How to Calculate Trigonometry Functions of Angles Using the Unit Circle
Calculating trig functions of angles within a unit circle is easy as pie. The figure shows a unit circle, which has the equation x^{2} + y^{2} = 1, along with some points on the circle and their coordinates.
Using the angles shown, find the tangent of theta.

Find the x and ycoordinates of the point where the angle’s terminal side intersects with the circle.
The coordinates are
The radius is r = 1.

Determine the ratio for the function and substitute in the values.
The ratio for the tangent is y/x, so you find that
Next, using the angles shown, find the cosine of sigma.

Find the x and ycoordinates of the point where the terminal side of the angle intersects with the circle.
The coordinates are
the radius is r = 1.

Determine the ratio for the function and substitute in the values.
The ratio for the cosine is x/r, which means that you need only the xcoordinate, so
Now, using the angles shown, find the cosecant of beta.

Find the x and ycoordinates of the point where the terminal side of the angle intersects with the circle.
The coordinates are x = 0 and y = –1; the radius is r = 1.

Determine the ratio for the function and substitute in the values.
The ratio for cosecant is r/y, which means that you need only the ycoordinate, so