Difference between revisions of "Product-to-sum identities"
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The product-to-sum identities are as follows: | The product-to-sum identities are as follows: | ||
| − | + | \begin{align*} | |
| − | + | \sin (x) \sin (y) = \frac{1}{2} (\cos (x-y) - \cos (x+y)) \\ | |
| − | + | \sin (x) \cos (y) = \frac{1}{2} (\sin (x-y) + \sin (x+y)) \\ | |
| − | + | \cos (x) \cos (y) = \frac{1}{2} (\cos (x-y) + \cos (x+y)) | |
| + | \end{align*} | ||
They can be derived by expanding out <math>\cos (x+y)</math> and <math>\cos (x-y)</math> or <math>\sin (x+y)</math> and <math>\sin(x-y)</math>, then combining them to isolate each term. | They can be derived by expanding out <math>\cos (x+y)</math> and <math>\cos (x-y)</math> or <math>\sin (x+y)</math> and <math>\sin(x-y)</math>, then combining them to isolate each term. | ||
Latest revision as of 10:49, 10 May 2024
The product-to-sum identities are as follows:
\begin{align*}
\sin (x) \sin (y) = \frac{1}{2} (\cos (x-y) - \cos (x+y)) \\
\sin (x) \cos (y) = \frac{1}{2} (\sin (x-y) + \sin (x+y)) \\
\cos (x) \cos (y) = \frac{1}{2} (\cos (x-y) + \cos (x+y))
\end{align*}
They can be derived by expanding out 
and 
or 
and 
, then combining them to isolate each term.
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