Small Angle Approximations | 小角度近似公式

📚 Small Angle Approximations | 小角度近似公式

When the angle θ is very small, certain trigonometric functions can be replaced by simpler algebraic expressions. These are called the small angle approximations, and they are a key tool in A-level mathematics, physics and engineering.

当角度 θ 很小时，某些三角函数可以用更简单的代数表达式代替。这就是所谓的小角度近似，它是 A-level 数学、物理和工程中的关键工具。

1. What Are Small Angle Approximations? | 什么是小角度近似？

For a small angle θ measured in radians, the following approximations are commonly used: sin θ ≈ θ, tan θ ≈ θ, and cos θ ≈ 1 − θ²/2.

对于以弧度为单位的小角度 θ，常用的近似有：sin θ ≈ θ，tan θ ≈ θ，以及 cos θ ≈ 1 − θ²/2。

These approximations become more accurate as θ approaches zero. They are derived from the Taylor (or Maclaurin) series of each function.

当 θ 趋近于零时，这些近似变得更加精确。它们由每个函数的泰勒（或麦克劳林）级数推导而来。

2. The Condition: θ Must Be Small | 条件：θ 必须很小

The approximations are valid only when θ is sufficiently close to zero. A common rule of thumb is θ < 0.1 radians, but the acceptable range depends on the required precision.

这些近似仅在 θ 足够接近零时有效。通常的经验法则是 θ < 0.1 弧度，但可接受的范围取决于所需的精度。

For example, when θ = 0.1 rad, sin θ ≈ 0.09983, which is very close to 0.1. The error is less than 0.2%.

例如，当 θ = 0.1 rad 时，sin θ ≈ 0.09983，非常接近 0.1，误差不到 0.2%。

3. sin θ ≈ θ | 正弦小角度近似

The Maclaurin series for sin θ is: sin θ = θ − θ³/3! + θ⁵/5! − … . For small θ, the higher power terms are negligible, so sin θ ≈ θ.

sin θ 的麦克劳林级数为：sin θ = θ − θ³/3! + θ⁵/5! − … 。对于小 θ，高次项可以忽略，所以 sin θ ≈ θ。

sin θ ≈ θ

Geometrically, on a unit circle, the arc length of angle θ is θ, while the vertical coordinate is sin θ. For small angles, the arc and the vertical height are almost equal.

从几何上看，在单位圆中，角 θ 对应的弧长为 θ，而纵坐标为 sin θ。在小角度时，弧长与垂直高度几乎相等。

4. tan θ ≈ θ | 正切小角度近似

The Maclaurin series for tan θ is: tan θ = θ + θ³/3 + 2θ⁵/15 + … . Therefore, for small θ, tan θ ≈ θ.

tan θ 的麦克劳林级数为：tan θ = θ + θ³/3 + 2θ⁵/15 + … 。因此，对于小 θ，tan θ ≈ θ。

tan θ ≈ θ

Note that tan θ ≈ θ is actually a better approximation than sin θ ≈ θ in some cases because the next term for tan is +θ³/3 while for sin it is −θ³/6. But for very small θ both work well.

注意，在某些情况下 tan θ ≈ θ 比 sin θ ≈ θ 更好，因为 tan 的下一项是 +θ³/3，而 sin 的下一项是 −θ³/6。但对于非常小的 θ，两者都很好用。

5. cos θ ≈ 1 − θ²/2 | 余弦小角度近似

The Maclaurin series for cos θ is: cos θ = 1 − θ²/2! + θ⁴/4! − … . Keeping the first two terms gives the standard approximation.

cos θ 的麦克劳林级数为：cos θ = 1 − θ²/2! + θ⁴/4! − … 。保留前两项即得到标准近似。

cos θ ≈ 1 − θ²/2

Unlike sin and tan, the first-order term is zero for cos, so the second-order term is essential. This is why the approximation includes θ².

与 sin 和 tan 不同，cos 的一阶项为零，因此二阶项至关重要。这就是近似中包含 θ² 的原因。

6. θ Must Be in Radians | θ 必须使用弧度

All small angle approximations require θ to be measured in radians, not degrees. If θ is given in degrees, you must convert first.

所有小角度近似都要求 θ 以弧度为单位，而不是角度。如果题目给出角度，必须先转换。

To convert: θ_rad = θ_deg × π/180. For example, 5° = 5 × π/180 ≈ 0.0873 rad.

