The Unit Circle
The unit circle is a circle whose center is the origin
and whose radius is 1. It is defined by the 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 knowledge of 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 question on the Math IIC
that tests this principle:
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 has points all of the form (cos
), we know that:
Tangent ranges from –∞ to ∞, but 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
the quotient, tan
, is undefined.
Radians and Degrees
Radians are another way to measure angles. Sometimes radians
will be used in questions, and other times you may choose to use
them since they can sometimes be more convenient to use than degrees.
Radians will never appear on the Math IC test, but for the Math
IIC, it is necessary to know how to convert between radians and
A degree is equal to 1
/360 of a
circle, while a radian is equal to the angle that intercepts an arc
of the same length as the radius of the circle. In the figure below,
arc AB has length r, and the central
angle measures one radian.
When converting between the two measurement systems, use
which can be simplified down to:
To convert from degrees to radians: multiply
the degree measure by π/
180. For example, 60º
is equal to 60π/
To convert from radians to degrees: multiply the degree
measure by 180/
π. For example,
equal to 180π/
4π = 45º.
Here are the most important angle measures in degrees
On the Math IIC, it is sometimes a better idea to work
solely in radians rather than convert back and forth between radians
and degrees. Using radians is especially easy on graphing calculators
that allow you to switch into radian mode.