Double-Angle Formulas

Posted on March 20, 2007 by drknuth | 1 Comment

To display equations, I recently installed LaTeX into WordPress by using a Latexrender plugin and it works great! I am going to try it out by blogging about a relatively easy way to derive the double angle formulas for the sine and cosine. The double angle formulas relate $\sin( 2\theta)$ and $\cos( 2\theta)$ to $\sin( \theta)$ and $\cos( \theta)$ .

Let’s be honest, who wants to waste their cortical space remembering formulas like this. Instead, you can derive them pretty easily by remembering a few more elementary facts. First, you can write any complex number as:

$e^{i\theta}=\cos(\theta)+i\sin(\theta)$ .

Next, you must remember how to work with exponents:

$x^{a} x^{b} = x^{a+b}$

and

$(x^a)^2 = x^{2a}$

Now we put it all together.
We start by writing

$e^{i\theta}=\cos(\theta)+i\sin(\theta)$

We then square both sides

$(e^{i\theta})^2=(\cos(\theta)+i\sin(\theta))^2$

Now use the rule for exponents on the left-hand side, and multiply out the right-hand side:

$(e^{i 2\theta})=\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)$

Now write the left-hand side in terms of sines and cosines again:

$\cos(2\theta)+i\sin(2\theta)$
$=\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)$

Last, equate the real parts and the imaginary parts and you have your double-angle formulas:

$\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)$
$\sin(2\theta) = 2\cos(\theta)\sin(\theta)$

Easy as $\pi$ !
Kevin Knuth
Albany NY

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One Response to Double-Angle Formulas

Anita | March 20, 2007 at 2:36 pm |

I keep thinking that I need to play around with LaTeX so that I can see where it might be useful in my world. Thanks for the demo!

…now to figure out what all of that math means!

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