Prerequisite Material

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Unit Circle Lessons

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Coordinates on the

Unit Circle

math pic

The second method to evaluate trig functions of angles on the unit circle is to use the coordinates of the point where the terminal side of the angle intersects the unit circle.

In order to use this method, we must first find the coordinates of the 16 angles on the unit circle and then apply the ratios for the six trig functions.


Coordinates on the unit circle

At pi over 6 or 30°, the reference triangle is the 30-60-90 below.

reference triangle 30 degrees

The coordinates of the point at pi over 6 will be the number of units the point is to the right of the origin and up from the origin. The coordinates are currently square root of 3 and 1. However, all the angles are on a circle and must have a common radius. Therefore, the hypotenuse of every triangle on the unit circle (its radius) is 1. To make the current hypotenuse 1 and keeping the same ratio of sides, divide each of the sides by 2. Thus, the coordinates at pi over 6 are square root of 3 over 2 and one half.


At pi over 4 or 45°, the reference triangle is the 45-45-90 below.

reference triangle 45 degrees

The coordinates of this angle would be (1, 1). However, the radius or hypotenuse is not 1 so divide everything by square root of 2 to make the radius 1.

coordinates pi over 4

At pi over 3 or 60°, the reference triangle is a 30-60-90 triangle just like at 30° except the angles and legs are flipped. Therefore, the coordinates are also flipped leaving us with coordinates pi over 3.

At pi over 2 or 90°, there will not be a reference triangle because the segment from 90° to the x-axis is the radius of the circle. Therefore, the coordinates would be (0, 1) since it is 1 unit above the x-axis.

So far, the first quadrant of the unit circle looks like the figure below.

quadrant 1

The other quadrants look similar; however, because the angles are in different quadrants some of the coordinates are negative.

quadrant signs

The coordinates in the other quadrants are also similar. Each of the angles in the other quadrants with the same reference angles as the angles in the first quadrants will have the same coordinates. For example, in the first quadrant the 45° angle has a 45° reference angle (angle to the x-axis). In the second quadrant, the 135° angle also has a 45° reference angle (135° is 45° away from the x-axis). Therefore, at the 135° angle, the coordinates are the same values as that of the coordinates at the 45° angle. The only difference there might be will be the signs of the x and y coordinates. Thus, when the entire unit circle is labeled with all of its coordinates, it will look like the figure below.

full unit circle

Click here to open a copy of the unit circle.

Evaluating Using the Coordinates

Once we have the coordinates of the angles on the unit circle, we must simply apply the ratios of the trig functions to find the desired values. However, we have to write the ratios in terms of x and y first to be able to use the coordinates.

trig ratios

trig ratios triangle

With the ratios, we can evaluate trig functions with the coordinates at the angles. Remember, the r stands for the radius and in the unit circle r is equal to 1.

Example: Evaluate cos 5pi over 3

Step 1: Locate the angle and coordinates on the unit circle above.

Step 2: Apply the appropriate trig ratio, which in this case is x/r (or x/1 since r = 1).

answer cos 5 pi over 3

Step 3: Simplify the fraction.

one half


Now you can evaluate trig functions using the unit circle.

Let's practice what we learned!

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Created by M. Riester Blahunka 2009
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