Focused Calculator
Cotangent Ratio Calculator
Calculate cot(A) from adjacent side b over opposite side a in a right triangle.
Calculate Cotangent Ratio
Esta calculadora segue e devolve cot(A).
Insira dados para calcular cot(A).
cot(A)
Resultado-
Passos da Solução
Fórmula:
How This Cotangent Ratio Calculator Works
Cotangent is the reciprocal of tangent. While tangent divides the opposite side by the adjacent side, cotangent flips that fraction — it divides the adjacent side by the opposite side. It shows up less often in basic geometry classes, but it plays an important role in advanced trigonometry and calculus.
This calculator takes the adjacent side b and the opposite side a, then returns cot(A) = b / a. If you are used to working with tangent, cotangent is simply the inverse fraction.
Enter adjacent side b and opposite side a in the fields above. The calculator returns cot(A) = b / a. Both values must be positive.
Formula
Cotangent of angle A equals the adjacent side divided by the opposite side. It is the reciprocal of tangent: cot(A) = 1 / tan(A).
Like tangent, cotangent uses only the two legs of the right triangle. The hypotenuse is not involved. Cotangent can take any positive value for acute angles.
Triangle Diagram
For angle A, side a is opposite, side b is adjacent, and side c is the hypotenuse.
Cotangent uses the adjacent side b (the leg next to angle A) as the numerator and the opposite side a (across from angle A) as the denominator. The hypotenuse c is not used.
Ratio Highlight
Cotangent uses the adjacent side b (the leg next to angle A) as the numerator and the opposite side a (across from angle A) as the denominator. The hypotenuse c is not used.
Side Key
- a = Opposite (height across from angle A) Used in cot(A)
- b = Adjacent (base next to angle A) Used in cot(A)
- c = Hypotenuse (slanted side) Not used in cot(A)
A cotangent of 1 means both legs are equal (a 45° angle). Values greater than 1 mean the adjacent side is longer than the opposite side — the triangle is wider and the angle is smaller. Values less than 1 mean the opposite side is longer — the angle is larger and the triangle is steeper.
How to Use
- Identify angle A in your right triangle.
- Find side b, the adjacent leg: the one next to angle A and the right angle.
- Find side a, the opposite leg: the one across from angle A.
- Enter b in the first input field.
- Enter a in the second input field.
- Click Calculate.
- The result is cot(A), which can be any positive number.
Step-by-Step Example
A right triangle has adjacent side b = 4 and opposite side a = 3.
The cotangent of angle A is approximately 1.3333. You can verify this: tan(A) = 3 / 4 = 0.75, and 1 / 0.75 = 1.3333.
What the Result Means
Cotangent tells you how many times the adjacent side fits relative to the opposite side. It is the run-over-rise version of the steepness ratio.
A cotangent of 1 means both legs are equal (a 45° angle). Values greater than 1 mean the adjacent side is longer than the opposite side — the triangle is wider and the angle is smaller. Values less than 1 mean the opposite side is longer — the angle is larger and the triangle is steeper.
Since cotangent is the reciprocal of tangent, you can always convert between them: cot(A) = 1 / tan(A) and tan(A) = 1 / cot(A).
When to Use This Ratio
Pick this cotangent ratio calculator when:
- You need the reciprocal of tangent without manually computing 1 / tan(A).
- You are working with formulas that specifically call for cotangent, such as certain calculus identities.
- You prefer to think in terms of run-over-rise rather than rise-over-run.
- You are solving problems in surveying, construction, or trigonometric proofs.
Common Mistakes
Keep an eye out for these cotangent pitfalls:
- Writing a / b instead of b / a. Cotangent puts the adjacent side on top, not the opposite side. Swapping them gives you tangent.
- Confusing cotangent with cosine. Cotangent is b / a (adjacent over opposite). Cosine is b / c (adjacent over hypotenuse). They are different ratios.
- Using the hypotenuse in the calculation. Cotangent only involves the two legs.
- Forgetting it is the reciprocal of tangent. If you already know tan(A), just take 1 / tan(A) to get cot(A).
- Mixing up which side is adjacent and which is opposite. Always identify sides relative to angle A.
Frequently Asked Questions
Answers to the most common right-triangle solving questions.
01 What is the cotangent formula for a right triangle? expand_more
It is cot(A) = b / a, where b is the adjacent side and a is the opposite side of angle A. This is the reverse of the tangent fraction.
02 How does cotangent relate to tangent? expand_more
Cotangent is the reciprocal of tangent. If tan(A) = 0.75, then cot(A) = 1 / 0.75 ≈ 1.3333.
03 When would I use cotangent instead of tangent? expand_more
Use cotangent when a formula specifically requires it, or when you want the adjacent-to-opposite ratio directly. In many calculus and physics contexts, cotangent appears in identities and derivative formulas.
04 Can cot(A) equal zero in a right triangle? expand_more
In a right triangle with acute angles only, cot(A) is always positive. Cotangent approaches zero as angle A approaches 90°, but the 90° angle is the right angle itself, not an acute angle you would measure.
05 Is cotangent the same as 1 divided by tangent? expand_more
Yes. cot(A) = 1 / tan(A) = b / a. Both expressions give the same result.