Graph the tangent function in radians

Ontario Grade 12 Mathematics

Trigonometric Functions

Radians, key points, periods, and vertical asymptotes

The tangent function makes a repeating curve with gaps where its value is undefined. To graph it, you need only a few features: where it crosses the horizontal axis, where it is undefined, how long one cycle is, and whether the curve rises or falls between its gaps. This lesson uses radians throughout.

What you will learn

Read tangent angles in radians and connect them to the unit circle.
Identify the zeros, vertical asymptotes, and period of the basic tangent function.
Sketch a tangent graph accurately using key features and a small value table.

1. Prerequisite bridge: radians and tangent

An angle can be measured in degrees or radians. Radians are the angle measure used in this lesson. A full turn is

2\pi

radians, and a half-turn is

\pi

radians. Familiar unit-circle angles include

\frac{\pi}{4}

,

\frac{\pi}{2}

, and

\pi

.

On the unit circle, the point at angle

x

has coordinates

(\cos x,\sin x)

. When

\cos x

is not zero, tangent is the ratio of the vertical coordinate to the horizontal coordinate:

\tan x=\frac{\sin x}{\cos x}

. This ratio explains why tangent is undefined when

\cos x=0

.

\tan x=\frac{\sin x}{\cos x}

Use radians on the horizontal axis, not degrees.
Tangent is undefined when its denominator, $\cos x$ , is zero.

2. Read the basic tangent graph

For the basic function

y=\tan x

, the graph crosses the horizontal axis whenever

\sin x=0

and

\cos x\ne 0

. These crossings occur at integer multiples of

\pi

.

The graph is undefined when

\cos x=0

. These locations are

x=\frac{\pi}{2}+k\pi

, where

k

is any integer. The graph approaches these vertical lines but does not touch or cross them. Such a boundary line is called a vertical asymptote.

The pattern repeats every

\pi

radians. This repeat length is the period. Between consecutive vertical asymptotes, the curve rises from very negative values to very positive values. At the midpoint of each such interval, it crosses the axis.

Useful points in one interval around zero are

\left(-\frac{\pi}{4},-1\right)

,

(0,0)

, and

\left(\frac{\pi}{4},1\right)

. These points help set the curve’s shape; the asymptotes determine the ends of the interval.

\tan(x+\pi)=\tan x

Zeros: $x=k\pi$ .
Vertical asymptotes: $x=\frac{\pi}{2}+k\pi$ .
Period: $\pi$ radians.
Each branch rises between its left and right asymptotes.

3. Build a sketch with a table and a repeat pattern

A value table gives exact points for plotting. Choose angles where sine and cosine are familiar, and calculate their ratio. Do not put a value in the table where cosine is zero, because tangent is undefined there.

Start with the interval between

-\frac{\pi}{2}

and

\frac{\pi}{2}

. Mark its asymptotes, then plot the three key points. Draw a smooth rising branch through the points, approaching each asymptote without touching it.

To extend the graph, copy the same branch left or right by exactly

\pi

radians. The repeated zeros and asymptotes provide checks that the copies are placed correctly.

\left(-\frac{\pi}{4},-1\right),\ (0,0),\ \left(\frac{\pi}{4},1\right)

Plot asymptotes as boundaries, not as points on the curve.
Use exact radian values for the key points.
Repeat the branch at intervals of $\pi$ .

4. Apply the features and check the shape

A correct sketch should show the same interval pattern wherever it is drawn. For example, the asymptotes at

-\frac{\pi}{2}

and

\frac{\pi}{2}

are separated by

\pi

, and the next branch has the same shape between

\frac{\pi}{2}

and

\frac{3\pi}{2}

.

Before finishing, check four features: the horizontal axis crossings, the vertical asymptotes, the spacing between repeated branches, and the rising direction within each branch. These checks catch most plotting errors without requiring many calculated points.

\text{period}=\pi

A zero is where the curve meets the horizontal axis.
An asymptote marks an excluded input where the function is undefined.
The period is the horizontal distance between matching parts of consecutive branches.

Useful values for one tangent branch

Angle $x$	$-\frac{\pi}{4}$	$0$	$\frac{\pi}{4}$
Value $\tan x$	$-1$	$0$	$1$

Worked example

Sketch the basic tangent graph from $-\pi$ to $\pi$

Sketch

y=\tan x

for

-\pi\le x\le\pi

. Label its zeros, vertical asymptotes, and key points.

