Tangent Function

Definition

Definition of the Tangent Function

In a right triangle, let $\theta$ be an acute angle, then tangent is defined as:

\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

where “opposite” refers to the side opposite to angle $\theta$ , and “adjacent” refers to the side adjacent to angle $\theta$ .

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Definition
$\theta$	Greek letter	Theta (theta)	Represents the measure of an angle (acute angle)

Properties

Domain: $x \neq \frac{\pi}{2} + k\pi$
Range: $\mathbb{R}$
Period: $\pi$
Parity: Odd function
Monotonicity: Increasing in each monotonic interval
Symmetry: Symmetric about the origin

Graph

Exercises

Exercise1

Find the domain and range of $y = \tan x$ .

Answer and Explanation (3 个标签)

tangent function domain range

Domain: $x \neq \frac{\pi}{2} + k\pi$ ; Range: $\mathbb{R}$ .

Exercise2

Determine the periodicity and parity of $y = \tan x$ .

Answer and Explanation (3 个标签)

tangent function period parity

Period is $\pi$ , it is an odd function.

Exercise3

Determine which of the following functions are trigonometric functions: $y = \sin x$ , $y = x^2$ , $y = \tan x$ , $y = e^x$ .

Answer and Explanation (3 个标签)

trigonometric function sine function tangent function

$y = \sin x$ , $y = \tan x$ are trigonometric functions, $y = x^2$ , $y = e^x$ are not.

Exercise4

Given $y = \tan x$ , write its domain and range.

Answer and Explanation (3 个标签)

tangent function domain range

Domain: $x \neq \frac{\pi}{2} + k\pi$ ; Range: $\mathbb{R}$ .

Exercise5

Which of the following functions are odd functions?
(A) $y = \sin x$
(B) $y = \cos x$
(C) $y = \tan x$
(D) $y = x^2$

Answer and Explanation (5 个标签)

sine function cosine function tangent function odd function even function

(A), (C) are odd functions, (B), (D) are not.

Exercise6

Given $y = \tan x$ , write its period and domain.

Answer and Explanation (3 个标签)

tangent function period domain

Period is $\pi$ , domain is $x \neq \frac{\pi}{2} + k\pi$ .

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\pi$	Greek letter	Pi (pie)	Pi, used to represent the period of the tangent function ( $\pi$ )

Bilingual Glossary

Chinese Term	English Term	Phonetic	Explanation
正切函数	tangent function	/ˈtændʒənt ˈfʌŋkʃən/	One of the trigonometric functions, denoted as $\tan x$
正切	tangent	/ˈtændʒənt/	Function name, abbreviated as $\tan$
对边	opposite side	/ˈɒpəzɪt saɪd/	The right side opposite to the angle in a right triangle
邻边	adjacent side	/əˈdʒeɪsənt saɪd/	The right side adjacent to the angle in a right triangle
周期	period	/ˈpɪəriəd/	The smallest interval at which function values repeat
定义域	domain	/dəʊˈmeɪn/	The range of values for the independent variable

PreviousCosine FunctionDetailed explanation of the cosine function's definition, properties, graph, and typical exercises. NextCotangent FunctionDetailed explanation of the cotangent function's definition, properties, graph, and typical exercises.

课程路线图

1
Exploring Functions in Advanced Mathematics
当前课程
Functions are a core idea of advanced mathematics. This course walks through foundational concepts, key properties, and classic constants so you can read, reason, and compute with confidence.
前往课程

进阶推荐

The World of Limits in Advanced Mathematics

Limits are the foundation of calculus and one of the most important ideas in advanced mathematics.

开始学习

进阶推荐

Sequences

Sequences bridge discrete thinking and calculus. This track covers core definitions, limits, convergence, and classic models.

开始学习