Lesson 12
Tangent
 Let’s learn more about 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

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 \(\frac{2\pi}{3}\) \(\frac{5\pi}{6}\) \(\pi\) \(\frac{7\pi}{6}\) \(\frac{4\pi}{3}\) \(\frac{3\pi}{2}\) \(\frac{5\pi}{3}\) \(\frac{11\pi}{6}\) \(2\pi\)  How are the values of \(\tan(\theta)\) like the values of \(\cos(\theta)\) and \(\sin(\theta)\)? How are they different?
 Where does the line \(x=1\) intersect the line that passes through the origin and the point corresponding to the angle \(\frac{\pi}{6}\)?
 Where does the line \(x=1\) intersect the line that passes through the origin and the point corresponding to the angle \(\theta\)?
 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.
 Explain why the graph of \(\tan(\theta)\) has a vertical asymptote at \(x = \frac{\pi}{2}\).
 Does the graph of \(\tan(\theta)\) have other vertical asymptotes? Explain how you know.
 For which values of \(\theta\) is \(\tan(\theta)\) zero? For which values of \(\theta\) is \(\tan(\theta)\) one? Explain how you know.
 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 \(\text2\pi, \text\pi, 0, \pi, \text{or } 2\pi\). 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 \(x\).