Online Cortex Thoughts on Science and Technology

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 [tex]\sin( 2\theta)[/tex] and  [tex]\cos( 2\theta)[/tex]  to [tex]\sin( \theta)[/tex] and  [tex]\cos( \theta)[/tex]. 

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:

[tex]e^{i\theta}=\cos(\theta)+i\sin(\theta)[/tex].

Next, you must remember how to work with exponents:

[tex]x^{a} x^{b} = x^{a+b}[/tex]

and

[tex](x^a)^2 = x^{2a}[/tex]

Now we put it all together.
We start by writing

[tex]e^{i\theta}=\cos(\theta)+i\sin(\theta)[/tex]

We then square both sides

[tex](e^{i\theta})^2=(\cos(\theta)+i\sin(\theta))^2[/tex]

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

[tex](e^{i 2\theta})=\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)[/tex]

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

[tex]\cos(2\theta)+i\sin(2\theta)[/tex]
      [tex]=\cos^2(\theta)+2i\cos(\theta)\sin(\theta)-\sin^2(\theta)[/tex]

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

[tex]\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta)[/tex]
[tex]\sin(2\theta) = 2\cos(\theta)\sin(\theta)[/tex]

Easy as [tex]\pi[/tex]!
Kevin Knuth
Albany NY

Posted under Mathematics