Thread: sum of cosines
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Old January 15th, 2009, 12:02 AM
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Default Sum of cosines
Hello qwerty321
Quote:
Originally Posted by qwerty321 View Post
Quote:
Originally Posted by qwerty321 View Post
I have the function: y(t) = (cos(300πt) + sin(500πt))^3

 I need too expand the expression of y to get a sum of cosines with positive frequencies. Using
 trigonometric identities find the frequencies of resulting sine waves.Can someone help me with that?
 Thank you
Using A to stand for 300\pi t and B for 500 \pi t:

(\cos A + \sin B)^3 = \cos^3A+3\cos^2A\sin B + 3\cos A \sin^2B+\sin^3B

Now make repeated use of the following identities:

\cos^2x = \frac{1}{2}(\cos 2x -1)

\sin^2x = \frac{1}{2}(1-\cos 2x)

\cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y))

\sin x \cos y = \frac{1}{2}(\sin(x+y) + sin(x-y))

So, for example: \cos^3A = \frac{1}{2}(\cos 2A -1)\cos A

= \frac{1}{2}\left(\frac{1}{2}(\cos3A + \cos A) -\cos A\right)

= \frac{1}{4}(\cos 3A - \cos A)

And the second term is: 3 \cos^2 A \sin B = \frac{3}{2}(\cos 2A -1) \sin B

= \frac{3}{2}\left(\frac{1}{2}(\sin(B+2A)+\sin(B-2A)) -\sin B \right)

Similarly with the last two terms. Finally, if you need to get an expression in terms of cosine only, you could use \sin x = \cos(\pi /2 - x).

Grandad
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