<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation
Our Terms of Use (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use.

Tangent and Cotangent Graphs

Involve asymptotes spaced pi radians apart.

Atoms Practice
Estimated7 minsto complete
Practice Tangent and Cotangent Graphs
Estimated7 minsto complete
Practice Now
Turn In
Tangent and Cotangent Graphs

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

Graph of various functions including tangent and cotangent

Credit: CK-12 Foundation
Source: CK12.org
License: CC BY-NC 3.0

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 θ through the unit circle so that it intersects with the tangent line, tanθ 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, tanθ=yx. So here, tanθ=tangent segment1=tangent segment

As the value of θ increases, the value of tanθ changes. As we rotate through the first quadrant, the value of tanθ will increase very slowly at first and then more rapidly.

As we get very close to the yaxis the segment gets infinitely large, until when the angle really hits 90, 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 (θ,tanθ) as (x,y).

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

Rotating past 90, the intersection of the extension of the angle and the tangent line is actually below the xaxis. This fits nicely with what we know about the tangent for a 2nd 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 3rd quadrant. The segment will again get infinitely large as it approaches 270. After being undefined at 270, the angle crosses into the 4th quadrant and once again changes from being infinitely negative, to approaching zero as we complete a full rotation.

The graph y=tanx over several rotations would look like this:

Notice the xaxis is measured in radians. Our asymptotes occur every π radians, starting at π2. The period of the graph is therefore π radians. The domain is all reals except for the asymptotes at π2,3π2,π2,etc. and the range is all real numbers.

Cotangent is the reciprocal of tangent, xy, 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 45 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 π{2π,π,0,π,2π} The range is all real numbers.



Sketching Graphs  

1. Sketch the graph of g(x)=2+cot13x over the interval [0,6π]

Starting with y=cotx, g(x) would be shifted down two and frequency is 13, which means the period would be 3π, instead of π. So, in our interval of [0,6π] there would be two complete repetitions. The red graph is y=cotx. 

2. Sketch the graph of y=3tan(xπ4) over the interval [π,2π]

If you compare this graph to y=tanx, it will be stretched and flipped. It will also have a phase shift of π4 to the right. The red graph is y=tanx.

3. Sketch the graph of h(x)=4tan(x+π2)+3 over the interval [0,2π]

The constant in front of the tangent function will cause the graph to be stretched. It will also have a phase shift of π2 to the left. Finally, the graph will be shifted up three. Here you can see both graphs, where the red graph is y=tanx.


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 y=1+13cot2x


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*}



Graph each of the following functions

  1. \begin{align*}f(x)=\tan(x)\end{align*}
  2. \begin{align*}h(x)=\tan(2x)\end{align*}
  3. \begin{align*}k(x)=\tan(2x+\pi)\end{align*}
  4. \begin{align*}m(x)=-\tan(2x+\pi)\end{align*}
  5. \begin{align*}g(x)=-\tan(2x+\pi)+3\end{align*}
  6. \begin{align*}f(x)=\cot(x)\end{align*}
  7. \begin{align*}h(x)=\cot(2x)\end{align*}
  8. \begin{align*}k(x)=\cot(2x+\pi)\end{align*}
  9. \begin{align*}m(x)=3\cot(2x+\pi)\end{align*}
  10. \begin{align*}g(x)=-2+3\cot(2x+\pi)\end{align*}
  11. \begin{align*}h(x)=\tan(\frac{x}{2})\end{align*}
  12. \begin{align*}k(x)=\tan(\frac{x}{2}+\frac{\pi}{4})\end{align*}
  13. \begin{align*}m(x)=3\tan(\frac{x}{2}+\frac{\pi}{4})\end{align*}
  14. \begin{align*}g(x)=3\tan(\frac{x}{2}+\frac{\pi}{4})-1\end{align*}
  15. \begin{align*}h(x)=\cot(\frac{x}{2})\end{align*}
  16. \begin{align*}k(x)=\cot(\frac{x}{2}+\frac{3\pi}{2})\end{align*}
  17. \begin{align*}m(x)=-3\cot(\frac{x}{2}+\frac{3\pi}{2})\end{align*}
  18. \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. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More


Circular Function

A circular function is a function measured by examining the angle of rotation around the coordinate plane.

Image Attributions

  1. [1]^ Credit: CK-12 Foundation; Source: CK12.org; License: CC BY-NC 3.0

Explore More

Sign in to explore more, including practice questions and solutions for Tangent and Cotangent Graphs.
Please wait...
Please wait...