Trig sum identities, the easy way

What?! Did I just use all of the words “trig”, “identities” and “easy” in the same sentence?

…Actually, I didn’t, since “Trig sum identities, the easy way” isn’t a sentence. That’s because it no verb, just like this sentence.

Anyway, remember those angle addition identities for sine and cosine?

\begin{array}{rcl} \sin (A + B) & = & \sin A \cos B + \cos A \sin B \\ \cos(A + B) & = & \cos A \cos B - \sin A \sin B \end{array}

If you’ve seen these before, you’ll know what I’m talking about; if you haven’t, trust me: they’re very useful, but a big pain in the butt to memorize. “Hmm, now wait, was that cosine A sine… um, cosine B cosine A… something like that… and, uh, cosine is the one where you switch the sign… or wait, was that sine…” It’s way too easy to get confused. And where do those equations even come from, anyway? If you’ve learned them in school, chances are they were simply presented to you as fact, without any derivation.

To the rescue comes this really nifty derivation: all you need to know is Euler’s Formula (which, if you recall, is also where we got The Most Beautiful Equation in the World from)! Just to remind you, Euler’s formula says that

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

Now let’s just replace \theta with (A + B) and see what happens!

\begin{array}{rcl} & \mathrel{\phantom{=}} & \cos (A + B) + i \sin (A + B) \\ & = & e^{i(A + B)} \\ & = & e^{iA + iB} \\ & = & e^{iA} e^{iB} \\ & = & (\cos A + i \sin A) (\cos B + i \sin B) \\ & = &  \cos A \cos B + i \sin A \cos B + i \cos A \sin B - \sin A \sin B \\ & = & (\cos A \cos B - \sin A \sin B) + i(\sin A \cos B + \cos A \sin B) \end{array}

Look at that! We can just equate the real parts and the imaginary parts to get both identities from this single equation. Nifty, eh?

(I got this from Trig Without Tears, found via Let’s Play Math!.)

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3 Responses to Trig sum identities, the easy way

  1. Pingback: Carnival of Mathematics IX « JD2718

  2. Technically, “That’s because it no verb, just like this sentence” does have one verb in the form of “‘s.” You’re clever anyway.

  3. Brent says:

    Darn it! You’re right, it does… =)

    I got the idea from a chapter in this book about self-referential sentences. The sentence in question was “This sentence no verb.” Which raises some interesting issues. Technically, since it has no verb, it’s not a sentence, hence not self-referential. And yet…

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