That look that says, “Trig makes sense!” (Source)

The Cosine Law Unmasked

A Geometric Proof

The Cosine Law. It looks like the Pythagorean Theorem. But what is that extra term?

Compare the two triangles in the figure below. Figure 1 a) shows a right triangle. The squares on sides a and b are equal in area to the square on side c.

Figure 1. a) On the left, a right triangle demonstrates the Pythagorean Theorem. b) On the right, angle C has become acute. The same squares are superimposed on the new triangle.

In Figure 1 b), the position of vertex C has been extended to the right. Angle C is now acute. Sides a and b have been stretched. Side c and the associated square remain unchanged.

We want to find an expression for c² in terms of sides a and b on the new triangle. To do so, we must make sense of the two orphaned squares. They still sum to c².

We’ll designate the areas of the squares as Q and P, since they are no longer a² and b² (Figure 2).

Drop a line from point A perpendicular to side a. This divides the triangle into two right triangles. The one on the left is the right triangle…