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The Unit Circle

Written by tutor ShuJen W.

The Unit Circle

The above drawing is the graph of the Unit Circle on the X – Y Coordinate Axis.
It can be seen from the graph, that the Unit Circle is defined as having a Radius ( r ) = 1.
Going from Quadrant I to Quadrant IV, counter clockwise, the Coordinate points on the axis of the Unit Circle are:

(1, 0), (0, 1), (-1, 0), and (0, -1)

This is important to remember when we define the X and Y Coordinates around the Unit Circle. The Unit Circle has 360°. In the above graph, the Unit Circle is divided into 4 Quadrants that split the Unit Circle into 4 equal pieces. Each piece is exactly 90°.

Question: Why is each section / Quadrant equal to 90°?

Also it can be shown that the Unit Circle is made up of four 90° angles, which total 360°:

The Four Quadrants

Now we are going to divide the Unit Circle into 30°, 45°, and 60° angles. These are the special angles and are very important to remember.

Special Angles: 30, 45, 90

Let’s start with Quadrant I, since this is the basics, and the X and Y Coordinates are both Positive. See below.

We then move on to Quadrant II, which starts at 90° and goes to 180°. From the diagram below, each angle in Quadrant II measures 30°, 45°, and 60° within that Quadrant. However, since the angles have a point of reference at the 0° mark in Quadrant I, they are labeled according to the angle they make from Quadrant I to Quadrant II. For example, 45° in Quadrant II is labeled 135° because that is the angle it makes from 0° in Quadrant I to the 45° angle in Quadrant II. Also, it can be seen from the graph that the 45° angle in Quadrant II falls between 90° and 180° on the Unit Circle. This is done for 30°, 45°, and 60° angles in each Quadrant. See below.

Quadrant II

Unit Circle with Angle Divisions

The graph below shows the degrees of the Unit Circle in all 4 Quadrants, from 0° to 360°.

Unit Circle with Angle Divisions

Now we will add Radians to the Unit Circle. Radians is the standard unit of angle measure.
The Formula for calculating Radians is:

Equation for Radians

We will calculate the Radians for each degree on the Unit Circle labeled above.

Degrees Formula Radians (simplified)
(0°)*(π/180°) 0
30° (30°)*(π/180°) = 30π/180° radians π/6
45° (45°)*(π/180°) = 45π/180° radians π/4
60° (60°)*(π/180°) = 60π/180° radians π/3
90° (90°)*(π/180°) = 90π/180° radians π/2
120° (120°)*(π/180°) = 120π/180° radians /3
135° (135°)*(π/180°) = 135π/180° radians /4
150° (150°)*(π/180°) = 150π/180° radians /6
180° (180°)*(π/180°) = 180π/180° radians π/1
210° (210°)*(π/180°) = 210π/180° radians /6
225° (225°)*(π/180°) = 225π/180° radians /4
240° (240°)*(π/180°) = 240π/180° radians /3
270° (270°)*(π/180°) = 270π/180° radians /2
300° (300°)*(π/180°) = 300π/180° radians /3
315° (315°)*(π/180°) = 315π/180° radians /4
330° (330°)*(π/180°) = 330π/180° radians 11π/6
360° (360°)*(π/180°) = 360π/180° radians /1

The graph below shows radian measure in all 4 Quadrants with their corresponding angles. This article explains an easy way to memorize points on the unit circle.

Unit Circle with Radians

Next, we will define the X and Y Coordinate points on the Unit Circle. In order to do this, we need to understand the relationship of the Special Right Triangles 30 – 60 – 90 and 45 – 45 – 90 degrees to the coordinate plane. These Right Triangles are very important to remember because they have certain properties that come in handy when solving Trigonometric functions.
Below shows the 30-60-90 and 45-45-90 degree Right Triangles in Quadrant I.

30-60-90 and 45-45-90 Triangles

These triangles can also be represented in the other 3 Quadrants, except that X and Y may change sign depending on the Quadrant. For example, the below graph shows the 45-45-90 degree Right Triangle in all 4 Quadrants. Notice that the angles of the triangles are still 45° regardless of which Quadrant they are in, but the X and Y coordinates change sign. For example, notice that Quadrant III, both X and Y is negative. Also notice that r = Radius of the Circle = Hypotenuse of the Triangle. This information is used to solve for the X and Y coordinates on the Unit Circle.

