The Addition Formulas

The fundamental identities are very important for the analysis of trigonometric expressions and functions but they are a direct result of the intimate relation between trigonometry and geometry. The power behind the algebraic nature of trigonometry is hidden and can be measured only with the addition formulas

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and

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Of course, we used the fact that

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Example. verify the identity

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Answer. We have

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which gives

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But

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and since

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and tex2html_wrap_inline149 , we get finally

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Remark. In general it is good to check whether the given formula is correct. One way to do that is to substitute some numbers for the variables. For example, if we take a=b = 0, we get

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or we may take tex2html_wrap_inline157 . In this case we have

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Example. Find the exact value of

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Answer. We have

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Hence, using the additions formulas for the cosine function we get

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Since

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we get

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Example. Find the exact value for

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Answer. We have

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Since

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we get

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Finally we have

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Remark. Using the addition formulas, we generate the following identities

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More identities may be proved similar to the above ones. The bottom line is to remember the addition formulas and use them whenever needed.

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

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

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