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What does the tangent function look like, and where are its period, asymptotes and zeros?

Topic 3.8 The Tangent Function: define the tangent function, graph it, and identify its period, vertical asymptotes, zeros and behavior between asymptotes.

A focused answer to AP Precalculus Topic 3.8, covering the definition of tangent as sine over cosine, its graph, period of pi, vertical asymptotes where cosine is zero, zeros where sine is zero, and its increasing behavior between asymptotes.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

The tangent function is the ratio $\tan x = \frac{\sin x}{\cos x}$ , so it is the slope of the terminal ray on the unit circle. Its graph has period $\pi$ , half that of sine and cosine, because the slope of the ray repeats every half-revolution. Vertical asymptotes occur wherever $\cos x = 0$ , namely at $x = \frac{\pi}{2} + \pi k$ , since dividing by zero is undefined there. Zeros occur wherever $\sin x = 0$ , namely at $x = \pi k$ . Between each pair of consecutive asymptotes the function increases without bound, sweeping from $-\infty$ up through $0$ to $+\infty$ , with an inflection (sign change of concavity) at the zero in the middle. Tangent has no amplitude because it grows without bound, which distinguishes it sharply from the sinusoids.

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What this topic is asking
Definition and the unit circle
Period, asymptotes and zeros
Behavior between asymptotes
How tangent differs from the sinusoids
Try this

What this topic is asking

The College Board (Topic 3.8) wants you to understand the tangent function: its definition as $\frac{\sin x}{\cos x}$ , the shape of its graph, its period of $\pi$ , the vertical asymptotes where cosine is zero, the zeros where sine is zero, and its steadily increasing behavior between consecutive asymptotes.

Definition and the unit circle

Because tangent measures slope, it grows without bound as the ray approaches vertical (where cosine, the run, shrinks to zero) and equals zero when the ray is horizontal (where sine, the rise, is zero).

Period, asymptotes and zeros

The asymptotes come from the denominator and the zeros from the numerator, exactly the rational-function logic of Topics 1.7 and 1.8 applied to $\frac{\sin x}{\cos x}$ .

Behavior between asymptotes

Describing tangent on one period

Describe $y = \tan x$ on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ : its asymptotes, zero, direction and concavity.

step 1 Identify the boundary asymptotes

At $x = -\frac{\pi}{2}$ and $x = \frac{\pi}{2}$ , $\cos x = 0$ , so these are vertical asymptotes bounding the interval.

step 2 Find the zero

$\tan x = 0$ where $\sin x = 0$ , which on this interval is $x = 0$ . The graph passes through the origin.

step 3 Determine the direction

As $x$ rises from just above $-\frac{\pi}{2}$ to just below $\frac{\pi}{2}$ , $\tan x$ rises from $-\infty$ through $0$ to $+\infty$ : the function is increasing throughout.

step 4 Determine the concavity

Left of the zero ( $-\frac{\pi}{2} < x < 0$ ) the graph is concave down; right of the zero ( $0 < x < \frac{\pi}{2}$ ) it is concave up. The concavity changes sign at the zero $x = 0$ , which is the inflection point of this branch.

How tangent differs from the sinusoids

A point worth stating once is that tangent is built from sine and cosine but behaves very differently. It is periodic, but with period $\pi$ rather than $2\pi$ ; it is increasing on every branch, never oscillating up and down; and it has no maximum, minimum or amplitude, because it runs off to infinity at each asymptote. The single feature it shares with the rational functions of Unit 1 is the asymptote: tangent blows up exactly where its cosine denominator hits zero, and you locate those points the same way you locate vertical asymptotes of any rational function. Seeing tangent as "a rational combination of sinusoids" ties Units 1 and 3 together and explains every feature of its graph.

Try this

Q1. What is the period of $y = \tan x$ ? [1 point]

Cue. $\pi$ , half the period of sine and cosine.

Q2. Where are the zeros of $y = \tan x$ ? [1 point]

Cue. Where $\sin x = 0$ , that is at $x = \pi k$ for integer $k$ ( $\ldots, -\pi, 0, \pi, \ldots$ ).

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part A (multiple choice, no calculator). Where does the graph of

y = \tan x

have a vertical asymptote on

\left(-\frac{\pi}{2}, \frac{3\pi}{2}\right)

? (A)

x = 0

(B)

x = \frac{\pi}{2}

(C)

x = \pi

(D)

x = \frac{\pi}{4}

Show worked answer →

The correct answer is (B), $x = \frac{\pi}{2}$ .

Since $\tan x = \frac{\sin x}{\cos x}$ , a vertical asymptote occurs where the denominator $\cos x = 0$ , that is at $x = \frac{\pi}{2} + \pi k$ . On the given interval that is $x = \frac{\pi}{2}$ . At $x = 0$ and $x = \pi$ , $\sin x = 0$ , so those are zeros of tangent, not asymptotes.

AP 2024 (style)3 marksSection II (free response, no calculator). Consider

y = \tan x

. (a) State the period and the location of the vertical asymptotes. (b) On one period between consecutive asymptotes, describe whether the function is increasing or decreasing, and identify its zero.

Show worked answer →

A 3-point question on the structure of the tangent graph.

(a) Period and asymptotes (2 points): the period of tangent is $\pi$ . Vertical asymptotes occur where $\cos x = 0$ , at $x = \frac{\pi}{2} + \pi k$ for integer $k$ .
(b) Behavior (1 point): on the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ between consecutive asymptotes, $\tan x$ is increasing throughout, rising from $-\infty$ to $+\infty$ , and it has a zero at $x = 0$ where $\sin x = 0$ .

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Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)