Trigonometry explores the relationships and calculations associated with triangle sides and angles. It is a crucial branch of Mathematics, which makes understanding it top on most students’ priority lists. If you are such a student, the Unit Circle Chart is your secret weapon.
The unit circle chart is used to describe all the angles within a circle through trigonometric functions. Note: Trigonometric functions are calculated through a circle because it contains all the angles (from 0-360, and all multiples, e.g., 720°). The same cannot be said for the triangle.
What Is a Unit Circle Chart?
A unit circle chart is a 1-unit radius circle with a center (0, 0) of the Cartesian plane used in trigonometry to determine the sine and cosine values of angles. In the circle, y (or the reading on the y axis) = sin (θ) and X (or the reading on the X-axis) = cos (θ). Some angles to the origin center share trigonometric values.
Unit Circle Charts
How to Make Unit Circle Chart (Guide)
Drawing a unit circle diagram will take some practice, but it is fairly easy to do. The following step-by-step guide should help you:
Note: The position of the points in a unit circle is obtained by dividing the circle into 8 and 12 parts and reading the coordinates from the corresponding special triangles.
Step 1: Draw the Circle
On a Cartesian plane, allot a point (0,0) that will be the origin and center of the unit circle. Measure a radius of one unit from this point and use it to draw a circle. The exact measurement of the radius will depend on how small or large you would like your circle to be, as long as you mark it as one unit. Next, draw a horizontal and vertical line through the origin to the points of the circle and label them the x and y axes. Mark the points where the circle intersects with the axes from the end closest to your right hand in an anticlockwise direction as 0° (1,0), 90° (0,1), 180° (-1,0), 270° (0, -1).
Step 2: Divide the Circle by Eight
From the origin, divide the unit circle into 8 equal parts to create more angles. Label these angles at the point where the lines meet the unit circle. Typically, the positive direction is anticlockwise, and you can label the angles in degrees or radians (or both, one inside the circle and the other outside). You should now have the following angles 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°.
You can draw special triangles for each of these angles to determine each point’s (x, y) coordinates. For example, for the 45° angle, draw a perpendicular line from the point the 45° angle line touches the circle to the x-axis. This gives you a 45-45-90-degree special triangle. Use the basic trigonometric identities to solve the x and y values of the triangle. Your calculations should give you the following coordinate (√2/2, √2/2).
Once you find a position, use the charge values to fill in the other 3 corresponding points in the 2nd, 3rd, and 4th quadrants. From our example, if 45° = (√2/2, √2/2), then:
- 135° = (-√2/2, √2/2)
- 225° = (-√2/2, -√2/2)
- 315° = (√2/2, -√2/2)
Step 3: Divide the Circle by Twelve
Next, divide the circle again, so it now has 12 equal parts. This should give you the following angles: 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°. Repeat step 2 for these points, labeling them by angle and position coordinates. For this set, you can find the positions by creating the 30-60-90-degree special triangle in the first quadrant and mirroring the value to corresponding angles in the other quadrants. Finally, join each of the angles (from steps 2 and 3) to the origin.
You have made a unit circle.
What Is the Easiest Way to Memorize the Unit Circle?
Once you master all the values for the first quadrant of the unit circle, you have mastered the entire diagram. This is because the values remain constant and only change their signs from one quadrant to the next. The following guide of the left-hand trick will help you memorize the first quadrant values.
Step 1: Hold your left hand so your little finger and thumb make a right angle. Your little finger will form the x-axis and your thumb the y axis. In turn, the x coordinate of an angle will give you the cosine and the y-coordinate the sine.
Step 2: Spread all your fingers so they each represent a first-quadrant angle. Here’s how:
- The little finger represents 0° and falls on the x-axis.
