# Math Help - sum of cosines

1. ## sum of cosines

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

2. Originally Posted by qwerty321
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
Why only cosine terms? Why not sine and cosine term?

3. because what I have to do is:
Expand the expression of
y(t) to get a sum of cosines with positive frequencies. Using trigonometric identities find the frequencies of resulting sine waves and compare with the frequency components obtained using MATLAB

4. ## Sum of cosines

Hello qwerty321
Originally Posted by qwerty321
Originally Posted by qwerty321
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 $\sin x = \cos(\pi /2 - x)$.