Graphing Tangent from the Unit Circle
This lesson provides instruction on how to use the unit circle to find the value of the tangent at certain common angle measures and graph the tangent function.
The Unit Circle
Did you know that a circle with a radius of one unit whose center is at the origin is called the unit circle? There’s a lot of information we can get from the simple unit circle. The unit circle helps us with trigonometry, which is the study of relationships of angles and sides in triangles.
To start, let’s look at the unit circle. Again the center is at the origin, or (0, 0), and the radius is one unit.
Basic unit circle showing four quadrants. The radius of the circle is one unit.
Basic unit circle showing four quadrants. The radius of the circle is one unit.
As you can see, the unit circle is naturally split into four sections by the x-axis and the y-axis. Since a whole circle measures 360 degrees, if we divide that by four, then each section measures 90 degrees. We can go ahead and label the angle measure for each section. All angles start on the positive x-axis and move counterclockwise. So at the point (0, 1), we have gone 90 degrees. At the point (-1, 0) we have gone 180 degrees. At (0, -1) we have traveled 270 degrees. When we get back to (1, 0), we’ve gone a full 360 degrees.
360-degree unit circle with each quadrant consisting of 90 degrees.
360-degree unit circle with each quadrant consisting of 90 degrees
We can then divide each quarter section in half, giving us angles that are multiples of 45 degrees. Each of these angles has coordinates for a point on the unit circle.
360-degree unit circle with each 90-degree quadrant divided in two to form two 45-degree sections of the quadrant within the trig circle.
360-degree unit circle with each 90-degree quadrant divided to form two 45-degree sections
The coordinates of the points on the unit circle help us to find the tangent of each angle. As you can see with this equation, the tangent of an angle is equal to the y-coordinate divided by the x-coordinate.
Trigonometry equation calculating the unit circle tangent of each angle by dividing the y-coordinate by the x-coordinate
For each angle, start by dividing each y-coordinate by the x-coordinate to get the tangent of that angle. For example, to find the tangent of 45 degrees, take the y-coordinate of the square root of two over two and divide it by the x-coordinate of the square root of two over two. Remember, to divide fractions, flip the second fraction and multiply the two together. The result is one.
Further breakdown of the trig equation showing calculation of tangent for a 45-degree angle within the unit circle.
We can do this same process for all of the angles on the unit circle. When we get to 90 degrees, we end up dividing by zero. Since we cannot divide by zero, the tangent of 90 degrees is undefined. The same happens when we get to 270 degrees, so the tangent of 270 degrees is also undefined.
Table showing the calculated tangent of each angle within the unit circle, in 45-degree increments.
We could continue in this manner to find the tangent for all the values on the unit circle. Now let’s graph these values on the coordinate plane.
Graphing the Tangent Function
Now that we know the tangent of each angle, we can use these values to graph the tangent function. The graph of the tangent function can be found by using the angle measures as the horizontal (x-axis), and the tangent as the vertical (y-axis).
First, we draw and label our axes. The y-axis is the dependent axis, which means that the y-values depend on the x-values. Since the tangent values depend upon the angle measure, the angle measure is independent and the tangent values are dependent. Since we are graphing from zero to 360 degrees, that’s how we’ll label the x-axis. If we look at the values of tangent, the lowest is -1 and the highest value is 1, so we know that we need to have these values on the y-axis. Tangent does continue to get larger and smaller than 1 and -1, respectively, so we can label more numbers on the y-axis.
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