Polar Coordinates: Key Points for AQA | AQA 数学：极坐标考点精讲

📚 Polar Coordinates: Key Points for AQA | AQA 数学：极坐标考点精讲

Polar coordinates offer a fresh lens through which to describe curves and areas beyond the reach of standard Cartesian thinking. In AQA A Level Further Mathematics, polar coordinates form a vital topic that combines geometry, calculus, and curve sketching into one unified framework. Mastering this area means you can confidently convert between coordinate systems, sketch elegant curves like cardioids and roses, compute enclosed areas, derive tangent slopes, and even tackle arc lengths. This revision guide walks you through every essential skill with clear explanations, worked examples, and exam-ready strategies.

极坐标为描述曲线和区域提供了一种超越标准笛卡尔思维的全新视角。在 AQA A Level 进阶数学中，极坐标是一个结合了几何、微积分和曲线绘制的核心考点。掌握这一领域，意味着你能够自信地在坐标系间转换、绘制如心形线和玫瑰线等优美曲线、计算封闭区域面积、推导切线斜率，甚至处理弧长问题。本复习指南将通过清晰的解释、丰富的示例和备考策略，带你逐一攻克所有必备技能。

1. What Are Polar Coordinates? | 极坐标是什么？

In the polar coordinate system, a point in the plane is located by its distance r from a fixed origin O (the pole) and the angle θ measured anticlockwise from the initial line (usually the positive x‑axis). We write a point as (r, θ), where r can be any real number, and θ is typically given in radians. This allows us to describe curves that would be cumbersome in Cartesian form, such as spirals or petals.

在极坐标系中，平面上一点由其到固定原点 O（极点）的距离 r 以及从初始线（通常为正 x 轴）逆时针测量的角度 θ 来确定。我们将点记为 (r, θ)，其中 r 可以是任意实数，θ 通常以弧度表示。这使我们能够描述在笛卡尔形式下显得繁琐的曲线，例如螺旋线或花瓣状曲线。

If r is negative, we interpret the point as lying on the ray opposite to the direction of θ, at a distance |r| from the pole. Usually we restrict ourselves to 0 ≤ θ < 2π for unique representation, but the domain often extends to accommodate multiple loops or symmetry.

如果 r 为负值，我们理解为该点位于与 θ 方向相反的射线上，距离极点为 |r|。通常我们限制在 0 ≤ θ < 2π 以获得唯一表示，但为了容纳多重环路或对称性，定义域常常有所扩展。

The polar grid is made of concentric circles centred at O and rays emanating from O at constant angles. This grid is perfect for capturing rotational symmetry and radial distances.

极坐标网格由以 O 为中心的同心圆以及从 O 出发的恒定角度射线组成。这种网格非常适合捕捉旋转对称性和径向距离。

2. Converting Between Polar and Cartesian Coordinates | 极坐标与直角坐标的互化

The bridge between polar and Cartesian systems rests on two fundamental equations: x = r cos θ and y = r sin θ. From these, we can also deduce r² = x² + y² and tan θ = y/x (taking care with the quadrant of θ). You must be completely fluent with these conversions, because AQA questions frequently demand switching between forms to simplify calculations.

极坐标与直角坐标之间的桥梁依赖于两个基本方程：x = r cos θ 和 y = r sin θ。由它们还可以推出 r² = x² + y² 以及 tan θ = y/x（需注意 θ 的象限）。你必须能熟练运用这些转换，因为 AQA 试题经常要求在不同形式之间切换以简化计算。

To convert from polar (r, θ) to Cartesian (x, y), simply substitute into the two equations. For example, the point (4, π/3) becomes (4 cos(π/3), 4 sin(π/3)) = (2, 2√3).

从极坐标 (r, θ) 转为直角坐标 (x, y)，只需代入这两个方程。例如，点 (4, π/3) 变为 (4 cos(π/3), 4 sin(π/3)) = (2, 2√3)。

To convert from Cartesian to polar, compute r = √(x² + y²) and find θ using arctan(y/x), adjusting the angle according to the quadrant of the point. For instance, (−1, 1) gives r = √2 and θ = 3π/4.

从直角坐标转为极坐标时，计算 r = √(x² + y²) 并用 arctan(y/x) 求 θ，再根据点所在的象限调整角度。例如，(−1, 1) 得到 r = √2 和 θ = 3π/4。

This conversion also lets us transform polar equations into Cartesian form. For example, the polar equation r = 2a cos θ can be rewritten as x² + y² = 2ax, which after completing the square gives a circle centred at (a, 0).

这种转换也让我们能将极坐标方程化为直角坐标形式。例如，极坐标方程 r = 2a cos θ 可改写为 x² + y² = 2ax，配方后得到一个圆心在 (a, 0) 的圆。

3. Sketching Polar Curves: Basic Principles | 极坐标曲线绘图：基本原理

Sketching a polar curve r = f(θ) requires a systematic approach. Start by considering key values of θ, such as 0, π/2, π, 3π/2, and multiples that make the trigonometric functions zero or reach extreme values. Build a table of θ versus r, marking where r = 0 (the pole) and where |r| is maximum. Then plot points and join them smoothly, respecting the direction of increasing θ.

绘制极坐标曲线 r = f(θ) 需要一套系统的方法。从关键的 θ 值入手，例如 0、π/2、π、3π/2 以及使三角函数为零或极值的倍数角度。建立 θ 与 r 的表格，标出 r = 0（极点处）和 |r| 最大时的位置。然后描点并光滑连接，注意遵循 θ 增大的方向。

Always check for symmetry to reduce work. A curve is symmetric about the initial line (θ = 0) if r(θ) = r(−θ) or if replacing θ by −θ yields the same r. It is symmetric about the vertical line θ = π/2 if r(θ) = r(π − θ). And it has symmetry about the pole if r(θ) = −r(θ) or r(θ) = r(θ + π). Recognising these symmetries allows you to sketch only a fraction of the θ‑range and reflect the rest.

务必检查对称性以减轻工作量。如果 r(θ) = r(−θ) 或者用 −θ 替换 θ 得到相同的 r，则曲线关于初始线 (θ = 0) 对称。如果 r(θ) = r(π − θ)，则关于竖直线 θ = π/2 对称。如果 r(θ) = −r(θ) 或 r(θ) = r(θ + π)，则关于极点对称。识别这些对称性可以让你只绘制 θ 范围的一部分，其余则通过反射得到。

For periodic functions like r = a cos(nθ) or r = a sin(nθ), you can predict the number of petals: if n is odd, there are n petals; if n is even, there are 2n petals. But always verify how r behaves over a full period 0 ≤ θ ≤ 2π.

对于周期性函数，如 r = a cos(nθ) 或 r = a sin(nθ)，你可以预测花瓣数量：若 n 为奇数，有 n 片花瓣；若 n 为偶数，有 2n 片花瓣。但务必验证 r 在一个完整周期 0 ≤ θ ≤ 2π 内的行为。

4. Common Polar Curves and Their Equations | 常见极坐标曲线及其方程

AQA expects you to be familiar with several families of polar curves. The simplest is a circle with centre at the pole, given by r = a (constant). Another circle passing through the pole is r = 2a cos θ (centred on the polar axis) or r = 2a sin θ (centred on θ = π/2).

AQA 期望你熟悉几类极坐标曲线。最简单的是圆心在极点的圆，方程为 r = a（常数）。另一种过极点的圆是 r = 2a cos θ（圆心在极轴上）或 r = 2a sin θ（圆心在 θ = π/2 线上）。

Cardioid: r = a(1 + cos θ) or r = a(1 + sin θ), producing a heart‑shaped curve with a single cusp at the pole. Limacons take the form r = a + b cos θ or r = a + b sin θ; when a < b they have an inner loop, when a = b it is a cardioid, and when a > b it is a dimpled or convex limacon.

心形线：r = a(1 + cos θ) 或 r = a(1 + sin θ)，生成一条在极点处有一个尖点的心形曲线。蚶线的形式为 r = a + b cos θ 或 r = a + b sin θ；当 a < b 时带有内环，当 a = b 时即为心形线，当 a > b 时为有凹痕或凸出的蚶线。

Rose curves: r = a cos(nθ) or r = a sin(nθ), with petal patterns as described earlier. Lemniscate: r² = a² cos(2θ) (figure‑eight shape) and spirals like r = aθ (Archimedean spiral). Memorising the standard shapes and their key angles will speed up your sketch work considerably.

