Angles and Radians

Angles and radians are two different unit systems for measuring angles. In daily life, we commonly use degrees (°) to describe the size of angles; in mathematics, especially when learning trigonometric functions, radians (rad) are more commonly used. Understanding the relationship between angles and radians is crucial for mastering trigonometric functions.

Definition of Angles

Historical Origin: The origin of the degree system can be traced back to the Babylonian sexagesimal counting system. They divided the circumference into 360 equal parts, which may come from their observation of 360 days in a year (approximate value), or because 360 can be divided by many numbers, making it convenient for partitioning.

Characteristics:

• Intuitive and easy to understand, convenient for daily use and measurement
• Widely used in daily life, engineering surveying, geographic navigation, etc.
• Unit symbol: °

Examples:

• Right angle: 90°
• Straight angle: 180°
• Perigon: 360°

Definition of Radians

When the arc length equals the radius ($s = r$), the angle is 1 radian, denoted as 1 rad.

Characteristics:

• Mathematical formulas are more concise and convenient for calculation
• Widely used in calculus, trigonometric functions, and other fields
• Unit symbol: rad (usually omitted in mathematical expressions)

Examples:

• When the arc length is twice the radius, the angle is 2 radians
• When the arc length is half the radius, the angle is 0.5 radians

Relationship Between Angles and Radians

Basic Relationship

A complete circumference has an arc length of $2\pi r$ (radius $r$), corresponding to an angle of 360°.

According to the radian definition, the radian measure of a complete circumference is:

$$\text{Radians} = \frac{2\pi r}{r} = 2\pi$$

Therefore:

Furthermore:

Conversion Formulas

Based on the basic relationship $180° = \pi$, we can obtain the conversion formulas:

Degrees to Radians:

Radians to Degrees:

Conversion Examples

Example 1: Convert 90° to radians

$$90° = 90 \times \frac{\pi}{180} = \frac{\pi}{2} \text{ radians}$$

Example 2: Convert $\frac{\pi}{3}$ radians to degrees

$$\frac{\pi}{3} \text{ radians} = \frac{\pi}{3} \times \frac{180}{\pi} = 60°$$

Common Correspondence Table

The following is the correspondence between common angles and radians:

Degrees	Radians	Description
$0°$	$0$	Starting position
$30°$	$\frac{\pi}{6}$
$45°$	$\frac{\pi}{4}$
$60°$	$\frac{\pi}{3}$
$90°$	$\frac{\pi}{2}$	Right angle
$120°$	$\frac{2\pi}{3}$
$135°$	$\frac{3\pi}{4}$
$150°$	$\frac{5\pi}{6}$
$180°$	$\pi$	Straight angle
$270°$	$\frac{3\pi}{2}$
$360°$	$2\pi$	Complete circumference

Why Use Radians in Mathematics

In mathematics, especially in calculus and trigonometric functions, the radian system is more commonly used than the degree system, for the following reasons:

1. Mathematical Formulas Are More Concise

When using radians, many important mathematical formulas are more concise and elegant:

Derivative Formulas:

If using degrees, the formula would become complex.

2. Limit Formulas Are Simpler

Important limit formulas are very concise when using radians:

This formula holds when $x$ is in radians and is the basis for many derivations in calculus.

3. Convenient for Integration and Series Expansion

Using radians makes the integration and Taylor series expansion of functions more concise:

Taylor Series Expansion:

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \quad (x \text{ in radians})$$ $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \quad (x \text{ in radians})$$

4. Natural Connection with Pi

In the radian system, a complete circumference naturally corresponds to $2\pi$, making many mathematical relationships more intuitive and unified.

Applications in Trigonometric Functions

Periods of Trigonometric Functions

The periods of trigonometric functions are usually expressed in radians:

• The period of $\sin x$ and $\cos x$ is $2\pi$ (radians)
• The period of $\tan x$ and $\cot x$ is $\pi$ (radians)

Correspondence on Graph Horizontal Axis

When drawing trigonometric function graphs, situations like this often occur:

• Graph horizontal axis may display angles (such as 0°, 90°, 180°, 360°)
• Text descriptions use periods in radians (such as $2\pi$)

Their correspondence is:

• 360° on the graph corresponds to period $2\pi$ (radians)
• 180° on the graph corresponds to period $\pi$ (radians)
• 90° on the graph corresponds to period $\frac{\pi}{2}$ (radians)

Conversion in Calculations

When using trigonometric functions in programming or calculations, angles usually need to be converted to radians:

// JavaScript example
const degrees = 90;
const radians = degrees * Math.PI / 180;
const sinValue = Math.sin(radians); // Calculate sin(90°)

This is because most programming language math library functions default to using radians as input units.

Exercises

Exercise 1

Convert the following angles to radians:

(1) $30°$
(2) $120°$
(3) $225°$

Exercise 2

Convert the following radians to degrees:

(1) $\frac{\pi}{4}$
(2) $\frac{3\pi}{2}$
(3) $\frac{7\pi}{6}$

Exercise 3

Given that the period of $\sin x$ is $2\pi$ (radians), explain that the repetition every 360° on the graph corresponds to the period $2\pi$.

---

Summary

Symbols Used in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\theta$	Greek letter	Theta (theta)	Represents the size of an angle (central angle)
$\pi$	Greek letter	Pi (pi)	Pi, used for angle-radian conversion (180° = π radians)

Chinese-English Glossary

Chinese Term	English Term	Pronunciation	Explanation
角度	degree	/dɪˈɡriː/	Dividing the circumference into 360 equal parts, each part being 1 degree, denoted as 1°
弧度	radian	/ˈreɪdiən/	Measuring the size of an angle using the ratio of arc length to radius
弧长	arc length	/ɑːk leŋθ/	The curved length between two points on a circle
半径	radius	/ˈreɪdiəs/	The distance from the center of a circle to any point on the circumference
圆心角	central angle	/ˈsentrəl ˈæŋɡəl/	An angle with the center of the circle as its vertex
换算	conversion	/kənˈvɜːʃən/	Converting between different units of measurement

Previous三角函数的起源与发现通过故事化场景，带领读者从实际问题出发，探索三角函数的诞生与应用。 NextSine FunctionDetailed explanation of the sine function's definition, properties, graph, and typical exercises.