Product and Sum Formulas

From the Addition Formulas, we derive the following trigonometric formulas (or identities)

displaymath113

Remark. It is clear that the third formula and the fourth are identical (use the property tex2html_wrap_inline115 to see it).

The above formulas are important whenever need rises to transform the product of sine and cosine into a sum. This is a very useful idea in techniques of integration.

Example. Express the product tex2html_wrap_inline117 as a sum of trigonometric functions.

Answer. We have

displaymath119

which gives

displaymath121

Note that the above formulas may be used to transform a sum into a product via the identities

displaymath123

Example. Express tex2html_wrap_inline125 as a product.

Answer. We have

displaymath127

Note that we used tex2html_wrap_inline129 .

Example. Verify the formula

displaymath131

Answer. We have

displaymath133

and

displaymath135

Hence

displaymath137

which clearly implies

displaymath131

Example. Find the real number x such that tex2html_wrap_inline143 and

displaymath145

Answer. Many ways may be used to tackle this problem. Let us use the above formulas. We have

displaymath147

Hence

displaymath149

Since tex2html_wrap_inline143 , the equation tex2html_wrap_inline153 gives tex2html_wrap_inline155 and the equation tex2html_wrap_inline157 gives tex2html_wrap_inline159 . Therefore, the solutions to the equation

displaymath145

are

displaymath163

Example. Verify the identity

displaymath165

Answer. We have

displaymath167

Using the above formulas we get

displaymath169

Hence

displaymath171

which implies

displaymath173

Since tex2html_wrap_inline175 , we get

displaymath177

[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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