📚 Solving Trigonometric Equations and Identities | 解三角方程及恒等式应用
Trigonometric equations and identities are a cornerstone of the Edexcel A-Level Mathematics specification. From simple sine and cosine equations to advanced methods such as the R‑formula, mastering this topic demands a firm grasp of the unit circle, exact values and algebraic manipulation. This article provides a structured revision guide covering key techniques, common pitfalls and practical modelling contexts, designed to help you solve any trigonometric equation with confidence and precision.
三角方程与恒等式是 Edexcel A-Level 数学大纲的核心内容。从简单的正弦、余弦方程到 R 公式等高级方法，掌握这一主题需要透彻理解单位圆、精确值以及代数变形。本文提供一份结构清晰的复习指南，涵盖主要解法、常见陷阱和实践建模背景，旨在帮助你精准、自信地求解各类三角方程。
1. Trigonometric Ratios and Basic Identities | 三角函数比与基本恒等式
The three primary trigonometric ratios — sine, cosine and tangent — can be defined on the unit circle for any real angle θ. For a right‑ oint at angle θ are (cosθ, sinθ), and tanθ = sinθ/cosθ. This geometric view leads directly to the fundamental identity: sin2θ + cos2θ ≡ 1. Dividing this identity by cos2θ yields 1 + tan2θ ≡ sec2θ, while dividing by sin2θ gives 1 + cot2θ ≡ cosec2θ.
三个基本三角比——正弦、余弦和正切——可以利用单位圆对任意实数角 θ 进行定义。在直角三角形中，sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。在单位圆中，角度 θ 对应点的坐标为 (cosθ, sinθ)，且 tanθ = sinθ/cosθ。这一几何观点直接导出基本恒等式：sin2θ + cos2θ ≡ 1。将此恒等式除以 cos2θ 得 1 + tan2θ ≡ sec2θ，除以 sin2θ 得 1 + cot2θ ≡ cosec2θ。
2. Special Angles and Exact Values | 特殊角与精确值
Exact trigonometric values for 30°, 45° and 60° (π/6, π/4 and π/3 radians) must be memorised. They frequently appear in equation solving and in proofs. The table below summarises these values, which can be derived from the equilateral triangle and the isosceles right triangle.
30°、45° 和 60°（π/6、π/4、π/3 弧度）的精确三角值必须熟记。它们经常出现在解方程和证明中。下表总结了这些值，它们可以通过等边三角形和等腰直角三角形推导得出。
| θ (degrees) | θ (radians) | sinθ | cosθ | tanθ |
|---|---|---|---|---|
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
When solving equations such as sinθ = √3/2, knowing that sin60° = √3/2 gives one principal solution immediately. The symmetry of the graph then provides the remaining solutions in the specified interval.
求解 sinθ = √3/2 等方程时，若熟知 sin60° = √3/2，便可立刻得到一个主解，再借助图像的对称性即可定出给定区间内的其他解。
3. Solving Simple Trigonometric Equations | 解简单三角方程
find all solutions in an interval, first determine the principal value using the inverse function (e.g. arcsin(k)) on a calculator, ensuring it lies in the correct quadrant. Because sine is positive in the first and second quadrants, a second solution is 180° − principal value (or π − α). For cosine, use the symmetry 360° − principal value (or 2π − α) for the second quadrant. Tangent has a period of 180° (π), so the general solution is α + 180°n.
形如 sinθ = k、cosθ = k 或 tanθ = k 的方程是基石。要在区间内找出所有解，先利用反三角函数（如 arcsin(k)）计算主值，并确保它落在正确的象限。由于正弦在第一、第二象限为正，另一个解为 180° − 主值（或 π − α）。余弦借助对称性可得 360° − 主值（或 2π − α）作为第二象限的解。正切的周期为 180°（π），因此通解为 α + 180°n。
35.6°. Hence θ ≈ 44.4°, 135.6°. Always check that all answers lie within the given domain.
例如，在 0° ≤ θ ≤ 360° 内解 sinθ = 0.7。主值 ≈ 44.4°。第二个解为 180° − 44.4° = 135.6°。因此 θ ≈ 44.4°, 135.6°。务必检查所有答案都落在指定范围内。
4. Using Trigonometric Identities to Solve Equations | 使用三角恒等式解方程
When an equation contains more than one trigonometric function, identities help rewrite it in terms of a single function. A classic strategy replaces sin2θ with 1 − cos2θ (or vice versa) to turn the equation into a quadratic. For instance, 2 sin2θ − cosθ = 1 becomes 2(1 − cos2θ) − cosθ − 1 = 0, which simplifies to 2 − 2cos2θ − cosθ − 1 = 0, then to 2cos2θ + cosθ − 1 = 0. Setting x = cosθ gives 2x2 )(x + 1) = 0. Thus cosθ = 1/2 or cosθ = −1, leading to θ = 60°, 300° or θ = 180° within 0°–360°.
