Graphing Tangent

Tangent is sine divided by cosine, so the graph is a quotient rather than a wave.

Where it breaks

Wherever cosine is zero the division is undefined. That happens at 90°, 270° and every 180° after, giving vertical asymptotes.

A shorter period

Tangent repeats every 180°, not 360°. Sine and cosine both flip sign in the opposite half, and the two flips cancel in the quotient.

No amplitude

Tangent runs from negative infinity to positive infinity, so it has no maximum, no minimum, and no amplitude to speak of.

Where it crosses

Tangent is zero wherever sine is zero: 0°, 180°, 360° and so on, midway between the asymptotes.

A graph with holes in it

tan x is undefined where cos x = 0, at π/2 plus multiples of π. At those points the graph has vertical asymptotes, so it comes in disconnected branches rather than one continuous curve.

Its period is π, not 2π

Tangent repeats every half turn, because a line through the origin has the same slope in opposite directions. That shorter period distinguishes it from sine and cosine and changes how its transformations work.

The range is everything

Unlike sine and cosine, tangent takes every real value, sweeping from −∞ to ∞ on each branch. So it has no amplitude — the word is meaningless for a function with no maximum.

Sketching a branch

Mark the asymptotes, put a point at the midpoint where the value is zero, and add the quarter points where the value is ±1. Those five features give the branch shape, and the rest repeats.