Functions
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Tangent Function

Discover asymptotes, the sin/cos ratio, and a period of π

The tangent function y = \tan(x) behaves very differently from sine and cosine. Instead of smooth waves, it produces curves that shoot off to infinity and reappear from the other side. Those vertical gaps are called asymptotes — places where the function is undefined.

Why does this happen? Because tangent is defined as \tan(x) = \frac{\sin(x)}{\cos(x)}. Wherever cosine equals zero, you're dividing by zero, and the function explodes. This creates a repeating pattern with a period of π (not 2π like sine and cosine).

In this lesson, you'll explore the tangent curve, understand its asymptotes, see why the period is π, and use sliders to stretch and compress the function.

Graph

FAQ

What is the tangent function?
The tangent function is defined as \tan(x) = \frac{\sin(x)}{\cos(x)}. It gives the ratio of sine to cosine at any angle. Unlike sine and cosine (which stay between -1 and 1), tangent can take any value from -∞ to +∞.
Why does tangent have asymptotes?
Tangent has vertical asymptotes wherever cos(x) = 0, because dividing by zero is undefined. This happens at x = π/2, 3π/2, 5π/2, ... (or ±π/2 + nπ for any integer n). Near these points, the function value grows without bound.
What is the period of tangent?
The tangent function repeats every π units (about 3.14), not every 2π like sine and cosine. Between consecutive asymptotes (e.g., from -π/2 to π/2), tangent completes one full cycle: rising from -∞ through 0 to +∞.
How is tangent different from sine and cosine?
Sine and cosine are bounded (between -1 and 1), continuous, and have period 2π. Tangent is unbounded (goes to ±∞), has vertical asymptotes (gaps in the graph), and has period π. Tangent also passes through the origin with slope 1, while sine passes through with slope 1 and cosine starts at its maximum.

Related Topics

Sine and CosineThe Unit CircleSlope of a Line