Trigonometry
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°:
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.
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.
The graph below shows the degrees of the Unit Circle in all 4 Quadrants, from 0° to 360°.
Now we will add Radians to the Unit Circle. Radians is the standard unit of angle measure.
The Formula for calculating Radians is:
We will calculate the Radians for each degree on the Unit Circle labeled above.
| Degrees | Formula | Radians (simplified) |
| 0° | (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 | 2π/3 |
| 135° | (135°)*(π/180°) = 135π/180° radians | 3π/4 |
| 150° | (150°)*(π/180°) = 150π/180° radians | 5π/6 |
| 180° | (180°)*(π/180°) = 180π/180° radians | π/1 |
| 210° | (210°)*(π/180°) = 210π/180° radians | 7π/6 |
| 225° | (225°)*(π/180°) = 225π/180° radians | 5π/4 |
| 240° | (240°)*(π/180°) = 240π/180° radians | 4π/3 |
| 270° | (270°)*(π/180°) = 270π/180° radians | 3π/2 |
| 300° | (300°)*(π/180°) = 300π/180° radians | 5π/3 |
| 315° | (315°)*(π/180°) = 315π/180° radians | 7π/4 |
| 330° | (330°)*(π/180°) = 330π/180° radians | 11π/6 |
| 360° | (360°)*(π/180°) = 360π/180° radians | 2π/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.
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.
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.
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:
| 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:
| 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:
Let’s solve Trigonometric Functions for the 30-60-90 Degree triangle and define the X – Y Coordinates:
| 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:
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 |
| 0° | (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.
