One way to find the values of the trig functions for angles is to use the coordinates of points on a circle that has its center at the origin. Letting the positive x-axis be the initial side of an angle, you can use the coordinates of the point where the terminal side intersects with the circle to determine the trig functions.

The figure shows a circle with a radius of r that has an angle drawn in standard position.

The equation of a circle is x2 + y2 = r2. Based on this equation and the coordinates of the point, (x,y), where the terminal side of the angle intersects the circle, the six trig functions for angle theta are defined as follows:

You can see where these definitions come from if you picture a right triangle formed by dropping a perpendicular segment from the point (x,y) to the x-axis. The following figure shows such a right triangle.

Remember that the x-value is to the right (or left) of the origin, and the y-value is above (or below) the x-axis — and use those values as lengths of the triangle’s sides. Therefore, the side opposite angle theta is y, the value of the y-coordinate. The adjacent side is x, the value of the x-coordinate.

Take note that for angles in the second quadrant, for example, the x-values are negative, and the y-values are positive. The radius, however, is always a positive number. With the x-values negative and the y-values positive, you see that the sine and cosecant are positive, but the other functions are all negative, because they all have an x in their ratios.

The signs of the trig functions all fall into line when you use this coordinate system, so no need to worry about remembering the ASTC rule here.