Double-Angle Formulas
Published on 20 Mar 2007 at 12:40 am.
1 Comment.
Filed under Mathematics.
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
Anita on 20 Mar 2007 at 2:36 pm: 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!