# 2.11: Tangent and Cotangent Graphs

**At Grade**Created by: CK-12

**Practice**Tangent and Cotangent Graphs

What if your instructor gave you a set of graphs like these:

and asked you to identify which were the graphs of the tangent and cotangent functions?

After completing this Concept, you'll be able to identify the graphs of tangent and cotangent.

### Watch This

James Sousa Animation: Graphing the Tangent Function Using the Unit Circle

### Guidance

The name of the tangent function comes from the tangent line of a circle. This is a line that is perpendicular to the radius at a point on the circle so that the line touches the circle at exactly one point.

If we extend angle \begin{align*}\theta\end{align*}

The dashed segment is 1 because it is the radius of the unit circle. Recall that the \begin{align*}\tan \theta = \frac{y}{x}\end{align*}

\begin{align*}\tan \theta &=\frac{y}{x}=\frac{t}{1}=t\\
\tan \theta &=t\end{align*}

So, as we increase the angle of rotation, think about how this segment changes. When the angle is zero, the segment has no length. As we rotate through the first quadrant, it will increase very slowly at first and then quickly get very close to one, but never actually touch it.

As we get ** very** close to the \begin{align*}y-\end{align*}

This means there is no finite length of the tangent segment, or the tangent segment is *infinitely large*.

Let’s translate this portion of the graph onto the coordinate plane. Plot \begin{align*}(\theta, \tan \theta)\end{align*}

In fact as we get infinitely close to \begin{align*}90^\circ\end{align*}

Rotating past \begin{align*}90^\circ\end{align*}

The graph \begin{align*}y=\tan x\end{align*}

Notice the \begin{align*}x-\end{align*}

Cotangent is the reciprocal of tangent, \begin{align*}\frac{x}{y}\end{align*}

When you overlap the two functions, notice that the graphs consistently intersect at 1 and -1. These are the angles that have \begin{align*}45^\circ\end{align*}

The cotangent function has a domain of all real angles except multiples of \begin{align*}\pi \left \{\ldots -2\pi, -\pi, 0, \pi, 2\pi \ldots \right \}\end{align*}

#### Example A

Sketch the graph of \begin{align*}g(x)=-2+ \cot \frac{1}{3}x\end{align*}

**Solution:** Starting with \begin{align*}y=\cot x\end{align*}

#### Example B

Sketch the graph of \begin{align*}y=-3 \tan \left(x-\frac{\pi}{4}\right)\end{align*} over the interval \begin{align*}[-\pi, 2\pi]\end{align*}.

**Solution:** If you compare this graph to \begin{align*}y=\tan x\end{align*}, it will be stretched and flipped. It will also have a phase shift of \begin{align*}\frac{\pi}{4}\end{align*} to the right. The red graph is \begin{align*}y=\tan x\end{align*}.

#### Example C

Sketch the graph of \begin{align*}h(x)=4 \tan \left(x+\frac{\pi}{2}\right) + 3 \end{align*} over the interval \begin{align*}[0, 2\pi]\end{align*}.

**Solution:** The constant in front of the tangent function will cause the graph to be stretched. It will also have a phase shift of \begin{align*}\frac{\pi}{2}\end{align*} to the left. Finally, the graph will be shifted up three. Here you can see both graphs, where the red graph is \begin{align*}y=\tan x\end{align*}.

### Vocabulary

**Circular Function:** A ** circular function** is a function that is measured by examining the angle of rotation around the coordinate plane.

### Guided Practice

1. Graph \begin{align*}y=-1+\frac{1}{3} \cot 2x\end{align*}.

2. Graph \begin{align*}f(x)=4+ \tan (0.5 (x - \pi))\end{align*}.

3. Graph \begin{align*}y=-2 \tan 2x\end{align*}.

**Solutions:**

1.

2.

3.

### Concept Problem Solution

As you can tell after completing this Concept, when presented with the graphs:

1 2 3

4 5 6

The tangent and cotangent graphs are the third and sixth graphs.

### Practice

Graph each of the following functions.

- \begin{align*}f(x)=\tan(x)\end{align*}.
- \begin{align*}h(x)=\tan(2x)\end{align*}.
- \begin{align*}k(x)=\tan(2x+\pi)\end{align*}.
- \begin{align*}m(x)=-\tan(2x+\pi)\end{align*}.
- \begin{align*}g(x)=-\tan(2x+\pi)+3\end{align*}.
- \begin{align*}f(x)=\cot(x)\end{align*}.
- \begin{align*}h(x)=\cot(2x)\end{align*}.
- \begin{align*}k(x)=\cot(2x+\pi)\end{align*}.
- \begin{align*}m(x)=3\cot(2x+\pi)\end{align*}.
- \begin{align*}g(x)=-2+3\cot(2x+\pi)\end{align*}.
- \begin{align*}h(x)=\tan(\frac{x}{2})\end{align*}.
- \begin{align*}k(x)=\tan(\frac{x}{2}+\frac{\pi}{4})\end{align*}.
- \begin{align*}m(x)=3\tan(\frac{x}{2}+\frac{\pi}{4})\end{align*}.
- \begin{align*}g(x)=3\tan(\frac{x}{2}+\frac{\pi}{4})-1\end{align*}.
- \begin{align*}h(x)=\cot(\frac{x}{2})\end{align*}.
- \begin{align*}k(x)=\cot(\frac{x}{2}+\frac{3\pi}{2})\end{align*}.
- \begin{align*}m(x)=-3\cot(\frac{x}{2}+\frac{3\pi}{2})\end{align*}.
- \begin{align*}g(x)=2-3\cot(\frac{x}{2}+\frac{3\pi}{2})\end{align*}.

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### Image Attributions

Here you'll learn to draw the graphs of the tangent and cotangent functions.