The unit circle is a powerful tool for learning and mastering trigonometry. Understanding how to work with the unit circle is essential for anyone studying trigonometry or higher mathematics. This step-by-step guide shows you how to use the unit circle to calculate angles and coordinates, how to determine quadrant placement, and how to apply the unit circle to problem-solving.
Understanding the Unit Circle
The unit circle is a circle with a radius of 1 and a center at the origin (0, 0) of the coordinate plane. Any point on the unit circle can be identified by its angle in reference to the positive x-axis. By convention, angles in the unit circle are measured in radians. At its foundation, radians measure the ratio of an arc length on a circle relative to its radius. To illustrate, if you travel one full circle (360 degrees) around the center of a circle, this is equal to 2π radians.
Fundamental Concepts of the Unit Circle
The unit circle can be used to easily reference sine, cosine, and tangent functions of angles. Additionally, certain special angles have coordinates that can be solved directly from the unit circle. These angles are commonly referred to as “sine-cosine angles” due to their easy reference with sine and cosine values. The sine-cosine angles include 30° (1÷2, √3÷2), 45° (√2÷2, √2÷2), and 60° (√3÷2, 1÷2).
Drawing the Unit Circle
To draw a unit circle, begin by plotting (0, 0) as its center point. To draw the outer circumference, know that circles like this are composed of 360 points, however, since the unit circle is measured in radians understanding the ratio of π radians will be important. To draw the circumference of the unit circle accurately divide the circle into 2π radians or 24π degrees. Take out a compass to ensure that each line segment is of equal distance and label each point or angle along with its measurement in radians.
Exploring Angles in the Unit Circle
Once you’ve labeled all 360 points and corresponding angles within your unit circle you can explore various relationships amongst them. When measuring and plotting angles in the unit circle always begin with 0° or 0 radians at the point on the positive x-axis and move in a counter clockwise direction. For example, if you wanted to measure 120 degrees you would start with 0° at 3 o’clock and move 120 degrees counter clockwise to 9 o’clock.
Identifying Coordinates of a Point on the Unit Circle
To determine coordinates of a point or a measure of an angle within the unit circle, remember that circles are composed of 360 points and each degree has a ratio of π radians (180 degrees and π radians). From there, you must use trigonometric ratios (sine and cosine) to determine an angle’s coordinate or vice versa. These trigonometric ratios will depend upon the type of angle being measured.
Determining Quadrant Placement of Points on the Unit Circle
The unit circle is divided into four quadrants. But it’s important to understand which points fall within which quadrant of the unit circle. As we discussed before, each point on the unit circle can be labeled with its angle in reference to the positive x-axis starting from 0° at 3 o’clock and moving counter clockwise. To determine which quadrant any point is in simply label all points in that quadrant starting at 0 degrees and then moving in a counter clockwise direction until all four quadrants are labeled.
Using Trigonometric Functions with the Unit Circle
It is essential to understand how to use trigonometric functions with the unit circle when solving for angles or coordinates in problems. To be able to do this successfully one must understand both right angle trigonometry (sine, cosine, and tangent) as well as inverse trigonometric functions (arcsines, arccosines, and arctangents). Trigonometric ratios must be used along with angle measurements in order to solve for missing angle measures or coordinates.
Converting Between Radians And Degrees
Since angles on a unit circle are measured in radians always remember that one radian is equivalent to 57.3°. The degree to radian conversion is as follows:
- To convert from degrees to radians multiply the degree measure by π and divide by 180.
Working with Special Angles in the Unit Circle
Discerning special angles on the unit ciricle is another great way to understand reference points when solving problems. Special angles refer to common measurements such as 30° (1÷2, √3÷2), 45° (√2÷2, √2÷2), and 60° (√3÷2, 1÷2). These special angles have coordinates that can be solved directly from the unit circle.
Applying the Unit Circle to Problem-Solving
Now that you have a working knowledge of the unit circle and how to use it for problem solving check out some sample problems to practice applying your knowledge and understanding of this valuable tool in mathematics.
- Use your knowledge of special angles on the unit circle to solve for the length of an arc with a 40° central angle.
- Solve for an angle measure given the coordinates (-√3÷2, 1÷2).
- Find the coordinates of an angle represented by θ = 300.
By familiarizing yourself with both hands-on activities such as drawing and labeling various angles on the unit circle as well as completing various practice activities you should gain a greater mastery of your understanding of how to use the unit circle effectively.
