How it works
What tangent actually measures
This tangent calculator works out tan(θ) for any angle, and its inverse — arctan — for finding an angle from a ratio of sides. Tangent is one of the three core trigonometric ratios, alongside sine and cosine, and the one most often used for slope, gradient and elevation problems because it directly relates rise to run.
The classic memory aid is SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Tangent is the only one of the three that doesn't involve the hypotenuse at all — which is exactly why it's the natural choice whenever a problem gives you two of the three sides of a right triangle but not the hypotenuse, or asks for a slope as "rise over run."
Three worked examples
1. Tangent of a known angle
tan(60°) = √3 ≈ 1.732. This is one of the "exact value" angles worth memorising alongside 30°, 45° and 90° — they turn up constantly in GCSE and A-Level trigonometry without a calculator.
2. Finding a missing side (roof pitch)
A roof rises 2.4 m over a horizontal run of 6 m. The pitch angle is arctan(2.4 / 6) = arctan(0.4) ≈ 21.8°. UK building regulations often specify minimum roof pitches in degrees, so this exact calculation — rise over run through arctan — is the one roofers and surveyors use daily.
3. Finding an angle from two known sides (arctan)
A ladder leans against a wall, reaching 4 m up with its base 1.5 m from the wall. The angle the ladder makes with the ground is arctan(4 / 1.5) = arctan(2.667) ≈ 69.4°. HSE guidance recommends a ladder angle close to 75° (a 1:4 ratio) for safety — this calculation is exactly how you'd check whether a ladder is positioned safely.
Reference table: exact tangent values
These angles have exact, memorable tangent values and appear constantly in exam questions and quick mental checks.
| Angle | Tangent | Decimal |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 1/√3 | 0.577 |
| 45° | 1 | 1 |
| 60° | √3 | 1.732 |
| 90° | undefined | — |
Why tangent is undefined at 90°
Tangent is opposite ÷ adjacent. At 90°, the adjacent side shrinks to zero, and dividing by zero is undefined — not "infinity" in the way a calculator display might round it, but genuinely undefined. This is the single most common point of confusion with tangent specifically (sine and cosine never have this problem, since neither ever divides by a side that reaches zero within 0–90°).
Common mistakes
- Degrees vs radians. The single most common tangent error: your calculator is in the wrong mode. tan(60°) ≈ 1.732, but tan(60 radians) ≈ 0.320 — a completely different number. Always check the mode before trusting the result.
- Confusing tangent with arctan. Tangent takes an angle and returns a ratio; arctan takes a ratio and returns an angle. Mixing them up is the second most common error, especially when a problem gives you a decimal and expects an angle back.
- Assuming tangent works outside a right triangle. SOH-CAH-TOA only applies directly to right-angled triangles. For any other triangle, you need the sine rule or cosine rule instead.
- Forgetting the undefined case at 90° (and its periodic repeats at 270°, 450°, etc.) — a calculator may return a very large number or an error depending on precision, not a clean "undefined."
Related formulas
- Sine: opposite ÷ hypotenuse — see the sine calculator.
- Cosine: adjacent ÷ hypotenuse — see the cosine calculator.
- Pythagoras' theorem: a² + b² = c², for finding a missing side when you have the other two — see the Pythagoras calculator.
- Identity: tan(θ) = sin(θ) / cos(θ) — tangent can always be derived from sine and cosine, which is how calculators compute it internally.