📚 Polar Coordinates: Key Exam Points | 极坐标：考点精讲
The polar coordinate system is a vital topic in IB Mathematics Analysis and Approaches (HL) and CIE A Level Mathematics (9709). It provides an alternative way to describe curves using a distance from the origin and an angle, offering elegant solutions to problems involving areas, tangents, and intersections. This revision guide highlights the key concepts, formulas, common pitfalls, and typical exam questions you need to master.
极坐标是 IB 数学分析与方法 (AA) 高等级 (HL) 和 CIE A Level 数学的重中之重。它用极径和极角来描述曲线，为解决面积、切线和交点等问题提供了优雅工具。本文精讲核心考点、公式、常见错误和典型考题，助你拿下高分。
1. Polar Coordinates Definition and Basic Relations | 极坐标的定义与基本关系
A point P in the polar coordinate system is represented by (r, θ), where r is the distance from the pole (origin) O to P, and θ is the angle measured anticlockwise from the initial line (positive x‑axis). r can be negative, indicating the point lies on the opposite ray. The polar coordinates of a point are not unique: (r, θ), (r, θ+2π), and (−r, θ+π) all represent the same point.
极坐标系中，点 P 表示为 (r, θ)，r 是极点 O 到 P 的距离，θ 是从极轴（正向 x 轴）逆时针量度的角度。r 可以为负，表示点在反向射线上。极坐标不唯一：(r, θ)、(r, θ+2π) 和 (−r, θ+π) 均表示同一点。
Basic identities: Many curves become simpler in polar form. The relationship between r and θ, r = f(θ), is the polar equation. The pole corresponds to r = 0 for any θ. When you see an equation like r = 2, it is a circle of radius 2 centred at the pole; a linear θ = constant represents a ray from the pole.
基本恒等式：许多曲线在极坐标下变得简洁。极坐标方程 r = f(θ) 描述了曲线。极点对应 r=0（任意 θ）。方程 r = 2 表示圆心在极点、半径为 2 的圆；θ = 常数表示从极点出发的一条射线。
2. Conversion Formulae | 坐标转换公式
When a point has polar coordinates (r, θ) and Cartesian coordinates (x, y), the following conversions hold:
当一点既有极坐标 (r, θ) 又有直角坐标 (x, y) 时，转换关系如下：
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Polar to Cartesian: x = r cosθ, y = r sinθ
极坐标转直角坐标：x = r cosθ, y = r sinθ
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Cartesian to Polar: r² = x² + y², tanθ = y/x (x ≠ 0). To determine the correct quadrant for θ, use the signs of x and y or the ATAN2 function. If x = 0, then θ = π/2 or 3π/2 depending on the sign of y.
直角坐标转极坐标：r² = x² + y², tanθ = y/x (x ≠ 0)。先用 x 和 y 的符号确定象限，再求 θ。若 x=0，则 θ = π/2 或 3π/2，根据 y 的正负决定。
These relations are essential for converting polar equations to Cartesian to identify familiar curves or for integrating in polar form. For example, r = 2 secθ converts to x = 2, a vertical line. Conversely, a circle x² + y² = 2x becomes r = 2 cosθ. Always check that the angle θ lies in the correct range (often 0 ≤ θ < 2π or −π < θ ≤ π) as required by the question.
这些关系是极坐标与直角坐标互化的基础，常用来识别曲线或进行极坐标积分。例如，r = 2 secθ 化为 x = 2，为一条竖直线。反之，圆 x² + y² = 2x 化为 r = 2 cosθ。注意根据题目要求确定 θ 的范围（常为 0 ≤ θ < 2π 或 −π < θ ≤ π）。
3. Sketching Simple Polar Curves | 常见极坐标曲线绘图
Mastering the sketches of standard polar curves saves time in exams. Key types include:
熟练画出标准极坐标曲线能大幅节省考试时间。常见类型包括：
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Circles: r = a (circle radius a centred at pole). r = 2a cosθ : circle radius a with centre at (a,0) on polar axis. r = 2a sinθ : circle radius a with centre at (0,a) (in Cartesian). To sketch, note the maximum r is 2a at θ=0 or π/2, and r=0 at θ=π/2 or π.
圆：r = a（圆心在极点，半径 a）。r = 2a cosθ：圆心在 (a,0)，半径 a。r = 2a sinθ：圆心在 (0,a)，半径 a。绘图时注意 r 的最大值出现在 θ=0 或 π/2，r=0 出现在 θ=π/2 或 π 处。
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Cardioid: r = a(1 + cosθ) or r = a(1 + sinθ). Heart‑shaped, symmetric about the polar axis (cos) or θ = π/2 (sin). The maximum r is 2a; it passes through the pole when cosθ = −1 (θ=π) or sinθ = −1 (θ=3π/2).
心形线：r = a(1 + cosθ) 或 r = a(1 + sinθ)。形状如心。cos 型关于极轴对称，sin 型关于 θ = π/2 对称。最大 r 为 2a；通过极点时 cosθ = −1 (θ=π) 或 sinθ = −1 (θ=3π/2)。
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Rose curves: r = a cos(nθ) or r = a sin(nθ). If n is odd, there are n petals; if n is even, there are 2n petals. Petal length a. For example, r = 2cos(3θ) has 3 petals, each extending to r=2. Find petal tips by setting cos(nθ) = ±1
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