Rational Expressions of Trigonometric Functions

Expressions like

displaymath37

are called rational expressions of sin and cos. Note that all the other trigonometric functions are rational functions of sin and cos. The main idea behind integrating such functions is the general substitution

displaymath39

In order to have better feeling how things do work, remember the trigonometric formulas

displaymath41

It is not hard to generate similar formulas for tex2html_wrap_inline43 , tex2html_wrap_inline45 , and tex2html_wrap_inline47 from the above formulas. Therefore, any rational function tex2html_wrap_inline49 will be transformed into a rational function of t via the above formulas. For example, we have

displaymath53

where tex2html_wrap_inline55 . Note that in order to complete the substitution we need to find dx as function of t and dt. Since tex2html_wrap_inline63 , we get

displaymath65

Now we are ready to integrate rational functions of sin and cos or at least transform them into integrating rational functions.

Check the following examples to see how this technique works:

[Calculus]
[Geometry] [Algebra] [Trigonometry ]
[Differential Equations] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Mohamed Amine Khamsi

Copyright 1999-2017 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour