Polar Coordinates: Revision Guide for WJEC IGCSE | 极坐标考点精讲

📚 Polar Coordinates: Revision Guide for WJEC IGCSE | 极坐标考点精讲

Polar coordinates provide an alternative way to describe the location of points in the plane using a distance and an angle. They are particularly useful for curves with circular or radial symmetry, and they appear in the WJEC IGCSE Mathematics specification (usually within the Additional or Further Mathematics content). Mastering the basics of polar coordinates, including conversion, graphing, and solving equations, is essential for exam success.

极坐标是描述平面上点位置的另一种方法，利用距离和角度。它们对于具有圆形或径向对称的曲线特别有用，并且出现在WJEC IGCSE数学大纲中（通常在附加或进阶数学内容里）。掌握极坐标的基础知识，包括坐标转换、图形绘制和方程求解，对于考试成功至关重要。

1. Polar Coordinate Basics | 极坐标基础

In polar coordinates, each point is represented by (r, θ), where r is the radial distance from the origin (called the pole) and θ is the angle measured anticlockwise from the positive x-axis (the polar axis). The angle can be given in degrees or radians; in WJEC IGCSE, degrees are commonly used, but you must be comfortable with both.

在极坐标系中，每个点用 (r, θ) 表示，其中 r 是到原点（极点）的径向距离，θ 是从正 x 轴（极轴）逆时针测量的角度。角度可以用度数或弧度表示；在 WJEC IGCSE 中，度数很常用，但你必须熟悉两者。

Negative values of r: ( -r, θ ) is equivalent to ( r, θ + 180° ) or ( r, θ + π ). This allows the same point to have multiple polar representations.

负 r 值：(-r, θ) 等价于 (r, θ + 180°) 或 (r, θ + π)。这使得同一点可以有多种极坐标表示。

The pole itself is represented by (0, θ) for any angle θ.

极点本身可以用任意角度 θ 的 (0, θ) 表示。

To uniquely define a point, we often restrict r ≥ 0 and 0° ≤ θ < 360° (or 0 ≤ θ < 2π), but be aware of other conventions.

为唯一定义点，我们通常限制 r ≥ 0 且 0° ≤ θ < 360°（或 0 ≤ θ < 2π），但要注意其他约定。

2. Plotting Polar Points | 极坐标点的绘制

To plot a point given in polar coordinates, start at the pole, rotate by the angle θ from the polar axis, and then move out a distance r along that ray. If r is negative, move in the opposite direction.

要绘制极坐标给出的点，从极点开始，从极轴旋转角度 θ，然后沿该射线向外移动距离 r。如果 r 为负值，则向相反方向移动。

The point (4, 120°) lies in the second quadrant, while (3, -45°) is plotted by rotating 45° clockwise. Always check the quadrant based on the angle and use a polar grid if possible.

点 (4, 120°) 位于第二象限，而 (3, -45°) 通过顺时针旋转 45° 绘制。始终根据角度确认象限，并尽可能使用极坐标网格。

It is useful to practice with a polar grid that features concentric circles for r and radial lines for θ. The concentric circles help you measure distances

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）