The Cotangent Function | Moosa Academy

Cotangent is the reciprocal of the tangent. The two graphs look related but sit in opposite places: wherever tangent has an asymptote, cotangent has a zero — and the other way round.

Concept Definition

\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}

Written as $\dfrac{\cos\theta}{\sin\theta}$ , the denominator is the sine — so cotangent is undefined where $\sin\theta = 0$ , that is at every multiple of 180°.

Its zeros are where $\cos\theta = 0$ , at $90° + 180°n$ . Note the contrast: $\cot 90° = 0$ is perfectly defined, while $\tan 90°$ runs off to infinity.

Graph Falling branches between asymptotes

Every branch decreases across its interval — the opposite of tangent, which increases. The branches repeat every 180°, so the period is 180° = π, and the range is all real numbers.

Example Reading values from the tangent

At 45°:

\tan 45° = 1

, so

\cot 45° = \dfrac{1}{1}

. Checking directly,

\dfrac{\cos 45°}{\sin 45°} = \dfrac{0.7071}{0.7071}

⟹ cot 45° = 1

At 135°:

\tan 135° = -1

, so

\cot 135° = \dfrac{1}{-1}

⟹ cot 135° = −1

At 90°:

\cos 90° = 0

and

\sin 90° = 1

, so

\cot 90° = \dfrac{0}{1}

⟹ cot 90° = 0, while tan 90° is undefined

Reference tan θ against cot θ

Property	tan θ	cot θ
Definition	sin θ ÷ cos θ	cos θ ÷ sin θ
Asymptotes	90° + 180°n	180°n
Zeros	180°n	90° + 180°n
Period	180°	180°
Range	all real numbers	all real numbers
Direction	increasing	decreasing

Summary

cot θ = 1 ÷ tan θ = cos θ ÷ sin θ.
Asymptotes at multiples of 180°, exactly where tangent has its zeros.
Zeros at 90° + 180°n, exactly where tangent has its asymptotes.
Period 180° and range all real numbers, like tangent — but each branch falls instead of rising.