The formulas of prostapheresis: memorization technique

This question is related purely for my students of an high school and indirectly for me. The formulas below are the formulas of prostapheresis,

\begin{cases} \sin\alpha+\sin\beta=2\,\sin \dfrac {\alpha+\beta}{2}\, \cos \dfrac {\alpha-\beta}{2} \\ \sin\alpha-\sin\beta=2\sin \dfrac {\alpha-\beta}{2} \,\cos \dfrac {\alpha+\beta}{2}\\ \cos\alpha+\cos\beta=2\cos \dfrac {\alpha+\beta}{2}\,\cos \dfrac {\alpha-\beta}{2}\\ \cos\alpha-\cos\beta=-2 \,\sin \dfrac {\alpha+\beta}{2} \,\sin \dfrac {\alpha-\beta}{2} \end{cases}

and while I am able to find them, I am not able to find a technique to memorize them.

Is there a technique to be able to memorize them?

Solution 1:

It is useful to know the principle that sum or difference to sine and cosine can be written in terms of products of sine and cosine. But I never memorize such identities per se. Whenever needed, you can derive them if you remember the formulas for $\sin(a\pm b)$ and $\cos(a\pm b)$. Or you can simply look at the known list: https://en.wikipedia.org/wiki/List_of_trigonometric_identities

If I need to take a close-book exam that requires memorizing these identities, some observations may be useful for a short-term memory.

• If you know $\sin(-x)=-\sin(x)$, then the second identity comes immediately from the first one.
• For the rest: $$ \begin{align} \color{green}{\sin}\alpha+\color{green}{\sin}\beta=2\,\color{green}{\sin}\dfrac {\alpha+\beta}{2}\, \color{green}{\cos}\dfrac {\alpha-\beta}{2} \\ \cos\alpha\color{red}{+}\cos\beta=2\color{red}{\cos} \dfrac {\alpha+\beta}{2}\,\color{red}{\cos} \dfrac {\alpha-\beta}{2}\\ \cos\alpha\color{blue}{-}\cos\beta=\color{blue}{-}2 \,\color{blue}{\sin} \dfrac {\alpha+\beta}{2} \,\color{blue}{\sin} \dfrac {\alpha-\beta}{2} \end{align} $$

Solution 2:

I am speaking of the angle summation formulas: $$ \eqalign{ & \cos \left( {a \pm b} \right) = \cos a\cos b \mp \sin a\sin b \cr & \sin \left( {a \pm b} \right) = \sin a\cos b \pm \cos a\sin b \cr} $$ Then e.g. summing the equations for $\cos$ $$\cos(a+b)+ \cos(a-b)=2\cos a \cos b$$ After which you can apply $$ \left\{ \matrix{ \alpha = {{a + b} \over 2} \hfill \cr \beta = {{a - b} \over 2} \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{ a = \alpha + \beta \hfill \cr b = \alpha - \beta \hfill \cr} \right. $$

Solution 3:

This answer is not apt to the general high school public, but it can be useful for particularly curious students.

I like very much how these formulas are derived by Feynman in his “Beats” lecture (Lectures on Physics, volume 1, https://www.feynmanlectures.caltech.edu/I_48.html, section 48-1). He uses complex exponential, something that has already been mentioned in comments.

I have always loved his explanation. These apparently obscure formulas actually express the adding and the subtracting of two waves. Since there is a real and an imaginary part, this amounts to four real formulas. The physical phenomenon behind them is the “Beats” one, and it can be heard easily by picking two strings of a guitar. It can actually be used to tune it.