How to Calculate Coordinates at the Origin on Any Unit Circle

You don’t need a unit circle to use this coordinate business when determining the function values of angles graphed in standard position on a circle. You can use a circle with any radius, as long as the center is at the origin. The standard equation for a circle centered at the origin is x² + y² = r².

Using the angles shown, find the sine of alpha.

1. Find the x- and y-coordinates of the point where the terminal side of the angle intersects with the circle.

   The coordinates are x = –5 and y = 12.

2. Determine the radius of the circle.

   The equation of the circle is x² + y² = r². Replacing the x and y in this equation with –5 and 12, respectively, you get (–5)² + (12)² = 25 + 144 = 169 = r². The square root of 169 is 13, so the radius is 13.

3. Determine the ratio for the function and substitute in the values.

   The ratio for sine is y/r, which means that you need only the y-coordinate and radius, so

   

Next, using the angles shown, find the cotangent of beta.

1. Find the x- and y-coordinates of the point where the terminal side of the angle intersects with the circle.

   The coordinates are x = –12 and y = –5.

   The cotangent function uses only the x- and y-coordinates, so you don’t need to solve for the radius.

2. Determine the ratio for the function and substitute in the values.

   The ratio of cotangent is x/y, so

   

Now, using the angles shown, find the secant of gamma.

1. Find the x- and y-coordinates of the point where the terminal side of the angle intersects with the circle.

   The coordinates are x = 0 and y = –13.

2. Determine the radius of the circle.

   Per the first example in this section, the radius is 13.

3. Determine the ratio for the function and substitute in the values.

   The ratio for secant is r/x, so you need only the x-coordinate; substituting in, you get

   

   This answer is undefined, which means that angle gamma has no secant.