How to Graph the Cotangent Function?

Cotangent is one of the trigonometric ratios. In this step-by-step guide, you will learn more about the graph of the cotangent function.

How to Graph the Cotangent Function?
Tutor-style math help

Graph the Cotangent Function: what to notice and how to work it

Trigonometry skill
Tangent and cotangent graphs repeat with vertical asymptotes. The period is \(\pi/|B|\), and the asymptotes are just as important as the curve.

What to notice first

For \(y=A\tan(Bx)+D\), find the period and asymptotes before plotting the center point.

Common student mistake

Do not draw tangent like sine. Tangent has branches separated by vertical asymptotes, not smooth waves with maximum and minimum values.

Key formulas and cues

\(y=A\tan(Bx)+D\)
\(\text{period}=\frac{\pi}{|B|}\)
\(\tan x\text{ asymptotes: }x=\frac{\pi}{2}+k\pi\)
\(\text{midline}=y=D\)
amplitude midline

A reliable path

  1. Choose the modelUse a right triangle, the unit circle, or a transformed graph.
  2. Track unitsConvert degrees and radians when needed.
  3. Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Read a tangent graph rule

Example: \(y=2\tan(3x)+1\)
  1. The vertical stretch is 2.
  2. The period is pi/3.
  3. The midline is y = 1.
Answer: Period \(\pi/3\), midline \(y=1\).

Find parent tangent asymptotes

Example: Graph one cycle of \(y=\tan x\).
  1. Place asymptotes at x = -pi/2 and x = pi/2.
  2. Plot the center point (0, 0).
  3. Draw an increasing branch between the asymptotes.
Answer: Asymptotes \(x=\pm\pi/2\), center \((0,0)\).
Open pop-up practice
Try one before moving on
Try: Find the period of \(y=\tan(4x)\).
Answer: \(\pi/4\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Step-by-step guide to graphing the cotangent function

Cotangent is one of the basic trigonometric ratios. It is usually represented as \(cot x\), where \(x\) is the angle between the base and the hypotenuse of a right triangle. Alternative names of cotangent are cotan and cotangent \(x\). 

The cotangent formula is:

\(\color{blue}{cot\:θ=\frac{Adjacent\:side}{Opposite\:side}}\)

Since the cotangent function is not defined for integer multiples of \(π\), the vertical asymptote exists at all multiples of \(π\) in the cotangent graph. Also, the cotangent is \(0\) at all odd multiples of \(\frac{π}{2}\). Also, in an interval say \((0, π)\), the values of the cot decrease as the angles increase. So \(cot\) is a decreasing function. Then, the graph of the cotangent function looks like this.

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