Using Trigonometric Identities in Parametric Equations | 参数方程中三角恒等式的运用

📚 Using Trigonometric Identities in Parametric Equations | 参数方程中三角恒等式的运用

Parametric equations define x and y separately in terms of a third variable, often t or θ. When that variable is an angle, trigonometric identities become powerful tools for eliminating the parameter, simplifying derivatives, and evaluating integrals.

参数方程将 x 和 y 分别表示为第三个变量（通常是 t 或 θ）的函数。当这个变量为角度时，三角恒等式就成为了消去参数、简化导数和计算积分的强大工具。

1. What Are Parametric Equations? | 什么是参数方程？

A parametric curve is written as x = f(t), y = g(t). Rather than relating x and y directly, both are expressed through a parameter t.

参数曲线通常写作 x = f(t)，y = g(t)。它不是直接建立 x 与 y 的关系，而是通过参数 t 将两者联系起来。

For example, the circle x² + y² = a² can be described as x = a cos θ, y = a sin θ with θ from 0 to 2π.

例如，圆 x² + y² = a² 可以表示为 x = a cos θ，y = a sin θ，其中 θ 在 0 到 2π 范围内变化。

2. Core Trigonometric Identities to Remember | 需要掌握的核心三角恒等式

The following identities are frequently used when working with parametric equations involving angles.

在处理含角参数的参数方程时，下列恒等式使用频率极高。

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Double-angle formulas help simplify expressions that arise after differentiation or integration:

倍角公式在微分或积分后化简表达式时也十分有用：

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ

3. Eliminating the Parameter: Circles | 消去参数：圆

Consider x = a cos θ, y = a sin θ. To eliminate θ, square both equations and use the Pythagorean identity.

考虑 x = a cos θ，y = a sin θ。为了消去 θ，可将两式平方，再利用勾股恒等式。

Because x² = a² cos²θ and y² = a² sin²θ, we have x² + y² = a²(cos²θ + sin²θ) = a².

因为 x² = a² cos²θ，y² = a² sin²θ，所以 x² + y² = a²(cos²θ + sin²θ) = a²。

This proves that the parametric curve is a circle with centre at the origin and radius a.

这证明了

Published by TutorHao | Mathematics Revision Series | aleveler.com

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