# Chapter 5, Section 8   Inverse Trigonometric Functions: Differentiation

Inverse Trig Functions

This section begins with a rather surprising statement: None of the six basic trigonometric functions has an inverse function.  This statement is true because all six trigonometric functions are periodic and hence not one-to-one.  In this section you will examine these six functions and see whether their domains can be redefined in such a way that they will have inverse functions on the restricted domains.

Under suitable restrictions, each of the six trigonometric functions is one-to-one and so has an inverse function, as indicated in the following definition.

Definition of Inverse Trigonometric Functions (links to tables and examples):

### Example 1 from our text : Evaluation Inverse Trig Function

(a) Evaluate arcsin (- ½)

(1)   By definition, y = arcsin (- ½) implies that sin y = (- ½).

(2)    In the interval [- π/2, π/2], the correct value of y is - π/6,

(3)   arcsin(- ½) = - π/6

(b) Evaluate arccos (0)

(1)   By definition, y = arccos (0) implies that cos y = (0).

(2)    In the interval [ 0 , π], the correct value of y is π/2,

(3)   arccos(0)=  π/2

Inverse functions have the properties

f ( f -1 (x)) = x and   f -1 (f (x)) = x

Theorem 5.18

Derivatives of Inverse Trigonometric Functions

Let u be a differentiable function x.

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### Example 4 from our text : Differentiating Inverse Trig Functions

(a)    (d/dx) [ arcsin(2x)] = 2 /((1-(2x)^(2))^2) = 2 /(1- 4x^(2))

(b)   (d/dx) [ arctan(3x)] = 3 /(1-(3x)^(2)) = 3 /(1 + 9x^(2)

(c)  (d/dx) [ arcsin(sqrt x)] = (1/2)*x(-1/2)/(sqrt(1-x)) = 1 / [(x(sqrt(x))*((sqrt(1-x) )] = [1 /  (sqrt(x-x^(2) )]

### Example 5 from our text : A Derivative That Can Be Simplified

#### Solution: (use this: [arssin u] = u' / (sqrt(1-u^(2)) and let u =x and let u' = 1)

y ' = 1 /(sqrt(1- x^(2) )) + (1/2)*(-2x)*(1- x^(2) )^(-1/2)(sqrt(1- x^(2) ))

= [1 / (sqrt(1- x^(2) ))] - [x^(2) / (sqrt(1- x^(2) ))] + [ (sqrt(1- x^(2) )]

= [ (sqrt(1- x^(2) )]  +  [ (sqrt(1- x^(2) )]

= 2*[ (sqrt(1- x^(2) )]

For homework examples, see Assignment #1