While it is convenient to describe the location of a point on a circle using an angle or a distance along the circle, relating this information to the x and y coordinates and the circle equation is an important application of trigonometry.
We begin by drawing a circle centered at the origin with radius r, and marking the point on the circle indicated by some angle θ. This point has coordinates (x, y). If we drop a line segment vertically down from this point to the x-axis, we would form a right triangle inside of the circle. No matter which quadrant our angle θ puts us in we can draw a triangle by dropping a perpendicular line segment to the x-axis, keeping in mind that the values of x and y may be positive or negative, depending on the quadrant. Additionally, if the angle θ puts us on an axis, we simply measure the radius as the x or y with the other value being 0, again ensuring we have appropriate signs on the coordinates based on the quadrant. Triangles obtained from different radii will all be similar triangles, meaning corresponding sides scale proportionally. While the lengths of the sides may change, as we saw in the last section, the ratios of the side lengths will always remain constant for any given angle. To be able to refer to these ratios more easily, we will give them names. Since the ratios depend on the angle, we will write them as functions of the angle θ.