If you're like me, you have a hard time remembering the addition of angles formulas for sin and cos:

$\displaystyle \cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B),$ and

$\displaystyle \sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B).$

The problem is, those formulas are extremely important ones to remember. So, how to do it?

Here's an alternative that I, for one, find easier to remember: rotation matrices.

Let's suppose you have a unit vector $\displaystyle \hat{n}$. Well then, it's at some angle $\displaystyle B$ with respect to the positive x-axis, so we can represent it as

$\displaystyle \hat{n}=\begin{bmatrix}\cos(B)\\ \sin(B)\end{bmatrix}.$

This is no different from the conversion from cartesian coordinates to polar coordinates: $\displaystyle x=r\cos(\theta)$ and $\displaystyle y=r\sin(\theta)$. Just set $\displaystyle r=1$ and $\displaystyle \theta=B$ in those equations to get my unit vector.

Now then, suppose we rotate this unit vector through an angle $\displaystyle A$? That is equivalent to multiplying by the rotation matrix

$\displaystyle R_{A}=\begin{bmatrix}\cos(A) &-\sin(A)\\ \sin(A) &\cos(A)\end{bmatrix}.$

What is the result going to be? Another unit vector, $\displaystyle \hat{n}_{\text{rot}}.$ And what will $\displaystyle \hat{n}_{\text{rot}}$ look like? Well, the matrix multiplication will look like the following:

$\displaystyle \begin{bmatrix}\cos(A) &-\sin(A)\\ \sin(A) &\cos(A)\end{bmatrix}\begin{bmatrix}\cos(B)\\ \sin(B)\end{bmatrix}=\begin{bmatrix}\cos(A)\cos(B)-\sin(A)\sin(B)\\ \sin(A)\cos(B)+\cos(A)\sin(B)\end{bmatrix}.$

But rotating a unit vector at angle $\displaystyle B$ through an angle $\displaystyle A$ is going to result in a unit vector at angle $\displaystyle A+B$. The representation of that vector is going to be

$\displaystyle \hat{n}_{\text{rot}}=\begin{bmatrix}\cos(A+B)\\ \sin(A+B)\end{bmatrix}.$

Hence,

$\displaystyle \begin{bmatrix}\cos(A+B)\\ \sin(A+B)\end{bmatrix}=\begin{bmatrix}\cos(A)\cos( B)-\sin(A)\sin(B)\\ \sin(A)\cos(B)+\cos(A)\sin(B)\end{bmatrix},$

which is equivalent to the addition of angles formulas.

Since matrix multiplication is in my blood to stay, and the conversion from cartesian coordinates to polar coordinates is in my blood,, which I think is easier to remember.I've reduced remembering the addition of angles formula down to remembering the rotation matrix form

To sum up, remember this:

$\displaystyle \begin{bmatrix}\cos(A+B)\\ \sin(A+B)\end{bmatrix}=\begin{bmatrix}\cos(A) &-\sin(A)\\ \sin(A) &\cos(A)\end{bmatrix}\begin{bmatrix}\cos(B)\\ \sin(B)\end{bmatrix},$

or "rotated vector equals rotation matrix times original vector."

As always with mnemonic devices, take this or leave it. If it's useful, great. If something else works better for you, then ignore this!

Regards.