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Cotangent

cotangent
cot
ctg

The trig function cotangent, written cot θ. cot θ equals 1 divided by tan θ, displayed as a fraction with 1 in the numerator and tan θ in the denominator. or cos θ divided by sin θ, shown as a fraction with cos 6 in the numerator and sin θ in the denominator.. For acute angles, cot θ can be found by the SOHCAHTOA definition, shown below on the left. The circle definition, a generalization of SOHCAHTOA, is shown below on the right. f(x) = cot x is a periodic function with period π.

 

Two diagrams defining cotangent: SOHCAHTOA triangle showing cot θ = adjacent/opposite; unit circle showing cot θ = x/y with...

 

See also

Inverse cotangent, angle

Key Formula

cot⁡θ=cos⁡θsin⁡θ=1tan⁡θ=adjacentopposite\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{1}{\tan\theta} = \frac{\text{adjacent}}{\text{opposite}}
Where:
  • θ\theta = The angle in question
  • cos⁡θ\cos\theta = The cosine of the angle
  • sin⁡θ\sin\theta = The sine of the angle (must not equal zero)
  • adjacent\text{adjacent} = The side of a right triangle next to the angle (not the hypotenuse)
  • opposite\text{opposite} = The side of a right triangle across from the angle

Worked Example

Problem: In a right triangle, the side adjacent to angle θ has length 4 and the side opposite angle θ has length 3. Find cot θ.
Step 1: Recall the right-triangle definition of cotangent.
cot⁡θ=adjacentopposite\cot\theta = \frac{\text{adjacent}}{\text{opposite}}
Step 2: Substitute the given side lengths.
cot⁡θ=43\cot\theta = \frac{4}{3}
Step 3: Verify using tangent: tan θ = opposite/adjacent = 3/4, so cot θ = 1/tan θ = 4/3. The results agree.
cot⁡θ=1tan⁡θ=134=43\cot\theta = \frac{1}{\tan\theta} = \frac{1}{\frac{3}{4}} = \frac{4}{3}
Answer: cot θ = 4/3

Another Example

This example uses an obtuse angle (beyond 90°) where the right-triangle definition doesn't directly apply, so we use the cos/sin ratio instead and must account for the sign based on the quadrant.

Problem: Find the exact value of cot 120°.
Step 1: Identify the reference angle. 120° is in the second quadrant, and its reference angle is 180° − 120° = 60°.
Reference angle=60°\text{Reference angle} = 60°
Step 2: Recall the sine and cosine of 120°. In the second quadrant, sine is positive and cosine is negative.
sin⁡120°=32,cos⁡120°=−12\sin 120° = \frac{\sqrt{3}}{2}, \quad \cos 120° = -\frac{1}{2}
Step 3: Apply the formula cot θ = cos θ / sin θ.
cot⁡120°=cos⁡120°sin⁡120°=−1232\cot 120° = \frac{\cos 120°}{\sin 120°} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}
Step 4: Simplify by dividing the fractions.
cot⁡120°=−12×23=−13=−33\cot 120° = -\frac{1}{2} \times \frac{2}{\sqrt{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
Answer: cot 120° = −√3/3

Frequently Asked Questions

What is the difference between cotangent and tangent?
Cotangent and tangent are reciprocals of each other. Tangent equals opposite over adjacent (or sin θ / cos θ), while cotangent equals adjacent over opposite (or cos θ / sin θ). Wherever tan θ is large, cot θ is small, and vice versa. Tangent is undefined when cos θ = 0, while cotangent is undefined when sin θ = 0.
When is cotangent undefined?
Cotangent is undefined whenever sin θ = 0, because cot θ = cos θ / sin θ and you cannot divide by zero. This occurs at θ = 0°, 180°, 360°, and in general at every integer multiple of π (i.e., θ = nπ where n is any integer).
What is the period of the cotangent function?
The cotangent function has a period of π (or 180°). This means cot θ = cot(θ + π) for all values of θ where cotangent is defined. Unlike sine and cosine, which have a period of 2π, cotangent repeats twice as often.

Cotangent (cot θ) vs. Tangent (tan θ)

Cotangent (cot θ)Tangent (tan θ)
Definitioncos θ / sin θ, or adjacent / oppositesin θ / cos θ, or opposite / adjacent
ReciprocalReciprocal of tangent: 1 / tan θReciprocal of cotangent: 1 / cot θ
Undefined whensin θ = 0 (at 0°, 180°, 360°, …)cos θ = 0 (at 90°, 270°, …)
Periodπ (180°)π (180°)
Value at 45°11
Behavior as angle increases from 0° to 90°Decreases from +∞ to 0Increases from 0 to +∞

Why It Matters

Cotangent appears frequently in trigonometry courses, calculus (its derivative is −csc² x), and physics problems involving slopes or angular relationships. Many trigonometric identities involve cotangent, such as the Pythagorean identity 1 + cot² θ = csc² θ. You will also encounter it when simplifying expressions or solving equations where dividing by tangent is more convenient.

Common Mistakes

Mistake: Confusing cotangent with the inverse tangent (arctan or tan⁻¹).
Correction: Cotangent is the reciprocal of tangent: cot θ = 1/tan θ. The inverse tangent, tan⁻¹(x), is a completely different function that returns an angle given a ratio. They are not the same thing.
Mistake: Using the wrong ratio: writing cot θ = opposite/adjacent instead of adjacent/opposite.
Correction: Remember that cotangent flips the tangent ratio. Since tan θ = opposite/adjacent, cot θ = adjacent/opposite. A helpful mnemonic: cotangent 'co-' reverses the tangent ratio.

Related Terms