📚 A-Level Mathematics: Using Trigonometric Identities in Parametric Equations | A-Level数学：参数方程中三角恒等式的运用
Parametric equations allow us to describe a curve by expressing both x and y in terms of a third variable, often called a parameter. When the parameter is an angle, trigonometric identities become powerful tools for simplifying, eliminating the parameter, and analysing the curve.
参数方程通过引入第三个变量（通常称为参数）来同时表示 x 和 y。当参数为角度时，三角恒等式便成为简化表达式、消去参数以及分析曲线性质的强大工具。
1. What Are Parametric Equations? | 参数方程是什么？
A parametric curve is defined by two equations, x = f(t) and y = g(t), where t is the parameter. Instead of writing y directly as a function of x, we track how both coordinates change as t varies.
参数曲线由两个方程 x = f(t) 和 y = g(t) 定义，其中 t 是参数。我们不再将 y 直接写成 x 的函数，而是追踪两个坐标如何随 t 的变化而变化。
For trigonometric parametrisation, the parameter is usually denoted θ (theta). Common examples include x = cos θ, y = sin θ, which trace out a circle as θ ranges from 0 to 2π.
对于三角参数化，参数通常记作 θ（theta）。常见例子包括 x = cos θ，y = sin θ，当 θ 从 0 到 2π 变化时，这一组方程描绘出一个圆。
Why use parametric equations? They can represent curves that are difficult or impossible to express as y = f(x), such as loops, cusps, and circles that fail the vertical line test. They also simplify many rate-of-change problems by connecting to the angle.
为什么要使用参数方程？因为它们可以表示难以或无法写成 y = f(x) 形式的曲线，例如环线、尖点以及不满足垂线检验的圆。同时，通过引入角度，许多变化率问题也得到了简化。
2. The Essential Trigonometric Identities | 必备三角恒等式
The most frequently used identity in this topic is the Pythagorean identity: sin² θ + cos² θ = 1. It holds for all real θ and forms the bridge between parametrised coordinates and Cartesian equations.
本主题中最常用的恒等式是毕达哥拉斯恒等式：sin² θ + cos² θ = 1。它对所有实数 θ 均成立，是连接参数坐标与直角坐标方程的桥梁。
Two other forms are equally important: sec² θ = 1 + tan² θ and csc² θ = 1 + cot² θ. These arise when secant, cosecant, or tangent functions appear in the parametrisation.
另外两个重要形式是：sec² θ = 1 + tan² θ 和 csc² θ = 1 + cot² θ。当参数化中出现正割、余割或正切函数时，这两个恒等式会起到关键作用。
Double-angle identities, such as cos 2θ = cos² θ − sin² θ, are occasionally useful in simplifying integrals or derivatives involving parametric forms. Still, for pure elimination, the Pythagorean identities are the most reliable.
二倍角公式，如 cos 2θ = cos² θ − sin² θ，在简化涉及参数形式的积分或导数时偶尔有用。但就单纯消去参数而言，毕达哥拉斯恒等式是最可靠的。
3. Circle Parametrisation: x = r cos θ, y = r sin θ | 圆的参数化：x = r cos θ, y = r sin θ
The simplest application is the circle of radius r centred at the origin. Let x = r cos θ and y = r sin θ. Squaring both equations and adding them gives x² + y² = r², the familiar Cartesian equation of a circle.
最简单的应用是以原点为圆心、半径为 r 的圆。设 x = r cos θ，y = r sin θ。将两个方程分别平方后相加，得到 x² + y² = r²，这正是熟悉的圆的直角坐标方程。
If the centre is (h, k), we modify the equations to x = h + r cos θ and y = k + r sin θ. Eliminating θ still yields (x − h)² + (y − k)² = r².
如果圆心在 (h, k)，我们将方程修改为 x = h + r cos θ 和 y = k + r sin θ。消去 θ 后仍然得到 (x − h)² + (y − k)² = r²。
When asked to eliminate the parameter, always check whether the parameterised expressions contain squared terms that can be combined directly through an identity.
当要求消去参数时，始终检查参数化表达式中是否含有可以通过恒等式直接合并的平方项。
4. Ellipse Parametrisation: x = a cos θ, y = b sin θ | 椭圆的参数化：x = a cos θ, y = b sin θ
An ellipse with semi-major axis a and semi-minor axis b can be written as x = a cos θ and y = b sin θ. Dividing x by a and y by b before squaring gives:
半长轴为 a、半短轴为 b 的椭圆可以写成 x = a cos θ 和 y = b sin θ。先将 x 除以 a、y 除以 b，再平方，得到：
x²/a² + y²/b² = 1
Notice the pattern: whatever constant multiplies cos θ appears in the denominator under x², and the constant multiplying sin θ appears under y².
注意规律：乘以 cos θ 的常数会出现在 x² 的分母中，而乘以 sin θ 的常数出现在 y² 的分母中。
This elegant use of the identity sin² θ + cos² θ = 1 is one of the most frequently tested A-Level problems. Students must not forget to divide by the coefficients before applying the identity.
