## The magic identity Trigonometry is the art of doing algebra over the circle. So it is a mixture of algebra and geometry. The sine and cosine functions are just the coordinates of a point on the unit circle. This implies the most fundamental formula in trigonometry (which we will call here the magic identity) where is any real number (of course measures an angle).

Example. Show that Answer. By definitions of the trigonometric functions we have Hence we have Using the magic identity we get This completes our proof.

Remark. the above formula is fundamental in many ways. For example, it is very useful in techniques of integration.

Example. Simplify the expression Answer. We have by definition of the trigonometric functions Hence Using the magic identity we get Putting stuff together we get This gives Using the magic identity we get Therefore we have Example. Check that Example. Simplify the expression The following identities are very basic to the analysis of trigonometric expressions and functions. These are called Fundamental Identities

Reciprocal identities Pythagorean Identities Quotient Identities  [Geometry] [Algebra] [Differential Equations]
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Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996