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High School Precalculus: Help and Review32 chapters | 297 lessons

Instructor:
*Melanie Olczak*

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will provide instruction on how to use the unit circle to find the value of the tangent at certain common angle measures, and how to use the unit circle to graph the tangent function.

Did you know that a circle with a radius of one unit, whose center is at the origin is called the **unit circle**? There is 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.

As you can see, the unit circle is naturally split into four sections, by the *x* and *y* axes. 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 have gone a full 360 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.

The coordinates of the points on the unit circle help us to find the tangent of each angle. The **tangent** of an angle is equal to the *y*-coordinate divided 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, too divide fractions, flip the second fraction and multiply the two together. The result is one.

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.

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.

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.

To graph, we are going to start with the values where the tangent function is undefined. It is undefined at 90 degrees and 270 degrees, so we will place an vertical asymptote at 90 and 270 degrees. An **asymptote** is a line at which a function is undefined. The function will get closer and closer to this line, but it will never touch or cross this line.

Next, we will plot the points from the table.

Finally, we connect the points to graph the function. Even though we only plotted points at one and negative one, each section of the graph continues to get closer and closer to the asymptotes without ever touching them.

We can use the unit circle to graph the tangent function. The **unit circle** is a circle whose center is at the origin, (0,0), and has a radius of one unit. The unit circle has many different angles that each have a corresponding point on the circle. The coordinates of each point give us a way to find the tangent of each angle. The **tangent** of an angle is equal to the *y*-coordinate divided by the *x*- coordinate. The tangent value is undefined at certain angle measures because we end up trying to divide by zero. The place where a function is undefined is called an **asymptote**.

To graph the tangent function:

- Find the tangent function from the unit circle, by dividing each
*y*-value, by the*x*-value. - Draw and label a set of
*x*and*y*axes. Let the*x*axis represent the angle measure and let the*y*axis represent the tangent value. - Draw dotted lines to represent the vertical asymptotes where the tangent function is undefined.
- Plot points, using the
*x*axis for angle measure and the*y*axis for the tangent. - Connect the points with curves that hug the asymptotes in either direction.

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High School Precalculus: Help and Review32 chapters | 297 lessons

- Trigonometry: Sine and Cosine 7:26
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