# Thread: Trigonometric Sum

1. ## Trigonometric Sum

I would like you to show me how to get the amplitude and phase shift of the sum of the trigonometric sum below.

8sin(50pi*x+pi/4)+7sin(50pi*x-pi/3)

2. ## Trigonometry

Hello magentarita
Originally Posted by magentarita
I would like you to show me how to get the amplitude and phase shift of the sum of the trigonometric sum below.

8sin(50pi*x+pi/4)+7sin(50pi*x-pi/3)

$8\sin(50\pi x +\pi/4) +7\sin(50\pi x -\pi/3)$

= $8\sin(50\pi x)\cos(\pi/4) +8\cos(50\pi x)\sin(\pi/4) + 7\sin(50\pi x)\cos(\pi/3)$ $-7\cos(50\pi x)\sin(\pi/3)$

$= \Big(\frac{8}{\sqrt2}+\frac{7}{2}\Big)\sin(50\pi x) + \Big(\frac{8}{\sqrt2}-\frac{7\sqrt3}{2}\Big)\cos(50\pi x)$

$= \frac{1}{2\sqrt2}\Big((16+7\sqrt2)\sin(50\pi x)-(7\sqrt6-16)\cos(50\pi x)\Big)$

$= r\sin(50\pi[x-\alpha])$, where

$r^2 = \tfrac18\Big((16+7\sqrt2)^2+(7\sqrt6-16)^2\Big)$

and $\tan(50\pi\alpha) = \frac{(7\sqrt6-16)}{(16+7\sqrt2)}$

Can you tidy up and simplify now?

Grandad

3. ## yes...

Originally Posted by Grandad
Hello magentarita $8\sin(50\pi x +\pi/4) +7\sin(50\pi x -\pi/3)$

= $8\sin(50\pi x)\cos(\pi/4) +8\cos(50\pi x)\sin(\pi/4) + 7\sin(50\pi x)\cos(\pi/3)$ $-7\cos(50\pi x)\sin(\pi/3)$

$= \Big(\frac{8}{\sqrt2}+\frac{7}{2}\Big)\sin(50\pi x) + \Big(\frac{8}{\sqrt2}-\frac{7\sqrt3}{2}\Big)\cos(50\pi x)$

$= \frac{1}{2\sqrt2}\Big((16+7\sqrt2)\sin(50\pi x)-(7\sqrt6-16)\cos(50\pi x)\Big)$

$= r\sin(50\pi[x-\alpha])$, where

$r^2 = \tfrac18\Big((16+7\sqrt2)^2+(7\sqrt6-16)^2\Big)$

and $\tan(50\pi\alpha) = \frac{(7\sqrt6-16)}{(16+7\sqrt2)}$

Can you tidy up and simplify now?

Grandad
Yes, I can. This question is not easy.

### sin50pi(3)

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