Why does the tangent graph have gaps?
Tangent equals sine divided by cosine. Wherever cosine is zero — at 90°, 270°, and every 180° apart — the value is undefined, creating vertical asymptotes the curve never crosses.
// maths › Trigonometric Functions & Graphs
y = tan x
Plot and evaluate the tangent function y = tan x, with its vertical asymptotes at 90°, 270° and every 180° thereafter, a period of just 180°, and an unbounded range, in degrees or radians, shown on a graph where the curve diverges towards each asymptote.
y = tan x = sin x / cos x (period 180°, asymptotes where cos x = 0)
Angle unit
Frequently asked questions
What is the period of y = tan x?
The tangent function repeats every 180° (π radians), half the 360° period of sine and cosine. Its range is all real numbers, from negative infinity to positive infinity.
Can you give a worked example?
At x = 45°, tan 45° = 1. As x approaches 90° (π/2 ≈ 1.5708 rad), tan x grows without bound toward infinity, and at exactly 90° it is undefined. The Degrees/Radians toggle marks the asymptotes at π/2, 3π/2.
What happens near an asymptote?
Just below 90° the tangent is very large and positive; just above 90° it is very large and negative. The curve shoots up to +∞ on one side and comes up from −∞ on the other.
Where is this used in real life?
Tangent describes slopes and gradients, the steepness of ramps and roofs, and angles of elevation; it appears in optics, surveying, and in computer graphics when working out viewing angles.