Prove the sine addition formula.

The sine addition formula states that sin(a+b) = sin(a)cos(b) + cos(a)sin(b).

To prove the sine addition formula, we start with the following diagram:


In this diagram, we have a unit circle with angle a and angle b. We draw a line from the origin to the point (cos(a), sin(a)), and another line from the origin to the point (cos(b), sin(b)). We then draw a line from the point (cos(a), sin(a)) to the point (cos(a+b), sin(a+b)).

Using the Pythagorean theorem, we can see that the length of the line from the origin to the point (cos(a), sin(a)) is 1. Therefore, the coordinates of this point are (cos(a), sin(a)).

Similarly, the length of the line from the origin to the point (cos(b), sin(b)) is also 1, so the coordinates of this point are (cos(b), sin(b)).

Using the angle addition formula, we can see that the coordinates of the point (cos(a+b), sin(a+b)) are:

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
sin(a+b) = sin(a)cos(b) + cos(a)sin(b)

Therefore, sin(a+b) = sin(a)cos(b) + cos(a)sin(b), which is the sine addition formula.