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)

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where tex2html_wrap_inline93 is any real number (of course tex2html_wrap_inline93 measures an angle).

Example. Show that

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Answer. By definitions of the trigonometric functions we have

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Hence we have

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Using the magic identity we get

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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

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Answer. We have by definition of the trigonometric functions

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Hence

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Using the magic identity we get

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Putting stuff together we get

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This gives

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Using the magic identity we get

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Therefore we have

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Example. Check that

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Answer.

Example. Simplify the expression

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Answer.

The following identities are very basic to the analysis of trigonometric expressions and functions. These are called Fundamental Identities

Reciprocal identities

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Pythagorean Identities

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Quotient Identities

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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