📚 Radians | 弧度制
Radians are the standard unit of angular measure in A-level Mathematics, especially in trigonometry, calculus and circular measure. Unlike degrees, radians link an angle directly to the length of an arc on a unit circle, which makes them essential for higher-level work.
弧度是 A-level 数学中角度的标准单位，尤其用于三角学、微积分和圆的度量。与度数不同，弧度把角度与单位圆上的弧长直接联系起来，因此对高阶数学至关重要。
1. What is a Radian? | 什么是弧度
A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius. This definition makes radian measure a natural ratio of two lengths, so it has no units. On a unit circle, the angle in radians is numerically equal to the arc length.
弧度是圆心角所对的弧长等于半径时的角度。这个定义使弧度成为两个长度的自然比值，因此没有单位。在单位圆上，弧度数在数值上等于弧长。
θ rad = s / r ; when s = r, θ = 1 rad
One full turn has arc length 2πr, so it sweeps 2π radians. This gives the fundamental identity 360° = 2π rad.
一整圈的弧长为 2πr，因此扫过 2π 弧度。由此得到基本恒等式 360° = 2π 弧度。
360° = 2π rad
2. Converting Between Degrees and Radians | 度与弧度的转换
Since 360° equals 2π radians, dividing by 2 gives 180° = π rad. The conversion factors follow: radians = degrees × π/180, and degrees = radians × 180/π.
因为 360° 等于 2π 弧度，除以 2 得 180° = π 弧度。换算公式为：弧度 = 度数 × π/180，度数 = 弧度 × 180/π。
radians = degrees × π/180 ; degrees = radians × 180/π
For example, convert 75° to radians: 75 × π/180 = 5π/12 rad. Convert 1.2 rad to degrees: 1.2 × 180/π ≈ 68.8°.
例如，将 75° 转换为弧度：75 × π/180 = 5π/12 弧度。将 1.2 弧度转换为度：1.2 × 180/π ≈ 68.8°。
3. Common Angles You Must Know | 必须熟记的常见角
You must be able to move instantly between common degree and radian measures, and to recall exact trigonometric values at these angles. These exact values are frequently tested in Edexcel exams.
你必须能在常见度数和弧度之间迅速转换，并记住这些角度的精确三角函数值。这些精确值在 Edexcel 考试中经常考查。
| Degrees | Radians | sin θ | cos θ | tan θ |
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
4. Arc Length | 弧长公式
For a circle of radius r, if a central angle θ is measured in radians, the arc length l is given by l = rθ. This is a direct consequence of the definition θ = l/r.
对于半径为 r 的圆，如果圆心角 θ 用弧度表示，则弧长 l 为 l = rθ。这是定义 θ = l/r 的直接结果。
l = rθ
For example, a circle has radius 10 cm and a central angle 2π/5. The arc length is 10 × 2π/5 = 4π cm.
例如，圆的半径为 10 cm，圆心角为 2π/5，弧长为 10 × 2π/5 = 4π cm。
5. Area of a Sector | 扇形面积公式
The area of a sector is the fraction θ/2π of the full circle area πr². Therefore A = (θ/2π) × πr² = ½r²θ, where θ must be in radians.
扇形面积占整个圆面积 πr² 的比例为 θ/2π，因此 A = (θ/2π) × πr² = ½r²θ，其中 θ 必须用弧度。
A = ½ r² θ
For example, radius 8 cm and angle π/4 give A = ½ × 64 × π/4 = 8π cm².
例如，半径 8 cm，角度 π/4，则 A = ½ × 64 × π/4 = 8π cm²。
6. Area of a Segment | 弓形面积
To find the segment area, subtract the isosceles triangle area ½r² sin θ from the sector area ½r²θ. This gives A = ½r²(θ − sin θ).
求弓形面积时，从扇形面积 ½r²θ 中减去等腰三角形面积 ½r² sin θ，得到 A = ½r²(θ − sin θ)。
A_segment = ½ r² (θ − sin θ)
For r = 6 and θ = π/3, the area is A = ½ × 36 × (π/3 − √3/2) = 18(π/3 − √3/2) cm².
当 r = 6，θ = π/3 时，面积为 A = ½ × 36 × (π/3 − √3/2) = 18(π/3 − √3/2) cm²。
7. Small Angle Approximations | 小角近似
When θ is small and measured in radians, sin θ and tan θ are approximately equal to θ, and cos θ is approximately 1 − θ²/2. These approximations come from Maclaurin series and are very useful in physics and further pure maths.
当 θ 很小且用弧度表示时，sin θ 和 tan θ 约等于 θ，cos θ 约等于 1 − θ²/2。这些近似来自麦克劳林级数，在物理和进阶纯数学中非常有用。
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