玫瑰线：r = a cos(nθ) 或 r = a sin(nθ)，形成如前所述的花瓣图案。双纽线：r² = a² cos(2θ)（8 字形），以及螺旋线如 r = aθ（阿基米德螺旋）。记住标准形状及其关键角度将极大加快你的绘图速度。

Here is a quick reference table:

Curve / 曲线	Polar Equation / 极坐标方程	Notes / 备注
Circle (centre pole)	r = a	半径 a
Circle through pole	r = 2a cos θ	圆心 (a, 0)
Cardioid	r = a(1 + cos θ)	心形
Limaçon with loop	r = a + b cos θ (a < b)	内环
Rose (n odd)	r = a cos(nθ)	n 片花瓣
Lemniscate	r² = a² cos(2θ)	8 字形

5. Finding the Area Enclosed by a Polar Curve | 求极坐标曲线围成的面积

The area swept out by a polar curve r = f(θ) from θ = α to θ = β is given by the integral A = ½ ∫_α^β r² dθ. This formula is derived by summing the areas of tiny circular sectors, each with area ½ r² Δθ.

极坐标曲线 r = f(θ) 从 θ = α 到 θ = β 所扫出的面积由积分 A = ½ ∫_α^β r² dθ 给出。这个公式源于对小扇形面积 ½ r² Δθ 的求和。

When asked to find the area enclosed by a single loop or the whole curve, you must carefully determine the limits of integration α and β. For a closed curve, these are often the angles where r = 0, that is, where the curve passes through the pole. Solve f(θ) = 0 to find successive roots, which bound one loop.

当要求求单环或整个曲线所围面积时，你必须仔细确定积分限 α 和 β。对于闭合曲线，这些通常是 r = 0 的角度，即曲线经过极点的位置。解 f(θ) = 0 求出相邻的根，它们界定了一个环路。

For example, to find the area of one loop of r = a cos(2θ), set cos(2θ) = 0, giving θ = π/4 and θ = 3π/4 as adjacent roots, though for a single petal you might use −π/4 to π/4 depending on symmetry. Always sketch or visualise to confirm the correct sector.

例如，求 r = a cos(2θ) 一个环的面积，令 cos(2θ) = 0，得到 θ = π/4 和 θ = 3π/4 作为相邻根，但对于单瓣你可能会根据对称性使用 −π/4 到 π/4。始终通过绘图或想象来确认正确的扇形区域。

The calculation then proceeds: A = ½ ∫_(-π/4)^(π/4) (a cos(2θ))² dθ = ½ a² ∫ cos²(2θ) dθ. Use the double‑angle identity cos²u = (1 + cos(2u))/2 to integrate. After evaluating, the area of one petal of r = a cos(2θ) simplifies to (π a²)/8.

然后进行计算：A = ½ ∫_(-π/4)^(π/4) (a cos(2θ))² dθ = ½ a² ∫ cos²(2θ) dθ。利用倍角恒等式 cos²u = (1 + cos(2u))/2 进行积分。经计算，r = a cos(2θ) 一瓣的面积化简为 (π a²)/8。

When a curve is symmetric, you can integrate over a smaller interval and multiply accordingly. This reduces the chance of algebraic error and saves time.

当曲线具有对称性时，你可以对更小区间积分并相应乘以倍数。这减少了代数出错的机会，也节省了时间。

6. Areas Between Two Polar Curves | 两曲线间的面积

To find the area of a region bounded by two polar curves r = f(θ) and r = g(θ) between θ = α and θ = β, where f(θ) ≥ g(θ) ≥ 0, we use the formula A = ½ ∫_α^β [f(θ)² − g(θ)²] dθ. This is a direct extension of the single‑curve area formula, subtracting the inner region’s area from the outer region’s area.

要求由两条极坐标曲线 r = f(θ) 与 r = g(θ) 在 θ = α 到 θ = β 之间围成区域的面积，且 f(θ) ≥ g(θ) ≥ 0，我们使用公式 A = ½ ∫_α^β [f(θ)² − g(θ)²] dθ。这是单曲线面积公式的直接扩展，即从外部区域面积减去内部区域面积。

The critical challenge here is to find the intersection points of the two curves, which define α and β. Set f(θ) = g(θ) and solve for θ. However, be aware that polar curves can intersect at the pole even when their equations are not equal for the same θ, so you must also check whether the region includes the pole separately.

这里的关键难点是找到两条曲线的交点，它们界定了 α 和 β。令 f(θ) = g(θ) 并求出 θ。但需要注意，极坐标曲线可以相交于极点，即使它们在相同 θ 下的方程不相等，因此你还必须单独检查区域是否包含极点。

For example, find the area outside the circle r = 2 and inside the cardioid r = 2(1 + cos θ). First, find intersections: 2 = 2(1 + cos θ) ⇒ cos θ = 0 ⇒ θ = π/2, 3π/2. By symmetry, the total area is twice the area from θ = 0 to θ = π/2. Hence A = 2 × ½ ∫_0^(π/2) [(2(1+cos θ))² − 2²] dθ. Expand and integrate to obtain A = 8 + π, a standard result worth remembering.

例如，求圆 r = 2 之外、心形线 r = 2(1 + cos θ) 之内的面积。首先求交点：2 = 2(1 + cos θ) ⇒ cos θ = 0 ⇒ θ = π/2, 3π/2。由对称性，总面积等于从 θ = 0 到 θ = π/2 面积的两倍。于是 A = 2 × ½ ∫_0^(π/2) [(2(1+cos θ))² − 2²] dθ。展开并积分可得 A = 8 + π，这是一个值得记住的标准结果。

Always draw a sketch before integrating; it helps you decide which curve is outer, confirm limits, and spot symmetries that halve your workload.

积分前务必画出草图；这有助于你判断哪条曲线在外部、确认积分限，并发现可以使工作量减半的对称性。

7. Tangents to Polar Curves | 极坐标曲线的切线

The gradient of a tangent to a polar curve r = f(θ) is given by dy/dx, but we cannot differentiate r directly with respect to x. Instead we use parametric differentiation with parameter θ. Write x = r cos θ, y = r sin θ. Then dy/dx = (dy/dθ) / (dx/dθ), provided dx/dθ ≠ 0.

极坐标曲线 r = f(θ) 的切线斜率由 dy/dx 给出，但我们不能直接将 r 对 x 求导。因此我们以 θ 为参数进行参数微分。写出 x = r cos θ, y = r sin θ。然后 dy/dx = (dy/dθ) / (dx/dθ)，前提是 dx/dθ ≠ 0。

Using the product rule, dy/dθ = (dr/dθ) sin θ + r cos θ, and dx/dθ = (dr/dθ) cos θ − r sin θ. Hence the tangent slope is:

dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ)

where r′ = dr/dθ. This formula lets you find the slope at any specific θ, and you can then write the equation of the tangent line in Cartesian form if required.

利用乘法法则，dy/dθ = (dr/dθ) sin θ + r cos θ，dx/dθ = (dr/dθ) cos θ − r sin θ。因此切线斜率为：dy/dx = (r′ sin θ + r cos θ) / (r′ cos θ − r sin θ)，其中 r′ = dr/dθ。该公式可让你求出任意特定 θ 处的斜率，若需要还能写出直角坐标形式的切线方程。

Parallel to the initial line occurs when dy/dθ = 0, and perpendicular to the initial line (vertical tangent) occurs when dx/dθ = 0. Watch out for points where both derivatives are zero; these may be cusps or self‑intersections, often requiring a limiting approach to evaluate the gradient.

当 dy/dθ = 0 时切线平行于初始线，当 dx/dθ = 0 时切线垂直于初始线（竖直切线）。注意两个导数同时为零的点；这些可能是尖点或自交点，通常需要利用极限方法来分析斜率。

AQA exam questions often ask you to find the equation of the tangent at a given point, or to find the points where the tangent is parallel to the initial line. Practise the full procedure: compute r′, substitute into the expression for dy/dx, evaluate at the given θ, then use the point‑slope form with Cartesian coordinates obtained from (r, θ).

AQA 考试题经常要求你求出给定点处的切线方程，或找出切线平行于初始线的点。请练习完整流程：计算 r′，代入 dy/dx 表达式，在给定 θ 处求值，然后用从 (r, θ) 得到的直角坐标通过点斜式写出方程。

8. Arc Length of a Polar Curve | 极坐标曲线的弧长

Although arc length is sometimes examined as a separate subtopic, it fits naturally within polar coordinates. The length L of a polar curve r = f(θ) from θ = α to θ = β is:

L = ∫_α^β √(r² + (dr/dθ)²) dθ

This formula comes from the parametric arc length expression √( (dx/dθ)² + (dy/dθ)² ) after substituting x = r cos θ, y = r sin θ and simplifying.

