18 min read

The Graph of Tangent

Unlike sine and cosine, the tangent graph repeats every 180° and has vertical asymptotes.

A Different Shape

tan(θ) = \frac{sin ( θ )}{cos ( θ )} . Whenever cos(θ) = 0 (at 90 ° and 270 °), tan(θ) is undefined — the graph shoots up or down without limit near these points. This is called an asymptote : a line the graph gets closer and closer to, but never touches. Between the asymptotes, tan(θ) climbs steadily from -\infty to +\infty. The whole pattern repeats every 180 °, not 360 ° like sine and cosine.

Example 1 — why tan is undefined at 90°

Explain why tan(90 °) is undefined. 1 tan(θ) = \frac{sin ( θ )}{cos ( θ )} 2 At θ = 90 °: sin(90 °) = 1, cos(90 °) = 0 3 tan(90 °) = \frac{1}{0}, and dividing by zero is undefined.

Practice

What happens to the tangent graph near θ = 90 °?

A Different Shape

tan(θ) = \frac{sin ( θ )}{cos ( θ )} . Whenever cos(θ) = 0 (at 90 ° and 270 °), tan(θ) is undefined — the graph shoots up or down without limit near these points. This is called an asymptote : a line the graph gets closer and closer to, but never touches. Between the asymptotes, tan(θ) climbs steadily from -\infty to +\infty. The whole pattern repeats every 180 °, not 360 ° like sine and cosine.

Example 1 — why tan is undefined at 90°

Explain why tan(90 °) is undefined. 1 tan(θ) = \frac{sin ( θ )}{cos ( θ )} 2 At θ = 90 °: sin(90 °) = 1, cos(90 °) = 0 3 tan(90 °) = \frac{1}{0}, and dividing by zero is undefined.

Practice

What happens to the tangent graph near θ = 90 °?