Tangent Calculator
Calculate tangent values in degrees and radians with step-by-step calculations, asymptote detection, and trigonometric identities. Perfect for trigonometry and calculus.
Calculate Tangent (tan)
Common Tangent Values
Click on these links to see instant calculations with common angles:
Tangent Function
The tangent function is one of the fundamental trigonometric functions. In a right triangle, tangent is the ratio of the opposite side to the adjacent side. It can also be expressed as sine divided by cosine: tan(x) = sin(x)/cos(x).
Tangent Properties
- Domain: All real numbers except odd multiples of π/2
- Range: All real numbers (-∞, ∞)
- Period: π radians (180°)
- Odd Function: tan(-x) = -tan(x)
- Asymptotes: At x = π/2 + nπ (90° + n×180°)
- Zeros: At x = nπ (n×180°)
Special Tangent Values
| Angle (Degrees) | Angle (Radians) | Tangent Value | Exact Value |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 | 0.577 | √3/3 |
| 45° | π/4 | 1 | 1 |
| 60° | π/3 | 1.732 | √3 |
| 90° | π/2 | undefined | ∞ |
| 120° | 2π/3 | -1.732 | -√3 |
| 135° | 3π/4 | -1 | -1 |
| 150° | 5π/6 | -0.577 | -√3/3 |
| 180° | π | 0 | 0 |
Tangent Identities
- Basic Identity: tan(x) = sin(x)/cos(x)
- Reciprocal: cot(x) = 1/tan(x)
- Pythagorean: 1 + tan²(x) = sec²(x)
- Addition: tan(A + B) = (tan A + tan B)/(1 - tan A × tan B)
- Double Angle: tan(2x) = 2tan(x)/(1 - tan²(x))
- Engineering: Slope calculations, angle measurements, and structural analysis
- Physics: Projectile motion, wave analysis, and optics calculations
- Mathematics: Calculus, trigonometry, and analytical geometry
- Navigation: Bearing calculations, GPS systems, and surveying
- Computer Graphics: 3D rotations, perspective projections, and animations
Frequently Asked Questions
What is tangent?
Tangent is a trigonometric function that represents the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as sine divided by cosine: tan(x) = sin(x)/cos(x).
When is tangent undefined?
Tangent is undefined when cosine equals zero, which occurs at 90°, 270°, 450°, etc. (or π/2, 3π/2, 5π/2, etc. in radians). These are called asymptotes.
What are the common tangent values?
Common tangent values include: tan(0°) = 0, tan(30°) = √3/3 ≈ 0.577, tan(45°) = 1, tan(60°) = √3 ≈ 1.732, tan(90°) = undefined.