Trigonometry: Common Exam Points and Solving Methods | 三角函数常见考察点与解题方法

📚 Trigonometry: Common Exam Points and Solving Methods | 三角函数常见考察点与解题方法

Trigonometry is a core topic in A-level Mathematics, appearing in pure mathematics, coordinate geometry, and even mechanics. Understanding the main exam patterns and mastering a structured solving approach is essential for high marks.

三角函数是A-level数学的核心内容，既出现在纯数学中，也出现在坐标几何甚至力学中。了解常见考察方式并掌握结构化的解题方法，是取得高分的关键。

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1. Basic Trigonometric Ratios and Exact Values | 基本三角比值与精确值

The definitions of sine, cosine, and tangent in a right-angled triangle form the foundation of all trigonometry. You must also recall the exact values for special angles: 0°, 30°, 45°, 60° and 90°.

直角三角形中正弦、余弦和正切的定义是整个三角函数的基础。你还需要记住特殊角（0°、30°、45°、60°、90°）的精确值。

• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3

  这些值经常在解三角形和求极限时直接使用。

• sin 45° = cos 45° = √2/2, tan 45° = 1

  45° 的正弦与余弦相等，常出现在对称图形中。

• sin 60° = √3/2, cos 60° = ½, tan 60° = √3

  注意 30° 与 60° 的正余弦值互换。

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2. Core Identities and Their Application | 核心恒等式及其应用

The two fundamental identities are used in nearly every trigonometric proof or simplification.

两个基本恒等式几乎出现在每一道三角证明或化简题中。

sin²θ + cos²θ = 1

The identity above connects sine and cosine. Its most common use is replacing 1, or converting between sin² and cos².

上述恒等式联系了正弦和余弦。最常见的用法是用它替换 1，或在 sin² 与 cos² 之间转换。

tan θ = sin θ / cos θ

This definition links all three basic ratios, and is essential when solving equations involving tan.

这个定义将三种基本比值联系起来，也是解含正切方程的重要工具。

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3. Solving Simple Trigonometric Equations | 求解简单三角方程

To solve an equation such as sin θ = 0.5 on a given interval, you need to find all solutions, not just the principal value.

求解如 sin θ = 0.5 在给定区间内的方程时，需要找出所有解，而不仅仅是主值。

• First find the acute reference angle using inverse sine: θ = sin⁻¹(0.5) = 30°.

  先用反正弦求出锐角参考角：θ = sin⁻¹(0.5) = 30°。

• Then use the ASTC / CAST rule to identify the quadrants where sine is positive: Quadrants I and II.

  然后用 ASTC / CAST 规则判断正弦为正的象限：第一和第二象限。

• Solutions in [0°, 360°): θ = 30° and θ = 180° − 30° = 150°.

  在 [0°, 360°) 内的解为 θ = 30° 和 θ = 180° − 30° = 150°。

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4. Equations of the Form sin(ax + b) = c | 形如 sin(ax + b) = c 的方程

When an angle is transformed, you must adjust the interval before solving.

当角度经过变换时，必须先在调整后的区间内求解。

For example, solve sin(2x) = 0.5 for 0° ≤ x ≤ 180°.

例如，在 0° ≤ x ≤ 180° 内求解 sin(2x) = 0.5。

• The transformed interval is 0° ≤ 2x ≤ 360°.

  变换后的区间为 0° ≤ 2x ≤ 360°。

• The solutions for 2x are 30°, 150°, 390°, 510°.

  2x 的解为 30°、150°、390°、510°。

• Thus x = 15°, 75°, 195°, 255°.

  因此 x = 15°、75°、195°、255°。

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5. Quadratic Trigonometric Equations | 二次三角方程

Equations involving sin²θ, cos²θ, or combinations such as 2cos²θ − cosθ − 1 = 0 are solved by factorising.

含 sin²θ、cos²θ 或如 2cos²θ − cosθ − 1 = 0 的组合方程，通常通过因式分解求解。

Treat the trigonometric function as the unknown variable.

把三角函数当成一个未知数来处理。

2cos²θ − cosθ − 1 = 0 ⇒ (2cosθ + 1)(cosθ − 1) = 0

Thus cosθ = −½ or cosθ = 1. Then solve each separately.

因此 cosθ = −½ 或 cosθ = 1。然后分别求解。

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6. Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切的图像

You must know the shape, key values, and period of each basic graph.

