Angle Addition Formula

This lesson will be a brief introduction to angle addition formulas by studying the one for the cosine. Angle addition formulas provide ways to find the sine or cosine of a sum of angles, which may be hard to find, in terms of the sine and cosine of each of the angles individually, which may be easy to find. For example, cos(105°) is hard to find on its own, but if you only need to know cos(45°) and cos(60°) to find it (60 + 45 = 105), then the task becomes much easier.

The angle addition and subtraction formulas for cosine are given by

So, if we want to find cos(105°), it's as simple as

But, of course, where did this formula come from? If you want to know, all you need is some geometry and a bit of elbow grease.

First, we can construct the unit circle with angles α and -β as shown below.

 The resulting triangle will have vertices at (0, 0), (cos(α), sin(α)), and (cos(-β), sin(-β)).

We can recall the distance formula, which states that the distance between two points (x₁, y₂) and (x₁, y₂) is

So, the distance between the two vertices on the unit circle is given by

where we made use of the fact that cosine is even and sine is odd in the second line and of the Pythagorean identity in the fifth line.

Now, to relate this to cos(α + β), we can simply rotate the triangle by an angle of β.

Using the distance formula again, we have

Now, as each expression is equal to AB, we can square both of them and equate them to each other. The math from there is fairly straightforward.

To get to the angle subtraction formula, simply replace β with -β and exploit the evenness of the cosine and the oddness of the sine.