Double-Angle Formulas

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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drknuth

Kevin Knuth was born in Fond du Lac, Wisconsin, USA. He received his Ph.D. in physics with a minor in mathematics at the University of Minnesota in 1995, and held postdoctoral positions studying neuroscience in New Orleans and New York City. From 2001 through 2005 he was a research scientist at NASA Ames Research Center in the Intelligent Systems Division developing machine learning techniques and their applications in the Autonomous Systems and Robotics Area. Currently he is an Assistant Professor in Physics at the University at Albany, State University of New York. His scientific interests include: probability theory, astrophysics, complex systems, earth science, and brain dynamics. For recreation, he enjoys hiking, birdwatching and poking around tidal pools.

One thought on “Double-Angle Formulas”

  1. 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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