若方程含有多个三角函数，可利用恒等式将其化为单一种类。经典策略是把 sin2θ 代换为 1 − cos2θ（或反之），将方程变为二次式。例如 2 sin2θ − cosθ = 1 变为 2(1 − cos2θ) − cosθ − 1 = 0，化简得 2cos2θ + cosθ − 1 = 0。令 x = cosθ，得 2x2 + x − 1 = 0，因式分解为 (2x − 1)(x + 1) = 0。于是 cosθ = 1/2 或 cosθ = −1，在 0°–360° 内得 θ = 60°, 300° 或 θ = 180°。
. This gives sinθ = 0 or 1/cosθ = −2, i.e. cosθ = −1/2. Always watch out for values that make a denominator zero.
另一个常用技巧是利用 tanθ = sinθ/cosθ。例如 tanθ + 2 sinθ = 0 可改写为 sinθ/cosθ + 2 sinθ = 0。提取 sinθ 得 sinθ (1/cosθ + 2) = 0，于是 sinθ = 0 或 1/cosθ = −2，即 cosθ = −1/2。注意要排除使分母为零的取值。
5. Quadratic Trigonometric Equations | 二次三角方程
Quadratic trigonometric equations are pervasive in Edexcel exams. They often appear after applying an identity, but they can also be presented directly, such as 3cos2 , factorise or use the quadratic formula, and remember that the variable (e.g. cosθ) must lie between −1 and 1. Any root outside this range must be discarded. After finding permissible values of the trigonometric function, solve the resulting simple equations as described earlier.
二次三角方程在 Edexcel 考试中无处不在。它们经常在应用恒等式后出现，但也可能直接给出，如 3cos2θ − cosθ − 2 = 0。把 cosθ 当作变量，进行因式分解或使用求根公式，并牢记变量（如 cosθ）必须在 −1 至 1 之间，超出该范围的根须舍去。得到合理的三角函数值后，再按前述方法解简单方程即可。
For 3cos2θ − cosθ − 2 = 0, let y = cosθ, so 3y2 0°, 360°; solving cosθ = −2/3 gives two further solutions based on the cosine symmetries.
对于 3cos2θ − cosθ − 2 = 0，令 y = cosθ，得 3y2 − y − 2 = 0 = (3y + 2)(y − 1)。故 cosθ = −2/3 或 cosθ = 1，二者均在 [−1, 1] 内。cosθ = 1 给出 θ = 0°, 360°；cosθ = −2/3 根据余弦对称性给出另外两个解。
6. Equations Involving Multiple Angles | 涉及多倍角的方程
lutions for 2θ in the new interval, then divide by 2. For sin2θ = 0.6, the principal value for 2θ is arcsin(0.6) ≈ 36.9°. The second solution for 2θ in 0°– 03.1°. Hence 2θ ≈ 36.9°, 143.1°, 396.9°, 503.1°. Finally θ ≈ 18.5°, 71.6°, 198.5°, 251.6°.
当角度为 θ 的倍数时，如 sin2θ = 0.6，必须相应地扩展区间。若 0° ≤ θ ≤ 360°，则 0° ≤ 2θ ≤ 720°。首先在新区间内找出 2θ 的全部解，再除以 2。对于 sin2θ = 0.6，2θ 的主值约为 36.9°，在 0°–360° 内的第二个解为 180° − 36.9° = 143.1°。因区间扩展至 720°，各加 360° 得 396.9° 和 503.1°。于是 2θ ≈ 36.9°, 143.1°, 396.9°, 503.1°。最终 θ ≈ 18.5°, 71.6°, 198.5°, 251.6°。
7. The R‑Formula and Its Applications | R 公式及其应用
of the form a sinθ + b cosθ can be combined into a single sine or cosine term using the R‑formula. Typically we write a sinθ + b cosθ = R sin(θ + α), where R = √a2 + b2 choice of form depends on whether you wish to use sine or cosine when solving an equation.
形如 a sinθ + b cosθ 的表达式可通过 R 公式合并为单一正弦或余弦项。通常写作 a sinθ + b cosθ = R sin(θ + α)，其中 R = √a2 + b2，tanα = b/a（α 需取在正确象限）。等价的余弦形式为 a sinθ + b cosθ = R cos
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图