Dear Uncle Colin,

I’m fine at remembering the formulas for things like $\sin(A+B)$, but I struggle to come up with the formulas for things like $\sin(A) + \sin(B)$. Any hints?

Have I Learnt Badly Equations Regarding Trig?

Hi, HILBERT, and thanks for your message!

I always used to loathe these ones, too – they don’t often show up at A-level, unless you’re integrating something horrid, so when they do, it’s an absolute bloodbath. (I don’t know what’s in the formula book these days, or if it still exists. They may be things you can look up.)

I do have a key insight to share, though: you can always write $A$ and $B$ as $M+D$ and $M-D$, where $M$ is the mean of $A$ and $B$, and $D$ is half the difference between $A$ and $B$.

This might not seem like much of an insight. In some ways, it looks like it complicates things rather than simplifies them. But, when I learned how to do this, it was a mess of trial and error – by which I mean the whole process was a trial, and I always made errors.

But! With your angle split, you can almost write down the formulas:

  • $\sin(A) = \sin(M+D) = \sin(M)\cos(D) + \sin(D)\cos(M)$
  • $\sin(B) = \sin(M-D) = \sin(M)\cos(D) - \sin(D)\cos(M)$
  • So $\sin(A) + \sin(B) = 2\sin(M)\cos(D)$

For completeness, you should probably add “where $M = \frac{A+B}{2}$ and $D = \frac{A-B}{2}$”.

A similar process works for the difference of sines, and sums and differences of cosines.

I hope that helps!

- Uncle Colin