Trigonometric functions and their properties

In earlier classes you met the trigonometric ratios of an angle only through a right triangle, which forces the angle to be acute — strictly between $0^\circ$ and $90^\circ$. Once we start using radian measure and want to talk about angles of any size at all — negative angles, angles larger than $360^\circ$, or an angle that has wrapped several times around a circle — the right-triangle picture no longer makes sense.

This section frees the six ratios from the right triangle. We redefine sine, cosine, and the rest for any angle whatsoever, using the coordinates of a point on the angle's terminal side instead of the sides of a triangle. Once defined this way for every real angle (in radian measure), they are called trigonometric functions rather than trigonometric ratios. The next four subsections build the idea up in stages: the coordinate definitions and their quadrant-wise signs (§3.4.1); stretching the domain from angles to all real numbers through the wrapping function (§3.4.2); the b …