If you have the value of one of a point’s coordinates on the unit circle and need to find the other, you can substitute the known value into the unit-circle equation and solve for the missing value.
You can choose any number between 1 and –1, because that’s how far the unit circle extends along the x- and y-axes. For example, say 2/5 is the x-coordinate of a point on the unit circle. You can find the y-coordinate like so:
Substitute the x-coordinate value into the unit-circle equation.
Square the x-coordinate and subtract that value from each side.
Take the square root of each side.
Note that the y-coordinate can have two values, because the unit circle has two different points for every particular x-coordinate (and for every y-coordinate). You can see how that happens:
Another example: Find the x-coordinate (or coordinates) if the y-coordinate is –7/25.
Substitute the y-coordinate value into the unit-circle equation.
Square the y-coordinate and subtract that value from each side.
Take the square root of each side.
As you can see, the x-coordinate here has two values, and the two points are
