Lesson_07.html

Math 174

Calculus II

Spring 2001

Dr. Constant J. Goutziers

Department of Mathematical Sciences

goutzicj@oneonta.edu

Lesson 7

Inverse Trigonometric Functions

7.1 Syntax and Graphs

The Maple syntax for inverse trigonom etric functions is: arcsin(x), arccos(x) and arctan(x).

Examples

Example 7.1.1
Plot the graphs of arcsin(x), arccos(x), and arctan(x).

> plot(arcsin(x), x=-1..1);

> plot(arccos(x), x=-1..1);

> plot(arctan(x), x=-10..10);

7.2 Derivatives of Inverse Trigonom etric Functions and integrals involving Inverse Trigonometric Functions.

Examples

Example 7.2.1
Compute the derivatives of arcsin(x), arccos(x) and arctan(x).

Derivatives of inverse trigonometric functions are computed similarly to those of other functions.

> diff(arcsin(x), x);

> diff(arccos(x), x);

> diff(arctan(x), x);

>

Example 7.2.2
Compute .

Many integrals involve inverse trigonometric functions and this is a prime example.

We first perfor m a u-substitution using the changevar command in the student package,

> e1:=Int(5/(4+3*x^2), x);

> with(student):

The u-substitution and its inverse are defined as follows.

> rr:=x=2*u/sqrt(3); invrr:=u=solve(rr, u);

> e2:=changevar(rr, e1, u);

> e3:=simplify(e2);

Clearly this integral represents an arctangent. We now integrate and translate the result back in terms of x.

> e4:=value(e3)+C ;

> e5:=subs(invrr, e4);

Of course Maple can handle this integral with a black box approach, yet it is illustrative to see all the steps.

> e1;

> value(e1)+C;

>