When students are first introduced to trigonometry, they are often shown a diagram like this one as a way of explaining the sine and cosine functions:

See, theta is an angle of a right triangle, and if the hypotenuse equals one then cosine and sine give us the lengths of the adjacent and opposite legs, respectively. As theta changes, the triangle changes, and sine and cosine return different values. All of this is correct and pretty easy to grasp. The problem is, as a conceptual aid, it starts crumbling the moment theta exceeds 90 degrees. Continue reading “Understanding Trigonometry: A Matter of Complex Circles” →