My Math Forum Express cos(4a) + sin(4a) using only cos(a) and sin(a)

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 August 6th, 2013, 04:25 AM #1 Senior Member   Joined: Sep 2010 From: Oslo, Norway Posts: 162 Thanks: 2 Express cos(4a) + sin(4a) using only cos(a) and sin(a) Hello, $\cos(3\theta) + i\,\sin(3\theta)= (\cos(\theta) + i\,\sin(\theta))^3$ Distribute and group terms containing i: $(\cos^3\theta - 3\cos\theta\sin^2\theta) + i(3\cos^2\theta \sin\theta - \sin^3\theta)$ 1. Now, just by seeing a glaring pattern, I am almost convinced that $\cos(3\theta)= \cos^3\theta - 3\cos\theta\sin^2\theta$ and $\sin(3\theta)= 3\cos^2\theta \sin\theta - \sin^3\theta$, but how can I be sure? 2. What is the name of the "method" used to arrive at the above conclusion? (I am sorry that I do not know a more precise word than 'method'.) Thank you for your time. Kind regards, Marius
 August 6th, 2013, 01:01 PM #2 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 Re: Express cos(4a) + sin(4a) using only cos(a) and sin(a) 1. When you have an equation involving complex quantities, the real and imaginary parts are separate equations. 2. I don't know if it has a specific name. Question - title seems to have little relation to the text.
 September 6th, 2013, 01:48 AM #3 Newbie   Joined: Jun 2013 Posts: 15 Thanks: 0 Re: Express cos(4a) + sin(4a) using only cos(a) and sin(a) Hi solved your problem here "Solution: Given cos (3?) + i sin (3?) = (cos ? + i sin ?)^3 we know by de Moivre's theorem (cos ? + i sin ?)^n = cos (n?) + i sin (n?) where cos (3?) = cos^3 ? - 3cos ? sin² ? sin (3?) = 3cos² ? - sin ^3 ? "
 September 6th, 2013, 12:09 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,757 Thanks: 2138 Similarly, cos(4a) + sin(4a) ? 1 + 4sin(a)cos³a - 8sin²a cos²a - 4sin³a cos(a).

 Tags cos4a, cosa, express, sin4a, sina

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# find cos^4a-sin^4a in terms of cosa

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