The Unit Circle

What is the unit circle?

A unit circle is an easy way to illustrate the two types of angle measurement that we have learned, degrees and radians, on a circle. A unit circle can be described as a circle with a radius of 1 (The radius of a unit circle is always equal to 1). In this lesson, we will discuss how to define cosines and sines as x- and y-coordinates on a unit circle. if you know how to use the unit circle to assist you in calculating angles, it can save you a great deal of frustration and hard work, especially if you are unable to use a calculator to make your calculations.

What does a unit circle look like, and how do sine and cosine relate to it?

A right triangle can be placed within the unit circle in order to illustrate how the cosine and sine can be used to assist with calculations. The right triangle will have an acute angle, the vertex of which is located at the center of the circle. The side opposite this angle, which we will refer to as y, will always be perpendicular to the x-axis, and the side adjacent to the angle will lie along the x-axis. This right triangle has legs x (adjacent to the angle θ) and y (opposite to the angle ) and a hypotenuse equal to 1. Remember that the hypotenuse is equal to the radius, and we know that the radius of the unit circle is always equal to 1.

The Unit Circle

Using the trigonometric ratios that we learned in a previous section, we know that if we were trying to solve for the length of side x we could make use of the “-cah” in soh cah toa, because x is the adjacent side, and we know that the length of the hypotenuse is equal to 1. So we can use the trigonometric ratio . We now perform the same operation to solve for the length of side y. We use the “soh-” in soh cah toa because side y is the opposite side of , and we know that the length of the hypotenuse is equal to 1. So we can determine that  We have now shown, using trigonometric ratios, that cos x can be defined by the x-coordinate, and sin x can be defined by the y-coordinate of point c on the unit circle.

The unit circle defined by cosine and sine as the x- and y-coordinates is shown below. Since the radius of the unit circle is always equal to 1, we have the coordinates shown below.

We will attempt a couple of examples on the sample unit circle above. Afterwards, you will be asked to complete some example problems on your own.

Using the unit circle, can you determine why ? In this case,  = 0. On the unit circle, we can see that at  = 0, the coordinates are (1, 0). Since cos  is the x-coordinate, and the x-coordinate is 1, we can determine that .

We will try one more example before you work an example by yourself, to make sure you have a good grasp on the unit circle and its applications.

Try to explain why . At = 90°, the coordinates are (0, 1). Since sin  is the y-coordinate, and the y-coordinate is equal to 1 at 90°, we can see that .

Try and do these examples on your own.

This next example is a little more difficult. First, we will learn a trick that will make angle measures greater than 360 degrees, or 2  radians, a little easier, using the unit circle. You may have seen this trick before, but we will go over the proper application of the trick, just to be sure.

For instance, you want to find x degrees on a unit circle, where x is greater than or equal to 360°. We know that there are 360° in a unit circle, so you can take x – 360°, and keep taking 360° off of your answer until you get an angle between 0° and 360°.

Where is 405° on a unit circle?

405° – 360° = 45°. Therefore, 405° on a unit circle is equal to 45° on a unit circle.

Where is 800° on a unit circle?

800° – 360° = 440°. Since 440° is not between 0° and 360°, we must repeat this process through one more iteration. So, 440° – 360°= 80°. Since 80° is between 0° and 360°, we know that a measurement of 800° goes all the way around the unit circle twice, and is ultimately equivalent to 80° on a unit circle.

For what do you use this trick going forward?

• The trick will simplify the process of determining where a point lies on the unit circle.
• If we are trying to find the cosine or sine of a large angle, we can use this rule to identify cosines and sines of angles we already know. The next example illustrates a fine example of this.

Review of New Vocabulary and Concepts

• A unit circle is an easy way to illustrate the two most common angle measurements, degrees and radians, on a circle.
• The radius of a unit circle is always equal to 1.
• You can insert a right triangle into a unit circle, with a radius point serving as the hypotenuse, to help you solve for sine and cosine by using the trigonometric ratios that we learned previously.