Cosines Past 90°

The cosine of an angle is the x coordinate of its point on the unit circle. Past 90° the point is left of the y axis, so the cosine is negative. The mirror image that kept the sine changes the sign of x:

cos(180° - theta) = -cos(theta)

So cos(120°) = -cos(60°), which is −1/2. This matters twice in this section. The law of cosines subtracts a term with cos C, so an obtuse C, with its negative cosine, makes the third side longer than Pythagoras gives. And unlike the sine, the cosine tells an acute angle from an obtuse one: between 0° and 180° each cosine belongs to exactly one angle.