Locate the undefined inputs
Tangent is undefined when cosine is zero. In the stated interval, this happens at the two half-turn angles shown. Draw dashed vertical lines there; they are not part of the graph.
$x=-\frac{\pi}{2},\quad x=\frac{\pi}{2}$
Mark the axis crossings
Tangent is zero when sine is zero and cosine is not zero. Within the interval, the crossings occur at the three integer multiples of $\pi$ shown.
$x=-\pi,\quad x=0,\quad x=\pi$
Plot useful points
Use the familiar angles on either side of zero. Their tangent values are $-1$ and $1$ , so these points show the scale and direction of the central branch.
$\left(-\frac{\pi}{4},-1\right),\quad (0,0),\quad \left(\frac{\pi}{4},1\right)$
Complete the branches
Draw a rising curve through the key points between each pair of asymptotes. The pattern repeats every $\pi$ , so the outer branches match the central branch after a horizontal shift. At $x=-\pi$ and $x=\pi$ , include the zero points because cosine is nonzero there.
$\tan(x+\pi)=\tan x$

Answer: The sketch has zeros at

-\pi

,

0

, and

\pi

, vertical asymptotes at

-\frac{\pi}{2}

and

\frac{\pi}{2}

, and rising branches on each interval between asymptotes.

Check: The two neighboring zeros are

\pi

radians apart, matching the period. The central branch passes through

\left(-\frac{\pi}{4},-1\right)

,

(0,0)

, and

\left(\frac{\pi}{4},1\right)

.

Common mistakes and how to avoid them

Using degrees on an axis labelled in radians.

Correction: Keep the angle units consistent. For this graph, label key positions with values such as

\frac{\pi}{4}

and

\frac{\pi}{2}

.

Drawing the curve through a vertical asymptote.

Correction: Cosine is zero there, so tangent is undefined. Leave a gap at each asymptote.

Using

2\pi

as the period.

Correction: Tangent repeats after a horizontal shift of

\pi

, so matching branches are

\pi

apart.

Drawing a branch that falls from left to right.

Correction: The basic tangent branch rises between consecutive asymptotes. Check it against the key points from

-1

through

0

to

1

.

Lesson summary

The basic graph is $y=\tan x=\frac{\sin x}{\cos x}$ wherever cosine is nonzero.
Its zeros are $x=k\pi$ , and its vertical asymptotes are $x=\frac{\pi}{2}+k\pi$ .
Its period is $\pi$ radians. Each branch rises between consecutive asymptotes.
A reliable sketch marks asymptotes and zeros, plots key points, and repeats the branch every $\pi$ .

Check your understanding

Question 1

Which statement correctly describes the basic tangent graph?

It has period $2\pi$ and is defined when $\cos x=0$ .
It has period $\pi$ and is undefined when $\cos x=0$ .
It has period $\pi$ and is undefined when $\sin x=0$ .
It has period $\frac{\pi}{2}$ and crosses the axis at $x=\frac{\pi}{2}+k\pi$ .

Show answer and explanation

It has period

\pi

and is undefined when

\cos x=0

.

Tangent repeats every

\pi

radians. Since

\tan x=\frac{\sin x}{\cos x}

, it is undefined wherever cosine is zero.

Question 2

What is the value of

\tan\left(\frac{\pi}{4}\right)

?

$0$
$-1$
$1$
It is undefined.

Show answer and explanation

1

At

\frac{\pi}{4}

, sine and cosine are equal and nonzero, so their ratio is

1

.

Question 3

Which pair gives the two vertical asymptotes around the central branch?

$x=-\pi$ and $x=\pi$
$x=-\frac{\pi}{4}$ and $x=\frac{\pi}{4}$
$x=-\frac{\pi}{2}$ and $x=\frac{\pi}{2}$
$x=0$ and $x=\pi$

Show answer and explanation

x=-\frac{\pi}{2}

and

x=\frac{\pi}{2}

Cosine is zero at

-\frac{\pi}{2}

and

\frac{\pi}{2}

, which bound the central branch.

Key terms

Radian: A unit for measuring angles. A full turn is $2\pi$ radians.
Period: The horizontal distance after which a repeating graph begins the same pattern again.
Zero: An input value where the function output is zero, so the graph meets the horizontal axis.
Vertical asymptote: A vertical boundary that the graph approaches but does not include; for tangent, it occurs where cosine is zero.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B2.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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