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

Written as , the denominator is the sine — so cotangent is undefined where , that is at every multiple of 180°.

Its zeros are where , at . Note the contrast: is perfectly defined, while 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°:

, so

. Checking directly,

⟹ cot 45° = 1

At 135°:

, so

⟹ cot 135° = −1

At 90°:

and

, so

⟹ 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.