# 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.

**X ^{2} + Y^{2} = r^{2}** (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: **X ^{2} + Y^{2} = r^{2}**

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.