Prove the addition formula for sine.

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

To prove this formula, we start with the following diagram:


In this diagram, we have two angles a and b, and we want to find the sine of their sum, a+b. We can use the fact that sin(x) is the y-coordinate of the point on the unit circle that is x radians counterclockwise from the positive x-axis.

Using this fact, we can see that sin(a) is the y-coordinate of point A, and sin(b) is the y-coordinate of point B. Similarly, cos(a) is the x-coordinate of point A, and cos(b) is the x-coordinate of point B.

Now, consider the point C, which is the endpoint of the vector that starts at the origin and goes to point B, and then turns counterclockwise by an angle of a. The coordinates of point C are (cos(a+b), sin(a+b)), since a+b is the angle that this vector makes with the positive x-axis.

We can also express the vector from the origin to point C as the sum of two vectors: one that starts at the origin and goes to point A, and another that starts at point A and goes to point C. The first vector has length cos(a) and direction a, and the second vector has length sin(b) and direction (a+b).

Using vector addition, we can find the coordinates of point C as follows:

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

These are the addition formulas for cosine and sine, respectively. Therefore, we have proven the addition formula for sine.