尽管弧长有时会作为一个独立子课题考查，但它自然地与极坐标融合在一起。极坐标曲线 r = f(θ) 从 θ = α 到 θ = β 的长度 L 为：L = ∫_α^β √(r² + (dr/dθ)²) dθ。此公式源自参数弧长表达式 √( (dx/dθ)² + (dy/dθ)² )，代入 x = r cos θ, y = r sin θ 并化简而来。

This integral can be challenging, often requiring trigonometric identities to simplify the square root. For example, the length of the cardioid r = a(1 + cos θ) from θ = 0 to θ = 2π involves r² + (r′)² = a²(1+cos θ)² + a² sin² θ = a²(2 + 2 cos θ) = 4a² cos²(θ/2). The square root becomes 2a cos(θ/2), which is easily integrable over the full range, yielding L = 8a after careful handling of the absolute value.

这个积分可能颇具挑战性，通常需要利用三角恒等式化简根号。例如，心形线 r = a(1 + cos θ) 从 θ = 0 到 θ = 2π 的长度涉及 r² + (r′)² = a²(1+cos θ)² + a² sin² θ = a²(2 + 2 cos θ) = 4a² cos²(θ/2)。开方后得到 2a cos(θ/2)，在全区间上容易积分，经妥善处理绝对值后得到 L = 8a。

Exam tip: before integrating, always check whether the curve retraces itself; if it does, adjust limits to avoid double‑counting length. Use symmetry to integrate over a half or quarter period and multiply.

考试技巧：积分前务必检查曲线是否自相重叠；若是，需调整积分限以免重复计算长度。利用对称性对半个或四分之一个周期积分再乘以倍数。

9. Strategies for AQA Exam Questions | AQA 考试题应对策略

AQA polar coordinate questions often combine multiple skills. A typical question might ask you to sketch a curve, find the area of one loop, determine the slope of the tangent at a point, and perhaps find an intersection with a line like θ = constant. Approach such questions in a logical order: sketch first (even a rough one) to understand the shape and key angles; then identify the limits for integration from the sketch; perform the integration carefully, showing clear steps; finally, apply the tangent formula if needed.

AQA 极坐标考题常常融合多种技能。一个典型的题目可能要求你绘制曲线、求一个环的面积、确定某点处的切线斜率，或许还会求与 θ = 常数等直线的交点。以逻辑顺序处理这类问题：先绘制草图（哪怕是简图），以理解形状和关键角度；接着由草图确定积分限；然后仔细计算积分，展示清晰步骤；最后，如有需要，应用切线公式。

When dealing with curves defined by r = a cos(nθ) or r = a sin(nθ), always state the period and number of petals explicitly. This demonstrates understanding and helps you set the correct limits for area or arc length. If asked for the area of a single petal, your limits are the nearest two angles where r = 0.

处理由 r = a cos(nθ) 或 r = a sin(nθ) 定义的曲线时，务必明确说明周期和花瓣数量。这展示了你的理解，并有助于为面积或弧长设置正确的积分限。如果要求单瓣面积，积分限就是 r = 0 的最近两个角度。

For integration, remember the key identities: cos²u = (1 + cos 2u)/2, sin²u = (1 − cos 2u)/2. Often the integrand simplifies to a constant plus a cosine or sine term of double angle, making integration straightforward. Keep your working neat, and always substitute back the limits carefully.

积分时，记住关键恒等式：cos²u = (1 + cos 2u)/2，sin²u = (1 − cos 2u)/2。被积函数通常化简为常数加双角余弦或正弦项，使积分变得简单。保持书写整洁，并始终小心地将积分限代回。

When asked to find the equations of tangents, double‑check your Cartesian coordinates for the point: x = r cos θ, y = r sin θ. Then use the point‑slope form y − y₁ = m(x − x₁). If the tangent is horizontal (m = 0) or vertical (dx/dθ = 0), you can state its equation directly by finding the y‑coordinate or x‑coordinate of the point.

当要求求切线方程时，仔细核对点的直角坐标：x = r cos θ, y = r sin θ。然后使用点斜式 y − y₁ = m(x − x₁)。如果切线为水平（m = 0）或竖直（dx/dθ = 0），你可以通过找出该点的 y 坐标或 x 坐标直接写出其方程。

10. Common Mistakes and How to Avoid Them | 常见错误与规避方法

One of the most frequent errors is misidentifying the limits of integration. Students often take 0 to 2π indiscriminately, leading to double the area or length. Remember: integrate only over the range that traces the curve exactly once without overlapping. For r = sin(2θ), a full trace from 0 to 2π covers the four‑petal rose twice; you typically need 0 to π to draw the entire curve once, and a single petal may be 0 to π/2.