这是 sin² θ + cos² θ = 1 的巧妙运用，也是 A-Level 考试中最常考的题型之一。学生务必记住在应用恒等式之前，先除以对应系数。
5. Hyperbola Parametrisation: x = a sec θ, y = b tan θ | 双曲线的参数化：x = a sec θ, y = b tan θ
For a hyperbola, the standard parametrisation is x = a sec θ and y = b tan θ. Using the identity sec² θ − tan² θ = 1, we obtain:
对于双曲线，标准参数化形式为 x = a sec θ 和 y = b tan θ。利用恒等式 sec² θ − tan² θ = 1，可以得到：
x²/a² − y²/b² = 1
This is the Cartesian equation of a horizontal hyperbola. The secant and tangent functions naturally create the subtraction sign in the identity.
这是水平双曲线的直角坐标方程。正割和正切函数天然地通过恒等式中的减号产生这一结构。
Similarly, x = a tan θ and y = b sec θ would produce y²/b² − x²/a² = 1, a vertical hyperbola. Always look at which function is paired with which coordinate.
类似地，x = a tan θ 和 y = b sec θ 会产生 y²/b² − x²/a² = 1，即垂直双曲线。务必仔细观察哪个函数与哪个坐标配对。
6. Eliminating the Parameter: Direct Substitution | 消去参数：直接代入法
The most direct method is to rearrange one equation to express the trigonometric function in terms of x or y, then substitute into the identity.
最直接的方法是先从其中一个方程中解出三角函数关于 x 或 y 的表达式，再代入恒等式。
For example, given x = 2 sin θ and y = 3 cos θ, we write sin θ = x/2 and cos θ = y/3. Substituting into sin² θ + cos² θ = 1 gives:
例如，已知 x = 2 sin θ 和 y = 3 cos θ，我们写出 sin θ = x/2 和 cos θ = y/3。代入 sin² θ + cos² θ = 1 得到：
x²/4 + y²/9 = 1
This is an ellipse centred at the origin. Direct substitution works well when the parameterised equations are already isolated for a single trigonometric function.
这是一个以原点为中心的椭圆。当参数化方程已经将单个三角函数分离出来时，直接代入法非常有效。
7. Eliminating the Parameter: Algebraic Manipulation | 消去参数：代数变换法
Sometimes the parameter appears in a more complex way, such as x = 1 + 2 cos θ and y = −1 + 2 sin θ. Here we must first isolate cos θ and sin θ, then square and add.
有时参数以更复杂的方式出现，例如 x = 1 + 2 cos θ 和 y = −1 + 2 sin θ。此时我们必须先分离 cos θ 和 sin θ，再平方相加。
From x = 1 + 2 cos θ, we get cos θ = (x − 1)/2. From y = −1 + 2 sin θ, we get sin θ = (y + 1)/2. Applying the identity gives:
由 x = 1 + 2 cos θ，得到 cos θ = (x − 1)/2。由 y = −1 + 2 sin θ，得到 sin θ = (y + 1)/2。应用恒等式得到：
(x − 1)²/4 + (y + 1)²/4 = 1
This simplifies to (x − 1)² + (y + 1)² = 4, a circle of radius 2 centred at (1, −1). Careful sign handling prevents common algebraic errors.
这可以简化为 (x − 1)² + (y + 1)² = 4，即半径为 2、圆心在 (1, −1) 的圆。小心处理符号可以避免常见的代数错误。
8. Finding dy/dx from Parametric Equations | 参数方程求导：dy/dx
The derivative of a parametric curve is found using the chain rule:
参数曲线的导数通过链式法则求得：
dy/dx = (dy/dθ) ÷ (dx/dθ), provided dx/dθ ≠ 0
For example, if x = 3 cos θ and y = 2 sin θ, then dx/dθ = −3 sin θ and dy/dθ = 2 cos θ. Therefore dy/dx = −(2 cos θ)/(3 sin θ) = −(2/3) cot θ.
例如，若 x = 3 cos θ 和 y = 2 sin θ，则 dx/dθ = −3 sin θ，dy/dθ = 2 cos θ。因此 dy/dx = −(2 cos θ)/(3 sin θ) = −(2/3) cot θ。
The gradient at a given point is obtained by substituting the specific value of θ. Notice that the derivative is expressed in terms of θ, which is often exactly what the exam question expects.
给定点的斜率可通过代入具体的 θ 值得到。注意导数是以 θ 表示的，这通常正是考试题目所要求的答案形式。
9. Tangents and Normals to Parametric Curves | 参数曲线的切线与法线
To find the tangent line at a point where θ = θ₀, first compute dy/dx at θ₀, then use the point-slope form:
要求 θ = θ₀ 处的切线，首先计算 dy/dx 在 θ₀ 处的值，然后使用点斜式：
y − y₀ = m(x − x₀)
Here m is the value of dy/dx at θ₀, and (x₀, y₀) is found by substituting θ₀ into the original parametric equations.
