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Wednesday, 30 July 2014

CHAPTER 6- 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
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