你需要掌握每个基本图像的形状、关键值和周期。

y = sin x: period 360°, range −1 ≤ y ≤ 1

y = cos x: period 360°, range −1 ≤ y ≤ 1

y = tan x: period 180°, range all real numbers

The graph of tan has vertical asymptotes at x = 90°, 270°, … and its range is all real numbers.

正切图像在 x = 90°、270°……处有垂直渐近线，值域为全体实数。

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7. Transformations of Trigonometric Graphs | 三角函数的图像变换

The transformations of y = sin x to y = a sin(bx + c) + d follow the standard rules.

从 y = sin x 到 y = a sin(bx + c) + d 的变换遵循标准规则。

• a affects amplitude: vertical stretch.

  a 影响振幅：纵向拉伸。

• b affects period: new period = 360°/b for sine and cosine.

  b 影响周期：正弦和余弦的新周期 = 360°/b。

• c causes a horizontal shift: careful with the sign inside the bracket.

  c 导致水平位移：注意括号内的符号。

• d translates the graph vertically.

  d 使图像垂直平移。

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8. Compound Angle and Double Angle Formulae | 和角与倍角公式

These formulae are essential for proving identities and solving advanced equations.

这些公式对于证明恒等式和求解进阶方程至关重要。

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Setting A = B gives the double angle formulae:

令 A = B 可以得到倍角公式：

sin 2θ = 2 sin θ cos θ

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

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9. The Form R sin(x ± α) | R sin(x ± α) 形式

Expressions like a sin x + b cos x can be written as a single sine or cosine function.

形如 a sin x + b cos x 的表达式可以写成单一正弦或余弦函数。

a sin x + b cos x ≡ R sin(x + α)

Where R = √(a² + b²) and tan α = b/a, with α chosen according to the quadrant.

其中 R = √(a² + b²)，tan α = b/a，α 根据象限确定。

This form simplifies solving equations and finding maximum/minimum values.

这种形式使方程求解和寻找最大/最小值变得简单。

For example, the maximum of y = 3 sin x + 4 cos x is 5, because R = √(3² + 4²) = 5.

例如，y = 3 sin x + 4 cos x 的最大值是 5，因为 R = √(3² + 4²) = 5。

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10. Proving Trigonometric Identities | 三角恒等式的证明

Proof questions require a logical sequence showing that the left-hand side equals the right-hand side.

证明题需要一系列逻辑步骤，说明左边等于右边。

• Start with the more complicated side.

  从较复杂的一边开始。

• Rewrite tan as sin / cos or use the Pythagorean identity to change squares.

  把 tan 写成 sin / cos，或利用勾股恒等式改变平方形式。

• Factorise or simplify using algebraic techniques.

  用代数技巧因式分解或化简。

Always state the key identity used at each step.

每一步都要说明使用的关键恒等式。

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11. Sine Rule and Cosine Rule | 正弦定理与余弦定理

For any triangle with sides a, b, c opposite angles A, B, C:

对于任意三角形，边 a、b、c 分别对应角 A、B、C：

a / sin A = b / sin B = c / sin C

c² = a² + b² − 2ab cos C

Use the sine rule when you know a side and its opposite angle. Use the cosine rule when you know two sides and the included angle, or three sides.

当知道一边及其对角时用正弦定理；当知道两边及其夹角，或三边时用余弦定理。

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12. Area of a Triangle and Common Pitfalls | 三角形面积与常见陷阱

The area of a triangle is given by ½ab sin C when two sides and the included angle are known.

当已知两边及其夹角时，三角形面积公式为 ½ab sin C。

Area = ½ ab sin C

Watch out for the ambiguous case of the sine rule: when using sin A = opposite/hypotenuse, two angles may satisfy the equation (acute and obtuse).

注意正弦定理的模糊情况：当用正弦关系求角时，可能有两个角（锐角和钝角）都满足方程。

• Always check if the angle is acute or obtuse based on the context.

  根据题目情境判断角是锐角还是钝角。

• When using inverse trig, the calculator gives only the principal value; you must add the period or use symmetry to find all solutions.

  使用反三角函数时，计算器只给出主值；你必须加上周期或利用对称性找到所有解。

• Do not mix degrees and radians without converting clearly.

  不要混淆角度制和弧度制，转换时要清楚标注。

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