转换公式：θ_rad = θ_deg × π/180。例如，5° = 5 × π/180 ≈ 0.0873 rad。

7. Derivation Using Limits | 利用极限推导

These approximations can also be derived from the standard limits: lim(θ→0) (sin θ)/θ = 1 and lim(θ→0) (tan θ)/θ = 1.

这些近似也可以由标准极限推导：lim(θ→0) (sin θ)/θ = 1 和 lim(θ→0) (tan θ)/θ = 1。

For cos, use the identity 1 − cos θ = 2 sin²(θ/2). When θ is small, sin(θ/2) ≈ θ/2, so 1 − cos θ ≈ θ²/2.

对于 cos，使用恒等式 1 − cos θ = 2 sin²(θ/2)。当 θ 很小时，sin(θ/2) ≈ θ/2，所以 1 − cos θ ≈ θ²/2。

8. Error and Higher-Order Terms | 误差与高阶项

For sin θ, the next term is −θ³/6. For tan θ, the next term is +θ³/3. For cos θ, the next term is +θ⁴/24. These determine the error.

对于 sin θ，下一项是 −θ³/6。对于 tan θ，下一项是 +θ³/3。对于 cos θ，下一项是 +θ⁴/24。这些决定了误差。

If more precision is needed, extra terms can be included:

如果需要更高精度，可以加入更多项：

sin θ ≈ θ − θ³/6
tan θ ≈ θ + θ³/3
cos θ ≈ 1 − θ²/2 + θ⁴/24

9. Approximations for Composite Expressions | 复合表达式的小角度近似

When expressions involve products or quotients, apply the approximations separately and simplify. For example, (sin θ × tan θ)/(1 − cos θ) ≈ θ × θ/(θ²/2) = 2.

当表达式涉及乘积或商时，分别对各部分应用近似并化简。例如：(sin θ × tan θ)/(1 − cos θ) ≈ θ × θ/(θ²/2) = 2。

For expressions like ln(1 + sin θ), use compound approximations: since sin θ ≈ θ, we get ln(1 + θ) ≈ θ − θ²/2 for small θ.

对于像 ln(1 + sin θ) 这样的表达式，可以使用复合近似：因为 sin θ ≈ θ，所以 ln(1 + θ) ≈ θ − θ²/2 当 θ 很小时。

10. Applications in Pendulum Motion | 在单摆运动中的应用

For a simple pendulum, the equation of motion involves sin θ. For small oscillations, sin θ ≈ θ, which linearises the equation and leads to simple harmonic motion.

对于单摆，运动方程包含 sin θ。对于小幅度摆动，sin θ ≈ θ，这使方程线性化，从而得到简谐运动。

The period of a pendulum is T = 2π√(L/g). This formula assumes the small angle approximation. For larger angles, the period becomes slightly longer.

单摆周期公式 T = 2π√(L/g) 就基于小角度近似。对于较大角度，周期会略微变长。

11. Worked Example | 例题解析

Example: Use small angle approximations to estimate the value of cos 0.1 − sin 0.1.

例题：使用小角度近似估算 cos 0.1 − sin 0.1 的值。

Solution: cos θ ≈ 1 − θ²/2 and sin θ ≈ θ. Thus cos 0.1 − sin 0.1 ≈ (1 − 0.1²/2) − 0.1 = 1 − 0.005 − 0.1 = 0.895.

解：cos θ ≈ 1 − θ²/2，sin θ ≈ θ。因此 cos 0.1 − sin 0.1 ≈ (1 − 0.1²/2) − 0.1 = 1 − 0.005 − 0.1 = 0.895。

Quantity	Approximation	Example θ=0.1
sin θ	θ	0.1
tan θ	θ	0.1
cos θ	1 − θ²/2	0.995

12. Common Mistakes to Avoid | 常见错误提醒

Do not use degrees. Do not replace cos θ by 1 without considering θ². Do not assume θ ≈ 0 when θ is not actually small.

不要使用角度制。不要将 cos θ 直接替换为 1 而不考虑 θ² 项。不要在 θ 并不真的很小时使用近似。

Not converting degrees to radians.
Forgetting that cos θ needs a second-order term.
Ignoring the sign of the next term in error analysis.
Using the approximation for θ > 0.5 without justification.

Always check that θ is in radians and that the approximation is valid for the required level of accuracy.

始终检查 θ 是否以弧度为单位，以及该近似是否满足所需的精度要求。

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Small Angle Approximations | 小角度近似公式