If you're like me, you have a hard time remembering the addition of angles formulas for sin and cos:
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 . Well then, it's at some angle with respect to the positive x-axis, so we can represent it as
This is no different from the conversion from cartesian coordinates to polar coordinates: and . Just set and in those equations to get my unit vector.
Now then, suppose we rotate this unit vector through an angle ? That is equivalent to multiplying by the rotation matrix
What is the result going to be? Another unit vector, And what will look like? Well, the matrix multiplication will look like the following:
But rotating a unit vector at angle through an angle is going to result in a unit vector at angle . The representation of that vector is going to be
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, I've reduced remembering the addition of angles formula down to remembering the rotation matrix form, which I think is easier to remember.
To sum up, remember this:
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!