Unit 7, Lesson 12
Tangent
Algebra 2
12.1
Warm-up
Notice and Wonder: An Unusual Function
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
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
We can also see the asymptotes of the tangent function:
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