2.11: 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
The dashed segment is 1 because it is the radius of the unit circle. Recall that the
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
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
In fact as we get infinitely close to
Rotating past
The graph
Notice the
Cotangent is the reciprocal of tangent,
When you overlap the two functions, notice that the graphs consistently intersect at 1 and 1. These are the angles that have
The cotangent function has a domain of all real angles except multiples of
Example A
Sketch the graph of
Solution: Starting with
Example B
Sketch the graph of
Solution: If you compare this graph to
Example C
Sketch the graph of
Solution: The constant in front of the tangent function will cause the graph to be stretched. It will also have a phase shift of
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
2. Graph
3. Graph
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.

f(x)=tan(x) . 
h(x)=tan(2x) .  \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)=23\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.