Triangles on the Unit Circle

Pythagorean Theorem

When solving for X, Y, or r in a 90° triangle, we can use the Pythagorean Theorem.
X2 + Y2 = r2 (Pythagorean Theorem)
To the right, the Pythagorean Theorem is used to solve for the radius of the 45° angle.
So, for the 45° angle, we have X = 1, Y = 1, and r = √2
Also, X and Y in terms of radius and angle can be written as:
X = r*cosΘ and Y = r*sinΘ
If r and Θ are given, then the X coordinate can be found.



Next we will define the Trigonometric Functions:

Trig Functions of a Triangle

cosΘ° = X/r = Adjacent/Hypotenuse sinΘ° = Y/r = Opposite/Hypotenuse tanΘ° = Y/X = Opposite/Adjacent
secΘ° = r/X = Hypotenuse/Adjacent cscΘ° = r/Y = Hypotenuse/Opposite cotΘ° = X/Y = Adjacent/Opposite

Let’s solve Trigonometric Functions for the 45-45-90 Degree triangle and define the X – Y Coordinates:

45 Degree Triangle

cosΘ° = X/r sinΘ° = Y/r tanΘ° = Y/X
cos45° = 1/√2 = √2/2 sin45° = 1/√2 = √2/2 tan45° = Y/X = 1/1 = 1

After solving for cos45° and sin45°, let's define the X and Y coordinate points for the Unit Circle.
Since X = r*cosΘ, Y = r*sinΘ, and r = 1
For Θ = 45°, we have X = 1*cos45° = √2/2 and Y = 1*sin45° = √2/2

Below is the graph of the X and Y Coordinates for the 45° angle:

X and Y coordinates for 45 Degree Triangle

Let’s solve Trigonometric Functions for the 30-60-90 Degree triangle and define the X – Y Coordinates:

X and Y coordinates for 30-60-90 Triangle

cosΘ° = X/r sinΘ° = Y/r tanΘ° = Y/X
cos30° = √3/2 sin30° = 1/2 tan30° = 1/√3 = √3/3
cos60° = 1/2 sin60° = √3/2 tan60° = √3/1

Below are the graphs of the X and Y Coordinates for the 30° and 60° angles:

Side lengths for 30-60-90 Triangle

The table below shows the X,Y coordinate points associated with the degrees on the Unit Circle.

Degrees = Θ (X,Y) coordinate Degrees = Θ (X,Y) coordinate
(1, 0) 210° (-√3/2, -1/2)
30° (√3/2, 1/2) 225° (-√2/2, -√2/2)
45° (√2/2, √2/2) 240° (-1/2, -√3/2)
60° (1/2, √3/2) 270° (0, -1)
90° (0, 1) 300° (1/2, -√3/2)
120° (-1/2, √3/2) 315° (-√2/2, -√2/2)
135° (-√2/2, √2/2) 330° (√3/2, -1/2)
150° (-√3/2, 1/2) 360° (1, 0)
180° (-1, 0)    

Key formulas to remember:
X = r*cosΘ
Y = r*sinΘ
On the Unit Circle, Radius (r) = 1
Pythagorean Theorem: X2 + Y2 = r2

Special Right Triangles:

The graph below shows the X and Y Coordinates on the Unit Circle. Note in Quadrant I, both X and Y coordinate points are positive. However in Quadrant II, the X coordinate is negative and the Y coordinate is positive. In Quadrant III, both X and Y are negative, and in Quadrant IV, X is positive, but Y is negative.

Unit Circle with Degrees and Radians

What is the radius of the unit circle?

A. √2/2
B. 1
C. √3/2
D. 1/2
The correct answer here would be B.

How many degrees are in the Unit Circle?

A. 360°
B. 180°
C. 270°
D. 90°
The correct answer here would be A.

In what quadrant is 135° located?

A. I
B. II
C. III
D. IV
The correct answer here would be B.

In what quadrant is 315° located?

A. I
B. II
C. III
D. IV
The correct answer here would be D.

What is the radian equivalency of 150°?

A. /3
B. π
C. /6
D. /4
The correct answer here would be C.

What is the radian equivalency of 240°?

A. /3
B. π
C. /6
D. /3
The correct answer here would be D.

What is the radian equivalency of 180°?

A. /3
B. π
C. /6
D. /3
The correct answer here would be B.

What are the X, Y coordinates for the 45° angle on the Unit Circle?

A. (√2/2, √2/2)
B. (-√2/2, √2/2)
C. (√2/2, -√2/2)
D. (0, 1)
The correct answer here would be A.

What are the X, Y coordinates for the 150° angle on the Unit Circle?

A. (√3/2, 1/2)
B. (-√3/2, 1/2)
C. (-1/2, √3/2)
D. (0, 1)
The correct answer here would be B.