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An overview of the unit circle, its concepts, and its use in calculating trigonometric functions such as sin(θ), cos(θ), tan(θ), sec(θ), csc(θ), and cot(θ). It explains the difference between degrees and radians, and provides conversion factors and examples for converting between the two. The document also introduces the triangle method for solving unit circle problems.
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11 Provided by the Academic Center for Excellence 1 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
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.
22 Provided by the Academic Center for Excellence 2 The Unit Circle
The Standard Unit Circle
π
π
2 π
3π
π
5π
7π
5π
π
4π
3π
5π
7π
11π
π
0π
2π
ξ
−ξ 2
ξ 2
−ξ 3
ξ
ξ
ξ
ξ
ξ 3
ξ 3
ξ
ξ
ξ
ξ
ξ 3
ξ 3
Y - Axis
X - Axis
44 Provided by the Academic Center for Excellence 4 The Unit Circle
Example 2 :
Use the triangle method to solve:
Step 1: Choose a triangle.
Because 45 is divisible by 45, use the 45°−45°−90° triangle.
Step 2: Draw the triangle in the correct quadrant.
This triangle will be in quadrant I because 45° is between 0 ° and 90°.
Step 3: Analyze the triangle.
Remember that cos(θ) represents
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
. Here, the adjacent side to θ (or 45°) is 1,
and the hypotenuse is ξ
1
ξ
2
Step 4: Rationalize the denominator.
The denominator is rationalized by removing the square roots. Do this by multiplying
the numerator and denominator of the resulting fraction
1
ξ
2
by the radical in the
denominator ξ 2.
Cos
ξ
ξ
ξ
Cos
ξ
ξ 2
55 Provided by the Academic Center for Excellence 5 The Unit Circle
Example 3 :
Use the triangle method to solve:
Step 1: Choose a triangle.
Because 240 is divisible by 30, use the 30°−60°−90° triangle.
Step 2: Draw the triangle in the correct quadrant.
This triangle will be in quadrant III because 240 ° is between 18 0° and 270 °. Additionally,
60 ° will be the angle near the origin because 240° is 60° more than 180°.
Step 3: Analyze the triangle.
Note that tan(θ) represents
𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡
. Here, the opposite side is − ξ
3 while the adjacent
side is − 1. This results in 𝑡𝑎𝑛( 240 °) =
−ξ 3
− 1
Step 4: Simplify.
The negatives cancel each other out to leave
ξ 3
1
, which is ξ 3.
= ξ 3
−ξ 3