How do trigonometric functions work for angles larger than 90 degrees?

To define trigonometric functions for angles larger than 90 degrees, mathematicians move beyond the right-triangle ratio method and use a coordinate system.12

In this approach, an angle $A$ is placed in standard position with its vertex at the origin and its initial side along the positive $x$-axis. For any point $Q$ with coordinates $(x,y)$ on the terminal side of the angle at a distance $r$ from the origin, the six trigonometric functions are defined as ratios of these coordinates. Because these functions are based on the position of the point $(x,y)$ rather than the sides of a triangle, they naturally extend to any angle, including those greater than 90 degrees.12

These functions are also known as circular functions because their values can be defined using the coordinates of points on a unit circle (a circle with radius 1). In this model, the $x$-coordinate corresponds to the cosine of the angle, and the $y$-coordinate corresponds to the sine. Furthermore, because these functions are periodic, they repeat their values for coterminal angles, such as $A\pm {360}^{\circ }$.21