📚 Mixed Exercise 10: Trigonometric Identities & Equations | 综合练习10：三角恒等式与方程
Mixed Exercise 10 from the Pearson Edexcel AS Pure Mathematics textbook is designed to consolidate your understanding of trigonometric identities and the methods for solving trigonometric equations. Mastery of this exercise signals strong readiness for the AS exam questions on this topic, where you must apply both algebraic manipulation and geometric insight.
来自 Pearson Edexcel AS 纯数教材的综合练习10旨在巩固你对三角恒等式以及解三角方程方法的理解。掌握该练习意味着你已充分准备好应对 AS 考试中有关这一主题的问题，这些问题既需要代数操作，也需要几何直观。
1. Understanding the Mixed Exercise | 理解综合练习
Mixed Exercise 10 brings together all question types from Chapter 10: Trigonometric identities and equations. You will encounter proof problems, simplifying expressions, and solving equations over specified intervals. The exercise tests both your algebraic manipulation and your knowledge of the unit circle.
综合练习10汇集了第10章“三角恒等式与方程”中的所有题型。你会遇到证明题、化简表达式以及在指定区间内求解方程的问题。该练习既考查你的代数操作能力，也考查你对单位圆的掌握程度。
The mixture of problem styles means you must be flexible: sometimes you factorise, other times you substitute, and you always need to consider the quadrant of the angle. Use this article to revise each technique systematically.
题型混合意味着你必须灵活：有时需要因式分解，有时需要代换，并且始终要考虑角度所在的象限。利用本文系统复习每一种方法。
2. Fundamental Trigonometric Identities | 基本三角恒等式
At the core of this exercise are two identities that must be committed to memory:
该练习的核心是必须牢记的两个恒等式：
sin²θ + cos²θ ≡ 1
tanθ ≡ sinθ / cosθ
The symbol '≡' means identically equal, so these relationships hold for all values of θ for which the expressions are defined.
符号“≡”表示恒等，因此对于表达式有定义的所有 θ 值，这些关系式都成立。
- sin²θ, which is particularly useful when reducing an expression to a single trigonometric function. The second identity allows you to swap between tanθ and a ratio of sine and cosine.
你可以对第一个恒等式变形，得到 sin²θ = 1 - cos²θ 或 cos²θ = 1 - sin²θ，这在将表达式归结为单一三角函数时特别有用。第二个恒等式则能让你在 tanθ 和正弦余弦之比率间相互转换。
3. Applying sin²θ + cos²θ = 1 | 应用 sin²θ + cos²θ = 1
diately becomes 3( sin²θ + cos²θ ) = 3. The factor 3 is irrelevant to the identity itself.
许多综合练习题要求化简诸如 3 sin²θ + 3 cos²θ 的表达式。利用恒等式，它立即变为 3(sin²θ + cos²θ) = 3。系数 3 与恒等式本身无关。
cific quadrant. For example, if cosθ = -4/5 and θ is in the second quadrant, then sinθ = +√1 - cos²θ = 3/5. The sign is determined by the quadrant: sine is positive in quadrant II.
另一常见任务是已知 cosθ 求 sinθ，或反之，并在特定象限中确定符号。例如，若 cosθ = -4/5 且 θ 在第二象限，则 sinθ = +√1 - cos²θ = 3/5。符号由象限决定：正弦在第二象限为正。
n²θ + cos²θ) = sin²θ. Always look for a common factor that reveals a hidden sin²θ + cos²θ.
当遇到像 sin⁴θ + cos²θ sin²θ 这种较复杂的表达式时，通过因式分解并应用恒等式可以将其巧妙地化简：sin²θ(sin²θ + cos²θ) = sin²θ。始终留意公因子，它可能揭示出隐藏的 sin²θ + cos²θ。
4. Working with the tanθ Identity | 运用 tanθ 恒等式
The identity tanθ ≡ stance, sinθ = tanθ cosθ is a direct consequence and is often used to eliminate fractions.
恒等式 tanθ ≡ sinθ / cosθ 使你能够将含 tanθ 的方程用 sinθ 和 cosθ 改写，反之亦然。例如，sinθ = tanθ cosθ 就是直接的推论，常用于消去分式。
sθ ≠ 0 is important to avoid losing potential solutions where cosθ = 0; here cosθ = 0 would force sinθ = 0, which violates the original equation.
在诸如 sinθ = 2 cosθ 的方程中，两边同除以 cosθ（假设 cosθ ≠ 0）得到 tanθ = 2，接着便可轻松求解。检查 cosθ ≠ 0 这一条件很重要，以免遗漏 cosθ = 0 时可能的解；此处 cosθ = 0 会迫使 sinθ = 0，与原方程矛盾。
Splitting the numerator gives sinθ/cosθ + cosθ/cosθ = tanθ + 1, which is a succinct proof.
该恒等式也有助于证明类似 (sinθ + cosθ) / cosθ = tanθ + 1 的关系式。拆分分子得到 sinθ/cosθ + cosθ/cosθ = tanθ + 1，这便是简洁的证明过程。
5. Solving Basic Trigonometric Equations | 解基本三角方程
Equations such as sinθ = 0.5 or cosθ = -√3/2 appear frequently. To solve, first find the reference angle α that satisfies the positive version of the trigonometric ratio. This α is always an acute angle between 0° and 90°.
诸如 sinθ = 0.5 或 cosθ = -√3/2 的方程频繁出现。求解时，首先找出满足三角比正值的参考角 α，它始终是一个介于 0° 到 90° 之间的锐角。
ive, solutions lie in the first and second quadrants: θ = α, 180° - α, giving 30° and 150°
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