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):

DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS

  More Definition of Inverse Trigonometric Functions chart 

Inverses of Trigonometric Functions on Visual Basic- This shows how they are derived.

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.

Additional link that also shows example problems:  

DERIVATIVES OF INVERSE TRIG FUNCTIONS

    ****** this chart was found at:

http://library.thinkquest.org/10030/2dfdoitf.htm  

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

Differentiate y = arcsin x + x*(sqrt (1-x^{^(2)}) 

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)} )] 

Loads of Derivatives

For homework examples, see Assignment #1