11Provided by the Academic Center for Excellence 1 The Unit Circle
Updated October 2019
The Unit Circle
The unit circle can be used to calculate the trigonometric functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ),
and cot(θ). It utilizes (x,y) coordinates to label the points on the circle, where x represents cos(θ) of a
given angle, y represents sin(θ), and 𝑦
𝑥 represents tan(θ). Theta, or θ, represents the angle in degrees or
radians. This handout will describe unit circle concepts, define degrees and radians, and explain the
conversion process between degrees and radians. It will also demonstrate an additional way of solving
unit circle problems called the triangle method.
What is the unit circle?
The unit circle has a radius of one. The intersection of the x and y-axes (0,0) is known as the origin. The
angles on the unit circle can be in degrees or radians.
The circle is divided into 360 degrees starting on the right side of the x–axis and moving
counterclockwise until a full rotation has been completed. In radians, this would be 2π. The unit circle
is shown on the next page.
Converting Between Degrees and Radians
In trigonometry, most calculations use radians. Therefore, it is important to know how to convert
between degrees and radians using the following conversion factors.
Conversion Factors
𝑫𝒆𝒈𝒓𝒆𝒆𝒔× 𝝅
𝟏𝟖𝟎°=𝑹𝒂𝒅𝒊𝒂𝒏𝒔
𝑹𝒂𝒅𝒊𝒂𝒏𝒔×𝟏𝟖𝟎°
𝝅=𝑫𝒆𝒈𝒓𝒆𝒆𝒔
Degrees
Degrees, denoted by °, are a
measurement of angle size that is
determined by dividing a circle into
360 equal pieces.
Radians
Radians are unit-less but are always
written with respect to π. They
measure an angle in relation to a
section of the unit circle’s
circumference.
Example 1:
Convert 120 to radians.
Step 1: If starting with degrees, 180 should be
on the bottom of the conversion factor so that
the degrees cancel.
120°×𝜋
180°=120°(𝜋)
1(180°)=2𝜋
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