Trig Functions

Tangent Graph

Graph y = tan(x): identify period π, vertical asymptotes at x = π/2 + kπ, and key points within one period.

Example

Describe tangent period, asymptotes, and key points exactly.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

y=tan⁡xy=\tan x

Step 2 — Period

Tangent repeats every pi.

period=π\text{period}= \hlmath{\pi}

Step 3 — Asymptotes

Vertical asymptotes occur at plus or minus pi over 2.

x=±π2x=\pm \hlmath{\frac{\pi}{2}}

Step 4 — Key points

Record the center and the pi-over-four key point.

xyrole00centerπ41key point\hlmath{\begin{array}{c|c|c}x&y&\text{role}\\0&0&\text{center}\\\frac{\pi}{4}&1&\text{key point}\end{array}}
tangent-graph For y = tan(x): period = π (half the period of sin/cos), vertical asymptotes where cos(x) = 0 (at x = π/2 + kπ). Key point: tan(π/4) = 1, tan(0) = 0.