3.6 The Tangent Function

Building tan from sine and cosine

The tangent function is defined as tan(θ) = sin(θ)/cos(θ), representing the ratio of the y-coordinate to x-coordinate on the unit circle. Since it depends on cosine in the denominator, tan(θ) is undefined whenever cos(θ) = 0, creating vertical asymptotes at θ = π/2 + πk for any integer k.

Period, symmetry, and behavior

Unlike sine and cosine, tangent has a period of π (not 2π) because it repeats every half-rotation. It's an odd function (tan(−θ) = −tan(θ)), so its graph has point symmetry about the origin. Between consecutive asymptotes, tan increases from −∞ to +∞, with no amplitude since the function is unbounded.

Transformations of tangent

For y = A·tan(B(θ − C)) + D, the period becomes π/|B|, C shifts the graph horizontally (shifting asymptotes too), and D shifts vertically. |A| stretches/compresses steepness but doesn't limit range since tangent has no max/min. Asymptotes occur where B(θ − C) = π/2 + πk.

Key terms

Definition of tan(θ)

tan(θ) = sin(θ)/cos(θ), the ratio of y to x coordinates on the unit circle.

Period of tangent

π (half the period of sine and cosine).

Vertical asymptotes of tan(θ)

Occur at θ = π/2 + πk, where cos(θ) = 0.

Symmetry of tangent

Odd function: tan(−θ) = −tan(θ), symmetric about the origin.

Behavior between asymptotes

Tangent increases continuously from −∞ to +∞ across each period.

Try an AP-style question

What is the period of y = tan x?

1. π/2
2. π
3. 1
4. 2π