最常见的错误之一是弄错积分限。学生常不加区分地使用 0 到 2π，导致面积或长度被加倍。请记住：只对恰好描绘曲线一次且不自相重叠的范围进行积分。对于 r = sin(2θ)，从 0 到 2π 的完整路径会重复绘制这条四瓣玫瑰线；你通常需要 0 到 π 才能将整条曲线画完一次，而单瓣可能为 0 到 π/2。

Another pitfall is forgetting the ½ factor in the area formula. The area is ½ ∫ r² dθ, not ∫ r² dθ. Missing that ½ will halve your final mark even with a perfect integration method. Write the ½ prominently and carry it through every step.

另一个陷阱是忘记面积公式中的 ½ 因子。面积是 ½ ∫ r² dθ，而不是 ∫ r² dθ。即使积分方法完美，漏掉这个 ½ 也会让你最终得分减半。将 ½ 写在校为显眼处，并贯穿每一步。

When using the tangent gradient formula, errors often arise from incorrectly differentiating r with respect to θ, especially when r involves a product or chain rule. Take your time: write r(θ) clearly, find dr/dθ before substituting, and double‑check using a simple test point. Also, don’t forget that y = r sin θ, not just sin θ – the r must be differentiated too.

使用切线斜率公式时，错误常源于对 r 关于 θ 的求导不正确，特别是当 r 涉及乘法或链式法则时。耐心对待：清楚地写出 r(θ)，在代入前求出 dr/dθ，并用一个简单的测试点进行核对。同时，别忘了 y = r sin θ，而不只是 sin θ——r 也必须求导。

In area‑between‑curves problems, assuming the outer curve is always the one with the larger coefficient can fail when the curves cross. Always sketch and determine, for each θ interval, which curve lies further from the pole. The integral may need to be split at the intersection angles.

在两曲线间的面积问题中，想当然地认为系数较大的曲线总是在外部可能会出错，因为曲线可能交叉。始终画草图，并对每一个 θ 区间确定哪条曲线离极点更远。积分可能需要在交点角度处分段进行。

Finally, rounding errors are seldom penalised if you show exact values. Whenever possible, keep your answers in exact form using π, √, and fractions. Decimal approximations should be given to three significant figures only when explicitly requested.

最后，只要给出精确值，舍入误差很少会被扣分。尽量保留 π、根号和分数的精确形式。只有在明确要求时，才将小数近似值保留到三位有效数字。

11. Practice Example: Full AQA‑Style Question | 练习题：完整 AQA 风格题目

Let’s consolidate everything with a worked problem: A curve C is defined by r = 2 cos(3θ) for 0 ≤ θ ≤ π/6. (a) Sketch the curve, stating the number of petals. (b) Show that the area enclosed by this loop is π/6. (c) Find the slope of the tangent to C at the point where θ = π/12. (d) Find the equation of this tangent.

我们来通过一个完整的问题巩固所学：曲线 C 由 r = 2 cos(3θ) 定义，0 ≤ θ ≤ π/6。（a）绘制曲线，说明花瓣数量。（b）证明该环围成的面积为 π/6。（c）求 θ = π/12 处 C 的切线斜率。（d）求此切线方程。

(a) Since n = 3 is odd, there are 3 petals. The given range 0 to π/6 is exactly one‑sixth of a full period 0 to 2π, but note the petal is traced once as θ runs from −π/6 to π/6; the question restricts to 0 ≤ θ ≤ π/6 to describe half a petal, then by symmetry we get the full loop. Sketch by plotting key points: θ = 0 ⇒ r = 2; θ = π/6 ⇒ r = 0. The curve starts at (2,0) and ends at the pole.

（a）因为 n = 3 是奇数，所以有 3 片花瓣。给定的范围 0 到 π/6 恰好是全周期 0 到 2π 的六分之一，但注意花瓣在 θ 从 −π/6 到 π/6 被完整描绘一次；题目限制在 0 ≤ θ ≤ π/6 描述了半个花瓣，然后根据对称性得到整个环路。通过关键点绘制草图：θ = 0 ⇒ r = 2；θ = π/6 ⇒ r = 0。曲线从 (2,0) 开始，终止于

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