Inverse Trig. Functions & Differentiation Section 5.8

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

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

Solve.

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 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

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). 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 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. INVERSE COSINE FUNCTIONSFormula 6

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

Inverse Trig. Functions None of the 6 basic trig. functions has an inverse unless you restrict their domains.

Function Domain Range y = arcsin x -1< x < 1 I & IV y = arccos x -1< x < 1 I & II y = arctan x < x < I & IV y= arccot x < x < I & I y = arcsec x I & II y = arccsc I & IV

The Inverse Trigonometric Functions Graphs of six inverse trigonometric functions :

The Inverse Trigonometric Functions Graphs of six inverse trigonometric functions :

Inverse Properties f (f –1 (x)) = x and f –1 (f (x)) = x Remember that the trig. functions have inverses only in restricted domains.

Table 11DERIVATIVES

Derivatives of Inverse Trig. Functions Let u be a differentiable function of x.

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

Find each derivative with respect to x.

Find each derivative with respect to the given variable.

Example 1

Example 2

Some homework examples: Write the expression in algebraic form Solution: Use the right triangle Now using the triangle we can find the hyp. Let then y 3x 1

Some homework examples: Find the derivative of: Let u =

Example

Find an equation for the line tangent to the graph of at x = -1 At x = -1 Slope of tangent line When x = -1, y =