Sine Function

Definition

Definition of the Sine Function

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

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}

where “opposite” refers to the side opposite to angle $\theta$ , and “hypotenuse” is the longest side of the triangle.

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: $\mathbb{R}$
Range: $[-1, 1]$
Period: $2\pi$
Parity: Odd function
Monotonicity: Increasing on $[-\frac{\pi}{2}, \frac{\pi}{2}]$
Symmetry: Symmetric about the origin

$\mathbb{R}$ (blackboard bold R): This is the standard mathematical symbol for the set of real numbers (Real numbers), representing all real numbers. Blackboard bold is a special font style used in mathematics to denote sets, distinguishing them from ordinary variables.

Graph

Divergent Thinking

1. What is a Chord?

Chord (xián): In a circle, a line segment connecting any two points on the circle is called a chord. A chord passing through the center is called the diameter, which is the longest chord.

Historical Background: As early as 1800 BCE, ancient Babylonians used chord tables to solve astronomical and surveying problems. Greek mathematicians Hipparchus and Ptolemy created detailed chord tables recording chord lengths for given central angles.

Relationship between Chord and Sine: The name “sine function” comes from the concept of “chord”. In a circle, drawing a perpendicular from the center to the midpoint of a chord divides the chord into two equal parts. Half the length of this chord is the prototype of the modern sine function. Later, Indian astronomers further developed the concept of “half-chord” to form the modern sine function.

2. How did “sine” get its name?

The English name “sine” for the sine function went through an interesting “translation journey”:

India: In the 5th century CE, Indian astronomer Aryabhata created the term “jya” (sine)
Arabic: “jya” was transliterated into Arabic as “jiba”, but misread as “jaib” (meaning “bay” or “gulf”)
Europe: In the 12th century, “jaib” was translated into Latin as “sinus” (also meaning “bay”)
Modern: Eventually evolved into the English “sine”

This is why the sine function’s name is a product of “transliteration + mistranslation”, originally related to “chord”, but during translation it became associated with “bay”.

Exercises

Exercise1

Given $y = \sin x$ , find its domain and range.

Answer and Explanation (3 个标签)

sine function domain range

Domain: $\mathbb{R}$ ; Range: $[-1, 1]$ .

Exercise2

Sketch the approximate graphs of $y = \sin x$ and $y = \cos x$ , and compare their periodicity and symmetry.

Answer and Explanation (4 个标签)

sine function cosine function period parity

Both $y = \sin x$ and $y = \cos x$ have period $2\pi$ . $\sin x$ is an odd function, $\cos x$ is an even function.

Exercise3

Given $y = \sin x$ , find its minimum positive period.

Answer and Explanation (2 个标签)

sine function period

The minimum positive period is $2\pi$ .

Exercise4

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.

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 sine function ( $2\pi$ )
$\mathbb{R}$	Mathematical symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, domain of the sine function

Bilingual Glossary

Chinese Term	English Term	Phonetic	Explanation
正弦函数	sine function	/saɪn ˈfʌŋkʃən/	One of the trigonometric functions, denoted as $\sin x$
正弦	sine	/saɪn/	Function name, abbreviated as $\sin$
对边	opposite side	/ˈɒpəzɪt saɪd/	The right side opposite to the angle in a right triangle
斜边	hypotenuse	/haɪˈpɒtənjuːz/	The longest side 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
值域	range	/reɪndʒ/	The range of values for the function
奇函数	odd function	/ɒd ˈfʌŋkʃən/	A function satisfying $f(-x) = -f(x)$
弦（xián）	chord	/kɔːd/	A line segment connecting any two points on a circle

PreviousAngles and RadiansDetailed explanation of the definitions of angles and radians, conversion relationships, and their applications in trigonometric functions NextCosine FunctionDetailed explanation of the cosine function's definition, properties, graph, and typical exercises.

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