Tan Calculator - Tangent Function Calculator Online

What is Tangent (Tan)?

The tangent function, denoted as tan(θ), is one of the six fundamental trigonometric functions. It represents the ratio of the opposite side to the adjacent side in a right-angled triangle, or equivalently, the ratio of sine to cosine. The tangent function is periodic, continuous, and has vertical asymptotes where the cosine function equals zero.

tan(θ) = sin(θ) / cos(θ) = opposite / adjacent

Definition of Tangent

In a right-angled triangle, the tangent of an angle is the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. The tangent function can also be defined as the ratio of the sine function to the cosine function.

Key Properties

Period: π radians (180°) - the function repeats every π radians
Domain: All real numbers except π/2 + nπ (where n is any integer)
Range: All real numbers (-∞, ∞)
Asymptotes: Vertical asymptotes at x = π/2 + nπ
Odd function: tan(-x) = -tan(x)
Continuous: Continuous everywhere except at its asymptotes

Frequently Asked Questions

Why is tangent undefined at 90° and 270°?

Tangent is undefined at 90° and 270° (and their multiples) because at these angles, the cosine function equals zero. Since tan(θ) = sin(θ)/cos(θ), dividing by zero makes the function undefined. These points correspond to vertical asymptotes on the tangent graph, where the function approaches positive or negative infinity.

What's the difference between tangent and sine/cosine?

While sine and cosine are bounded functions with ranges [-1, 1], tangent is unbounded with range (-∞, ∞). Tangent has a period of π (180°) compared to sine and cosine which have periods of 2π (360°). Tangent also has vertical asymptotes where sine and cosine do not, making it discontinuous at certain points.

How do I find the tangent of an angle in different quadrants?

The sign of tangent depends on the quadrant: Quadrant I (0°-90°): positive, Quadrant II (90°-180°): negative, Quadrant III (180°-270°): positive, Quadrant IV (270°-360°): negative. This follows the pattern "All Students Take Calculus" where A=All positive, S=Sine positive, T=Tangent positive, C=Cosine positive.

What are the practical applications of the tangent function?

Tangent is widely used in physics for wave analysis and harmonic motion, in engineering for signal processing and control systems, in computer graphics for rotations and transformations, in navigation and GPS systems, in architecture for structural analysis, and in mathematics for calculus and differential equations. It's essential for understanding periodic phenomena and angular relationships.