# Using the unit circle to prove the double angle formulas for sine and cosine?

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

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

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:

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