The addition formulas for sine and cosine look like this:

\begin{aligned}
\cos(\alpha + \beta) &= \cos \alpha \cos \beta – \sin \alpha \sin \beta, \\
\sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta. \\
\end{aligned}

I can never remember them.

Abramowitz and Stegun: Handbook of Mathematical Functions

One solution is of course to look them up in a book or search the internet. What I usually do, though, is derive them using complex arithmetic. Apart from the usual rules known from real-valued arithmetic, all that is needed is knowing e^{i \phi} = \cos \phi + i \sin \phi and i^2 = -1. Then you get

\begin{aligned}
\cos(\alpha + \beta) + i \sin(\alpha + \beta) &= e^{i (\alpha+\beta)} \\
&= e^{i \alpha} e^{i \beta} \\
&= (\cos \alpha + i \sin \alpha)(\cos \beta + i \sin \beta) \\
&= (\cos \alpha \cos \beta – \sin \alpha \sin \beta) + i (\sin \alpha \cos \beta + \cos \alpha \sin \beta).
\end{aligned}

By equating the real and imaginary parts you have the answer.

Flicking through Proofs Without Words II by Roger B. Nelsen, I saw the following wonderful figure. It could be a contender to the easiest way to remember the addition formulas.

Figure 1

Figure 1

(Attributed to the author of the book.) It should be pretty much self-explanatory. Apart from using sine and cosine to assign side-lengths to the four relevant right-angled triangles, all you need know is that the sum of the angles in a triangle is equal to two right angles (to realize that the two \alpha-angles are indeed equal).

Nelsen: Proofs without Words II
Nelsen: Proofs without Words

Both volumes of Proofs Without Words contain other “visual proofs” of the addition formulas. Some of these can also be found online.

How do you remember the addition formulas for sine and cosine?

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