📚 The Polar Form of a Complex Number | 复数的极坐标形式
, we describe it by its distance from the origin and the angle it makes with the positive real axis. This article provides a thorough, exam-focused guide to everything you need to know about the polar form for AQA A-Level Mathematics.
复数的极坐标形式是 A-Level 数学中最强大的表示方法之一。我们不再将复数写作 z = x + yi，而是通过它到原点的距离以及与正实轴所成的角度来描述它。本文将提供一份针对 AQA A-Level 数学考试、全面的极坐标形式学习指南。
1. The Modulus and Argument | 模与辐角
Before we can write a complex number in polar form, we must understand its two components: the modulus and the argument.
在将复数写成极坐标形式之前，我们必须理解它的两个组成部分：模与辐角。
number z = x + yi is its distance from the origin on the Argand diagram. It is denoted by |z| and is always non-negative.
复数 z = x + yi 的模是它在阿甘图（复平面）上到原点的距离，记作 |z|，始终为非负数。
|z| = √x² + y²
sured anticlockwise. The principal argument is the unique value of θ such that -π < θ ≤ π.
复数 z 的辐角是原点与 z 连线与正实轴之间的夹角 θ，按逆时针方向测量。主辐角是满足 -π < θ ≤ π 的唯一 θ 值。
Example: For z = 1 + i, we have |z| = √1² + 1² = √2, and arg(z) = π/4.
例如：对于 z = 1 + i，有 |z| = √1² + 1² = √2，且 arg(z) = π/4。
Special cases to memorise: the modulus of a real number a is its absolute value |a| a) = π. Pure imaginary numbers have argument π/2 for positive yi and -π/2 for negative yi.
需要记忆的特殊情况：实数 a 的模是其绝对值 |a|；若 a 为正，则 arg(a) = 0；若 a 为负，则 arg(a) = π。纯虚数中，正 yi 的辐角为 π/2，负 yi 的辐角为 -π/2。
2. Expressing a Complex Number in Polar Form | 用极坐标形式表达复数
Once the modulus r and argument θ are known, we can write the complex number in polar form (also called modulus-argument form).
一旦已知模 r 与辐角 θ，我们就可以将复数写成极坐标形式（也称为模-辐角形式）。
z = r(cos θ + i sin θ)
: x = r cos θ and y = r sin θ. Substituting these into z = x + yi gives the formula above.
这可以直接从笛卡尔坐标推导得出：x = r cos θ，y = r sin θ。将这些代入 z = x + yi 即可得到上述公式。
For example, the complex number with modulus 2 and argument π/3 can be written as:
例如，模为 2、辐角为 π/3 的复数可以写为：
z = 2(cos(π/3) + i sin(π/3)) = 2(½ + i√3/2) = 1 + i√3
= cos θ + i sin θ. Remember this abbreviation; it is widely used in examination solutions.
一种常用的紧凑写法是 r cis θ，其中 cis θ = cos θ + i sin θ。请记住这个缩写，它在考试解答中被广泛使用。
3. Converting Between Cartesian and Polar Forms | 笛卡尔形式与极坐标形式的相互转换
Converting from Cartesian to polar form requires care, especially when determining the argument. The modulus is straightforward, but the argument must be adjusted for the correct quadrant.
从笛卡尔形式转换为极坐标形式需要特别小心，尤其是在确定辐角时。模的计算很直接，但辐角必须根据所在象限进行调整。
r = √x² + y², θ = arctan(y/x) (adjusted for the correct quadrant)
lator gives a value in the range (-π/2, π/2). However, the correct argument may differ by π or 2π depending on which quadrant the complex number lies in. The following table is essential.
当使用 θ = arctan(y/x) 时，计算器给出的值在 (-π/2, π/2) 范围内。然而，正确的辐角可能相差 π 或 2π，具体取决于复数所在的象限。下表至关重要。
| Quadrant 象限 | Argument 辐角 |
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