# Lesson 12

Tangent

### 12.1: Notice and Wonder: An Unusual Function

What do you notice? What do you wonder?

$$\theta$$ $$\cos(\theta)$$ $$\sin(\theta)$$ $$\tan(\theta)$$
$$\text-\frac{\pi}{2}$$ 0 -1
$$\text-\frac{\pi}{3}$$ 0.5 -0.87
$$\text-\frac{\pi}{6}$$ 0.87 -0.5
0 1 0
$$\frac{\pi}{6}$$ 0.87 0.5
$$\frac{\pi}{3}$$ 0.5 0.87
$$\frac{\pi}{2}$$ 0 1

### 12.2: A Tangent Ratio

1. 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.

$$\theta$$ $$\cos(\theta)$$ $$\sin(\theta)$$ $$\tan(\theta)$$
$$\text-\frac{\pi}{2}$$ 0 -1
$$\text-\frac{\pi}{3}$$ 0.5 -0.87
$$\text-\frac{\pi}{6}$$ 0.87 -0.5
0 1 0
$$\frac{\pi}{6}$$ 0.87 0.5
$$\frac{\pi}{3}$$ 0.5 0.87
$$\frac{\pi}{2}$$ 0 1
$$\frac{2\pi}{3}$$
$$\frac{5\pi}{6}$$
$$\pi$$
$$\frac{7\pi}{6}$$
$$\frac{4\pi}{3}$$
$$\frac{3\pi}{2}$$
$$\frac{5\pi}{3}$$
$$\frac{11\pi}{6}$$
$$2\pi$$
2. How are the values of $$\tan(\theta)$$ like the values of $$\cos(\theta)$$ and $$\sin(\theta)$$? How are they different?

1. Where does the line $$x=1$$ intersect the line that passes through the origin and the point corresponding to the angle $$\frac{\pi}{6}$$?
2. Where does the line $$x=1$$ intersect the line that passes through the origin and the point corresponding to the angle $$\theta$$?
3. Where do you think the name “tangent” of an angle comes from?

### 12.3: The Tangent Function

Before we graph $$y= \tan(\theta)$$, let’s figure out some things that must be true.

1. Explain why the graph of $$\tan(\theta)$$ has a vertical asymptote at $$x = \frac{\pi}{2}$$.
2. Does the graph of $$\tan(\theta)$$ have other vertical asymptotes? Explain how you know.
3. For which values of $$\theta$$ is $$\tan(\theta)$$ zero? For which values of $$\theta$$ is $$\tan(\theta)$$ one? Explain how you know.
4. Is the graph of $$\tan(\theta)$$ periodic? Explain how you know.

### Summary

The tangent of an angle $$\theta$$, $$\tan(\theta)$$, is the quotient of the sine and cosine: $$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$$. Here is a graph of $$y = \tan(\theta)$$.

We can see from the graph that $$\tan(\theta) = 0$$ when $$\theta$$ is $$\text-2\pi, \text-\pi, 0, \pi, \text{or } 2\pi$$. This makes sense because the sine is 0 for these values of $$\theta$$. Since sine and cosine are never 0 at the same $$\theta$$, we can say that tangent has a value of 0 whenever sine has a value of 0.

We can also see the asymptotes of tangent $$\text-\frac{3\pi}{2}, \text-\frac{\pi}{2}, \frac{\pi}{2}, \text{and }\frac{3\pi}{2}$$. Let’s look more closely at what happens when $$\theta = \frac{\pi}{2}$$. We have $$\sin \frac{\pi}{2} = 1$$ and $$\cos \frac{\pi}{2} = 0$$. This means $$\tan \left(\frac{\pi}{2}\right) = \frac{1}{0}$$, which is not defined. Whenever $$\cos(\theta) = 0$$, the 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 sine and cosine. The period of tangent is only $$\pi$$ while the period of sine and cosine is $$2\pi$$.

### Glossary Entries

• periodic function

A function whose values repeat at regular intervals. If $$f$$ is a periodic function then there is a number $$p$$, called the period, so that $$f(x + p) = f(x)$$ for all inputs $$x$$.