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。

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。

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。

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°。

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°。

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。

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。

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。

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

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。

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。

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。

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。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

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