Suppose that a triangle has sides of length a and b adjacent to an angle C. What is the length c of the remaining side? By putting the triangle in standard position with the side of length a aligned with the x-axis and the angle C at the origin we can use coordinates to prove:
PROOF:
The coordinates of the three vertices of the triangle are (0,0),(a,0), and. The distance formula in the plane tells us that the length c is
PROOF:
Let us consider a diagram with a unit circle and the angle s with its initial side on the positive x-axis and the angle t also having its initial side on the positive x-axis:
The points of intersection with the unit circle are
We can calculate the square of the distance between the last two of these points using either the law of cosines or the distance formula to get
Thus
PROOF:
For the cosine of a sum we use
For the formulas involving
we use the complement relation:
PROOF:
From the formula for the cosine of a sum we get
We can then use
to obtain either
or
In calculus this result is usually used in one of the forms
