Unit 7, Lesson 12
Tangent
Algebra 2
12.1
Warm-up
What do you notice? What do you wonder?
| 0 | -1 | ||
| 0.5 | -0.87 | ||
| 0.87 | -0.5 | ||
| 0 | 1 | 0 | |
| 0.87 | 0.5 | ||
| 0.5 | 0.87 | ||
| 0 | 1 |
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Complete the table. For each positive angle in the table, add the corresponding point and the segment between it and the origin to the unit circle.
| 0 | -1 | ||
| 0.5 | -0.87 | ||
| 0.87 | -0.5 | ||
| 0 | 1 | 0 | |
| 0.87 | 0.5 | ||
| 0.5 | 0.87 | ||
| 0 | 1 | ||
The tangent of an angle , , is the quotient of sine and cosine: . Here is a graph of .
A graph. Horizontal axis, theta, scale negative 2 pi to 2 pi, by pi over 2. Vertical axis, y, from negative 1.5 to 1.5, by 0.5’s. A curve begins at negative 2 pi comma 0 and curves upward towards an invisible vertical line at negative 3 pi over 2 but never touches it. Another curve crosses the x axis at negative pi. It continues in both directions, up and down, approaching invisible vertical lines at negative 3 pi over 2 and negative pi over 2, but never touches them. Another curve crosses the x axis at 0. It continues in both directions, up and down, approaching invisible vertical lines at negative pi over 2 and pi over 2, but never touches them. Another curve crosses the x axis at pi. It continues in both directions, up and down, approaching invisible vertical lines at pi over 2 and 3 pi over 2, but never touches them. Another curve crosses the x axis at 2 pi. It continues down only approaching an invisible vertical line at 3 pi over 2 but never touches it.
We can see from the graph that when is . This makes sense because sine is 0 for these values of . Since sine and cosine are never 0 at the same , we can say that tangent has a value of 0 whenever sine has a value of 0.
We can also see the asymptotes of the tangent function: . Let’s look more closely at what happens when . We have and . This means that , which is not defined. Whenever , tangent is not defined and has a vertical asymptote.
Like the sine and cosine functions, the tangent function is periodic. This makes sense because it is defined using the sine and cosine functions. The period of tangent is only , while the period of sine and cosine is .