其中 m 是 dy/dx 在 θ₀ 处的值，(x₀, y₀) 通过将 θ₀ 代入原始参数方程得到。
The normal line is perpendicular to the tangent, so its gradient is −1/m. If m = 0, the normal is vertical; if m is undefined, the normal is horizontal. Trigonometric functions often produce exactly these edge cases, so test them separately.
法线与切线垂直，因此其斜率为 −1/m。若 m = 0，法线为竖直方向；若 m 不存在，法线为水平方向。三角函数经常会产生这些边界情况，所以应单独检验。
10. Sketching Parametric Curves Using Trigonometric Behaviour | 利用三角函数性质绘制参数曲线
Understanding the range of sin θ and cos θ helps sketch parametric curves quickly. For instance, with x = a cos θ and y = b sin θ, we know x is always between −a and a, and y is always between −b and b.
理解 sin θ 和 cos θ 的值域有助于快速绘制参数曲线。例如，对于 x = a cos θ 和 y = b sin θ，我们知道 x 始终在 −a 和 a 之间，y 始终在 −b 和 b 之间。
As θ increases from 0 to π/2, cos θ decreases from 1 to 0 while sin θ increases from 0 to 1. This traces the first quadrant portion of the curve. Repeating for each quadrant builds the full shape.
当 θ 从 0 增加到 π/2 时，cos θ 从 1 减小到 0，而 sin θ 从 0 增加到 1。这描绘出曲线的第一象限部分。对每个象限重复此过程，即可构建完整图形。
For more unusual parametrisations, draw a small table of θ, x, and y values at key angles such as 0, π/4, π/2, π, 3π/2. This provides enough points to sketch the curve accurately.
对于更特殊的参数化形式，可以制作一个关于 θ、x、y 的小表格，选取 0、π/4、π/2、π、3π/2 等关键角度。这样可以获得足够多的点来精确绘制曲线。
11. Area Under a Parametric Curve | 参数曲线下方的面积
The area under a parametric curve is given by the integral:
参数曲线下方的面积由以下积分给出：
A = ∫ y dx = ∫ y (dx/dθ) dθ
For an ellipse x = a cos θ, y = b sin θ from θ = 0 to 2π, we compute dx/dθ = −a sin θ, so the signed area is ∫₀²π b sin θ (−a sin θ) dθ = −ab ∫₀²π sin² θ dθ.
对于椭圆 x = a cos θ，y = b sin θ，θ 从 0 到 2π，计算 dx/dθ = −a sin θ，因此有向面积为 ∫₀²π b sin θ (−a sin θ) dθ = −ab ∫₀²π sin² θ dθ。
To evaluate sin² θ, use the double-angle identity sin² θ = (1 − cos 2θ)/2. The integral becomes −ab [θ/2 − sin 2θ/4] from 0 to 2π, giving −abπ. Taking the absolute value gives the actual area abπ, which matches the known area of an ellipse.
为了计算 sin² θ，使用二倍角恒等式 sin² θ = (1 − cos 2θ)/2。该积分变为 −ab [θ/2 − sin 2θ/4] 从 0 到 2π，得到 −abπ。取绝对值得到实际面积 abπ，这与椭圆面积公式一致。
Remember the absolute value: the integral may give a negative result if the curve is traced clockwise. Always interpret the sign in the physical context of the problem.
记住绝对值：如果曲线沿顺时针方向绘制，积分结果可能为负。务必结合问题的实际情境来解释符号。
12. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧
One common mistake is forgetting to divide by the coefficient before applying the identity. For x = 5 cos θ and y = 5 sin θ, do not write x² + y² = 1; instead write x²/25 + y²/25 = 1, which simplifies to x² + y² = 25.
一个常见错误是忘记在应用恒等式前先除以系数。对于 x = 5 cos θ 和 y = 5 sin θ，不要写成 x² + y² = 1；而应写成 x²/25 + y²/25 = 1，化简为 x² + y² = 25。
Another pitfall is misidentifying which identity to use. If the equation contains sec θ and tan θ, use sec² θ − tan² θ = 1. If it contains cos θ and sin θ, use sin² θ + cos² θ = 1.
另一个陷阱是误判该用哪个恒等式。如果方程包含 sec θ 和 tan θ，使用 sec² θ − tan² θ = 1；如果包含 cos θ 和 sin θ，则使用 sin² θ + cos² θ = 1。
In differentiation, make sure dx/dθ is not zero at the point of interest. If dx/dθ = 0, the tangent is vertical and dy/dx is undefined. State this explicitly rather than trying to divide by zero.
在求导时，确保所关注的点上 dx/dθ 不为零。如果 dx/dθ = 0，则切线为竖直方向，dy/dx 不存在。应明确指出这一点，而不是试图除以零。
Finally, practice linking parametric equations to their geometric interpretations. Recognising the underlying circle, ellipse, or hyperbola from the parametrisation saves time and reduces errors in examinations.
最后，反复练习将参数方程与其几何意义联系起来。从参数化形式中识别出潜在的圆、椭圆或双曲线，不仅节省时间，还能减少考试中的错误。
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