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Wednesday, 30 July 2014

CHAPTER 3 - 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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