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?

### Tangent and Cotangent Graphs

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*} through the unit circle so that it intersects with the tangent line, \begin{align*}\tan \theta\end{align*} will be equal to the length of the red segment. Below, this segment is labeled the "tangent segment".

Why? The dashed segment is 1 because it is the radius of the unit circle. Recall that in general, \begin{align*}\tan \theta = \frac{y}{x}\end{align*}. So here, \begin{align*}\tan \theta = \frac{\text{tangent segment}}{1}=\text{tangent segment}\end{align*}

As the value of \begin{align*}\theta\end{align*} increases, the value of \begin{align*}\tan \theta\end{align*} changes. As we rotate through the first quadrant, the value of \begin{align*} \tan \theta\end{align*} will increase very slowly at first and then more rapidly.

As we get ** very** close to the \begin{align*}y-\end{align*}axis the segment gets infinitely large, until when the angle really hits \begin{align*}90^\circ\end{align*}, at which point the extension of the angle and the tangent line will actually be parallel and therefore never intersect.

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*} as \begin{align*}(x, y)\end{align*}.

In fact as we get infinitely close to \begin{align*}90^\circ\end{align*}, the tangent value increases without bound, until when we actually reach \begin{align*}90^\circ\end{align*}, at which point the tangent is undefined. Recall there are some angles (\begin{align*}90^\circ\end{align*} and \begin{align*}270^\circ\end{align*}, for example) for which the tangent is not defined. Therefore, at these points, there are going to be vertical asymptotes.

Rotating past \begin{align*}90^\circ\end{align*}, the intersection of the extension of the angle and the tangent line is actually below the \begin{align*}x-\end{align*}axis. This fits nicely with what we know about the tangent for a \begin{align*}2^{nd}\end{align*} quadrant angle being negative. At first, it will have very large negative values, but as the angle rotates, the segment gets shorter, reaches 0, then crosses back into the positive numbers as the angle enters the \begin{align*}3^{rd}\end{align*} quadrant. The segment will again get infinitely large as it approaches \begin{align*}270^\circ\end{align*}. After being undefined at \begin{align*}270^\circ\end{align*}, the angle crosses into the \begin{align*}4^{th}\end{align*} quadrant and once again changes from being infinitely negative, to approaching zero as we complete a full rotation.

The graph \begin{align*}y=\tan x\end{align*} over several rotations would look like this:

Notice the \begin{align*}x-\end{align*}axis is measured in radians. Our asymptotes occur every \begin{align*}\pi\end{align*} radians, starting at \begin{align*}\frac{\pi}{2}\end{align*}. The period of the graph is therefore \begin{align*}\pi\end{align*} radians. The domain is all reals except for the asymptotes at \begin{align*}\frac{\pi}{2}, \frac{3\pi}{2}, -\frac{\pi}{2}, etc.\end{align*} and the range is all real numbers.

Cotangent is the reciprocal of tangent, \begin{align*}\frac{x}{y}\end{align*}, so it would make sense that where ever the tangent had an asymptote, now the cotangent will be zero. The opposite of this is also true. When the tangent is zero, now the cotangent will have an asymptote. The shape of the curve is generally the same, so the graph looks like this:

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*} as reference angles, which always have tangents and cotangents equal to 1 or -1. It makes sense that 1 and -1 are the only values for which a function and it’s reciprocal are the same. Keep this in mind as we look at cosecant and secant compared to their reciprocals of sine and cosine.

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*} The range is all real numbers.

#### Sketching Graphs

1. Sketch the graph of \begin{align*}g(x)=-2+ \cot \frac{1}{3}x\end{align*} over the interval \begin{align*}[0, 6\pi]\end{align*}

Starting with \begin{align*}y=\cot x\end{align*}, \begin{align*}g(x)\end{align*} would be shifted down two and frequency is \begin{align*}\frac{1}{3}\end{align*}, which means the period would be \begin{align*}3\pi\end{align*}, instead of \begin{align*}\pi\end{align*}. So, in our interval of \begin{align*}[0, 6\pi]\end{align*} there would be two complete repetitions. The red graph is \begin{align*}y=\cot x.\end{align*}

2. 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*}

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*}

3. 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*}

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*}.

### Examples

#### Example 1

Earlier, you were asked to identify which graphs are tangent and cotangent.

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

1 2 3

4 5 6

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

#### Example 2

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

#### Example 3

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

#### Example 4

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

### Review

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*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.11.