The Unit Circle
The unit circle is a circle whose center
is the origin and whose radius is 1. It is defined by equation x2 + y2 =
The most useful and interesting property of the unit circle
is that the coordinates of a given point on the circle can be found
using only the measure of the angle.
Any radius of the unit circle is the hypotenuse of a right
triangle that has a (horizontal) leg of length cos
and a (vertical) leg of length sin
. The angle
is defined as the radius measured
in standard position. These relationships are easy to see using
the trigonometric functions:
As you can see, because the radius of the unit circle
is 1, the trigonometric functions sine and cosine are simplified:
. This means
that another way to write the coordinates of a point (x
on the unit circle is (cos
is the measure of the angle in standard
position whose terminal side contains the point.
Here’s an example of a typical Math IC question that tests
are the coordinates of the point P pictured below?
is the endpoint of a radius of
the unit circle that forms a 30º angle with the negative x
This means that an angle of 210º in standard position would terminate
in the same position. So, the coordinates of the point are (cos
210º, sin 210º) = (–/2
Both coordinates must be negative, since the point is in the third
The unit circle also provides a lot of information about
the range of trigonometric functions and the values of the functions
at certain angles.
For example, because the unit circle has a radius of one
and its points are all of the form (cos
), we know that:
Tangent ranges from –∞ to ∞, but it is undefined at every
angle whose cosine is 0. Can you guess why? Look at the formula
. If cos
= 0, then division by 0 occurs, and
so the quotient, tan
, is undefined.
The Unit Circle and Important Angles
Using the unit circle makes it easy to find the values
of trigonometric functions at quadrantal angles. For example, a
90º rotation from the positive x-axis puts you
on the positive y-axis, which intersects the unit
circle at the point (0, 1). From this, you know that (cos 90º, sin
90º) = (0, 1). Here is a graph of the values of all three trigonometric
functions at each quadrantal angle:
There are a few other common angles besides the quadrantal
angles whose trigonometric function values you should already know.
Listed below are the values of sine, cosine, and tangent taken at
30º, 45º, and 60º. You might recognize some of these values from
the section on special triangles.