERSE TANGENT FUNCTIONS

This fact is expressed by these limits: INVERSE TANGENT FUNCTIONSEquations 8

Evaluate:  Since the first equation in Equations 8 gives: INVERSE TANGENT FUNCTIONSExample 4

Since tan is differentiable, tan -1 is also differentiable. To find its derivative, let y = tan -1 x.  Then, tan y = x. INVERSE TANGENT FUNCTIONS

Differentiating that latter equation implicitly with respect to x, we have: Thus, INVERSE TANGENT FUNCTIONSEquation 9

The remaining inverse trigonometric functions are not used as frequently and are summarized as follows. Equations 10INVERSE TRIG. FUNCTIONS

Equations 10INVERSE TRIG. FUNCTIONS

The choice of intervals for y in the definitions of csc -1 and sec -1 is not universally agreed upon. INVERSE TRIG. FUNCTIONS

For instance, some authors use in the definition of sec -1.  You can see from the graph of the secant function that both this choice and the one in Equations 10 will work. INVERSE TRIG. FUNCTIONS

In the following table, we collect the differentiation formulas for all the inverse trigonometric functions.  The proofs of the formulas for the derivatives of csc -1, sec -1, and cot -1 are left as Exercises 19–21. DERIVATIVES OF INVERSE TRIG. FUNCTIONS

Table 11DERIVATIVES

Each of these formulas can be combined with the Chain Rule.  For instance, if u is a differentiable function of x, then DERIVATIVES

Differentiate: DERIVATIVESExample 5

DERIVATIVESExample 5 a

DERIVATIVESExample 5 b

Prove the identity  Although calculus is not needed to prove this, the proof using calculus is quite simple. INVERSE TRIG. FUNCTIONSExample 6

If f(x) = tan -1 x + cot -1 x, then for all values of x.  Therefore f(x) = C, a constant. INVERSE TRIG. FUNCTIONSExample 6

To determine the value of C, we put x = 1.  Then,  Thus, tan -1 x + cot -1 x = π/2. INVERSE TRIG. FUNCTIONSExample 6

Each of the formulas in Table 11 gives rise to an integration formula.  The two most useful of these are: INTEGRATION FORMULASEquations 12 & 13

Find:  If we write the integral resembles Equation 12 and the substitution u = 2x is suggested.  This gives du = 2dx; so dx = du/2. INTEGRATION FORMULASExample 7

When x = 0, u = 0. When x = ¼, u = ½. Thus, Example 7INTEGRATION FORMULAS

Evaluate:  To make the given integral more like Equation 13, we write:  This suggests that we substitute u = x/a. Example 8INTEGRATION FORMULAS

Then, du = dx/a, dx = a du, and Example 8INTEGRATION FORMULAS

Thus, we have the formula E.g. 8—Formula 14INTEGRATION FORMULAS

Find:  We substitute u = x 2 because then du = 2x dx and we can use Equation 14 with a = 3: INTEGRATION FORMULASExample 9