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How do you use the unit circle to prove the double angle formulas for sine and cosine?

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I'd suggest deriving the formula for sums of angles. –  user21436 Apr 25 '12 at 13:00
Use complex numbers. –  The Chaz 2.0 Apr 25 '12 at 13:01
I once saw a direct geometric proof of $\cos\,u=2\cos^2 u-1$, but I don't remember where I saw it... –  J. M. Apr 25 '12 at 13:36
@J.M. Here is one geometric proof of $\cos 2\theta=1-2\sin^2\theta$. –  David Mitra Apr 25 '12 at 13:56
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2 Answers

Look at this figure:


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Ingenious -- but the unit circle is mostly decoration here, isn't it? –  Henning Makholm Apr 25 '12 at 15:22
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Here's one possibility. Say we want to find $\sin 2\theta$ and $\cos 2\theta$. Draw the unit circle in an ordinary $x$-$y$ coordinarte system, and also introduce a new coordinate system $x'$-$y'$ that has been turned $\theta$ clockwise around the origin. It is important that the unit circle in the $xy$ system and in the $x'y'$ system is the same:

(a diagram)

The relation between the two coordinate systems is $$ x' = x\cos\theta - y\sin\theta \qquad y'=x\sin\theta + y\cos\theta $$

The point $P$ on the diagram has coordinates $(x,y)=(\cos\theta,\sin\theta)$ in the $xy$-system, but in the $x'y'$ system is is $2\theta$ above the $x'$-axis and so its coordinates there must be $(x',y')=(\cos2\theta, \sin2\theta)$. Substituting this into the known relation between the coordinate systems yields: $$ \cos2\theta = (\cos \theta)^2 - (\sin\theta)^2 \qquad \sin2\theta = \cos(\theta)\sin(\theta) + \sin(\theta)\cos(\theta)$$

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How did you produce your diagram? Did you draw it with pen and then scan it? –  MJD Apr 25 '12 at 14:41
I drew it with a pen, photographed it with a digital camera, and then cleaned it up (for background and contrast) in Gimp. –  Henning Makholm Apr 25 '12 at 14:43
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