- The ring finger represents 30°
- The middle finger represents 45°
- The index finger represents 60°
- The thumb represents 90°
Step 3: To read the cosine value of an angle, fold the finger it is represented by and count the fingers remaining to its left. Find the square root of this number and divide it by 2: this will give you the cosine coordinates. For example, fold your index finger to find the cosine of 60°. To the left of the folded finger, you have one finger (your thumb), making the cosine coordinate √1/2.
Step 4: Repeat step 3, this time counting the fingers on the right to get the sine coordinates.
Step 5: The left-hand trick gives you the values for angles in the first quadrant. Because the coordinates remain constant, all you need to change are the charges as follows:
- Quadrant 1 (+, +) coordinates
- Quadrant 2 (-, +) coordinates
- Quadrant 3 (-, -) coordinates
- Quadrant 4 (+, -) coordinates
Understanding a Unit Circle Chart
You can draw and memorize the Unit Circle Chart, but you can’t use it if you don’t understand it. Here is a breakdown of the main concepts:
A radian is an angle measurement and an alternative to the degree (°). A full circle (360°) is equal to 2π radians, and you will use this designation to identify special angles.
Converting Degrees to Radians
As already mentioned, a full circle has 2π radians, and you will use the radian measurement to designate special angles. This means you will need to learn how to convert from degree to radian and vice versa. The following are some quick conversions to keep in mind:
- 360 degrees = 2π radians
- 360/2π degrees = 1 radian
- 180/π degree = 1 radian
Basic Trig Ratios
A unit circle allows you to calculate sine and cosine values from which you can evaluate other trigonometric ratios using the following relationships:
- Cos θ = adjacent/hypotenuse
- Sin θ = opposite/hypotenuse
- Tan θ = opposite/adjacent
- Sec θ = 1/Cos θ
- Csc θ = 1/Sin θ
- Cot θ = 1/Tan θ
The unit circle contains special angles whose sine and cosine values correspond to their multiples in other quadrants. These angles are π/6 (30°), π/3 (60°), π/4 (45°), π/2 (90°), π (180°).
Trigonometric Identities and Functions
You can read the 6 basic trigonometric functions from the unit circle through the following identities:
- Cos θ = x
- Sin θ = y
- Tan θ = y/x
- Sec = 1/x
- Csc = 1/y
- Cot = x/y
Angles On the Axes
For angles whose terminal side falls along the x-axis, the sine is equal to 0 and the cosine 1 or -1. If the angle’s terminal side falls along the y axis, the sine will be 1 or -1 and the cosine 0.
Reference angles are the special angles discussed earlier that fall in the first quadrant π/6 (30°), π/3 (60°), π/4 (45°). To calculate the value of a larger or smaller angle, you can use these special angles by determining in what family the angle belongs. You can do this by simplifying the angle as far as is possible then checking the denominator.
If there is no denominator, the angle belongs to the π family
- Denominator 2 = π/2 family
- Denominator 3 = π/3 family
- Denominator 4 = π/4 family
- Denominator 6 = π/6 family
How Do You Read a Unit Circle Chart?
Through the unit circle, you can determine the sine and cosine value of any natural angle by drawing and reading the diagram as follows:
- Step 1: Stand at (1,0); this is the point where the X-axis touches the unit circle in the first quadrant.
- Step 2: Move along the circle’s radius in the counterclockwise direction and stop where the angle formed between the origin (0,0), your position, and the X-axis is equal to θ.
- Step 3: Read the y-coordinate at this point, that is, sin θ. The x-coordinate is equal to cos θ.
- Step 4: Determine other trigonometric functions (tan, sec, etc.) through their relation with cosine and sine.
The Unit Circle Chart is an important trigonometry tool that helps you find sine and cosine values of all possible angles. The diagram itself is a 1-unit radius circle with the center as the (0,0) origin of the Cartesian plane and the y and x-axes for diameters. To use it, you must understand basic concepts such as radian to degree conversion, special angles, and trigonometric angles. You can also easily memorize it through the easy-to-remember left-hand trick.