📚 Worked Example 4.3.2: Solving a Quadratic Trigonometric Equation | 例4.3.2：解二次型三角方程
onometric on the surface, but its core structure is quadratic in sin x. Mastering this pattern is essential for high marks in AQA Pure Maths, especially in trigonometry and equation-solving questions.
在本 AQA A-Level 纯数学例题中，我们在 0° ≤ x ≤ 360° 范围内解三角方程 2sin²x − sin x − 1 = 0。该方程表面上是三角问题，但其核心结构是关于 sin x 的二次式。掌握这一模式对在 AQA 纯数学中获得高分至关重要，尤其是在三角学和方程求解题中。
We will follow a clear step-by-step method: identify the hidden quadratic, substitute a single variable, factorise, solve the basic trigonometric equations, and then find every solution inside the given interval. This mirrors the method expected in AQA mark schemes.
我们将采用清晰的逐步方法：识别隐藏的二次结构、代换为单个变量、因式分解、解基本三角方程，然后找出给定区间内的所有解。这与 AQA 评分标准中要求的方法一致。
1. The Problem Statement | 题目陈述
the equation 2sin²x − sin x − 1 = 0, giving all values of x in the interval 0° ≤ x ≤ 360°.
解方程 2sin²x − sin x − 1 = 0，并给出区间 0° ≤ x ≤ 360° 内的所有 x 值。
This is a classic AQA Pure Maths question. Although the unknown x is wrapped inside sin x, the expression 2sin²x − sin x − 1 has exactly the same shape as a quadratic expression such as 2u² − u − 1. Recognising this is the key to a quick and accurate solution.
这是一道经典的 AQA 纯数学题。尽管未知数 x 被包在 sin x 中，但表达式 2sin²x − sin x − 1 与 2u² − u − 1 这样的二次表达式具有完全相同的结构。认识到这一点是快速、准确求解的关键。
In an exam, you should write down the interval again next to your working, because many marks are lost when candidates find valid angles but forget to check whether they lie inside the requested range.
在考试中，你应该在解题步骤旁再次写下区间，因为许多考生虽然求出了有效角度，却忘记检查它们是否位于题目要求的范围内，从而丢分。
2. Why This Equation Is Quadratic in Form | 为什么该方程是二次型
The expression 2sin²x − sin x − 1 can be rewritten as 2(sin x)² − (sin x) − 1. If we replace sin x with a temporary variable u, the expression becomes 2u² − u − 1, which is clearly a quadratic. This is called a quadratic trigonometric equation.
表达式 2sin²x − sin x − 1 可以改写为 2(sin x)² − (sin x) − 1。如果我们用临时变量 u 代替 sin x，该表达式就变成 2u² − u − 1，显然是一个二次式。这被称为二次型三角方程。
Many AQA questions deliberately hide a quadratic inside a trigonometric function to test whether you can transfer algebraic skills into a trigonometric context. The substitution method is
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