Sines Past 90°

A triangle can have an angle over 90°, so the law of sines needs the sine of an obtuse angle. On the unit circle, the sine of an angle is the height of its point. The point at 150° is the mirror image of the point at 30° across the y axis, at the same height.

So sin(150°) = sin(30°), and in general sin(180° - theta) = sin(theta). Every angle strictly between 0° and 180° is above the x axis, so every angle of a triangle has a positive sine.

The catch: a sine does not pin down a triangle's angle. A sine of 1/2 belongs to both 30° and 150°, and both fit in a triangle. When the law of sines hands you an angle, it can hand you two.