Cotangent Graph: y = cot x

The cotangent function is one of the three reciprocal trigonometric ratios, defined in terms of sine and cosine:

\( \cot x = \dfrac{\cos x}{\sin x} = \dfrac{1}{\tan x} \)

Because cotangent is built from the same building blocks as the tangent ratio, its graph shares many features with the tangent graph, but flipped and shifted in a specific way. Understanding \( y = \cot x \) is easiest once you already know where sine and cosine equal zero, since those points control the entire shape of the curve.

Since \( \cot x = \dfrac{\cos x}{\sin x} \), the function is undefined anywhere \( \sin x = 0 \). That happens at every integer multiple of \( \pi \):

\( x = 0, \pm\pi, \pm 2\pi, \pm 3\pi, \dots \)

At each of these values the graph has a vertical asymptote, and the curve is never actually defined there. So:

• Domain: all real numbers except \( x = n\pi \), where \( n \) is any integer
• Range: all real numbers, from \( -\infty \) to \( \infty \)
• Period: \( \pi \) (the pattern repeats every \( \pi \) radians)

If you need a refresher on switching between radians and degrees before working through asymptote locations, see the lesson on how to convert between degrees and radians.

\( \cot x = 0 \) exactly when \( \cos x = 0 \), which occurs at:

\( x = \dfrac{\pi}{2} + n\pi \)

So the curve crosses the x-axis at \( \dfrac{\pi}{2}, \dfrac{3\pi}{2}, -\dfrac{\pi}{2}, \) and so on, exactly halfway between each pair of asymptotes.

Between any two consecutive asymptotes, the cotangent curve decreases from positive infinity down to negative infinity, passing through zero at the midpoint. This is the opposite behavior to tangent, which increases between its asymptotes.

Notice the key values along one branch, from \( x = \dfrac{\pi}{4} \) to \( x = \dfrac{3\pi}{4} \):

• At \( x = \dfrac{\pi}{4} \): \( \cot x = 1 \)
• At \( x = \dfrac{\pi}{2} \): \( \cot x = 0 \)
• At \( x = \dfrac{3\pi}{4} \): \( \cot x = -1 \)

These three points confirm the decreasing pattern on each branch.

Suppose you are asked to find \( \cot\left(\dfrac{5\pi}{4}\right) \) using the graph rather than a calculator.

1. Locate \( \dfrac{5\pi}{4} \) on the x-axis. It sits between the asymptotes at \( x = \pi \) and \( x = \dfrac{3\pi}{2} \).
2. Since the period is \( \pi \), \( \dfrac{5\pi}{4} \) is exactly one period past \( \dfrac{\pi}{4} \), because \( \dfrac{5\pi}{4} - \pi = \dfrac{\pi}{4} \).
3. The value at \( \dfrac{\pi}{4} \) is \( 1 \), so by the periodicity of cotangent, \( \cot\left(\dfrac{5\pi}{4}\right) = 1 \) as well.

This shortcut, using the repeating period instead of recalculating from scratch, works for any angle once you know the pattern within a single branch.

Cotangent belongs to the same family as secant and cosecant, all built from reciprocals of the basic sine and cosine ratios, though each has its own asymptote pattern and shape. If you want to see how the reciprocal of cosine behaves, check out the cosecant graph lesson for a side-by-side comparison. Once you're comfortable with the base shape of \( y = \cot x \), you can also explore how shifting, stretching, or reflecting the graph changes its appearance in the lesson on transformations of trigonometric functions.

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