📚 Trigonometry Essentials for IGCSE Mathematics | IGCSE数学核心三角函数攻略
Trigonometry is a fundamental part of the IGCSE Mathematics Extended syllabus. It connects geometric intuition with algebraic precision and appears in exam questions every year. This article covers all the essential rules, formulas and problem-solving techniques you need to master.
三角函数是IGCSE数学（扩展部分）的基础内容。它将几何直观与代数精确性结合起来，每年考试都会出现。这篇文章将涵盖你所需掌握的所有重要规则、公式和解题技巧。
1. The Three Basic Trigonometric Ratios | 三个基本三角函数
For a right-angled triangle, the three ratios relate an angle to the lengths of two sides. The labels are Opposite (O), Adjacent (A) and Hypotenuse (H), where H is the side opposite the right angle.
在直角三角形中，三个三角比将一个角与两条边的长度联系起来。它们分别定义为对边（O）、邻边（A）和斜边（H），其中斜边是直角所对的边。
sin θ = O/H, cos θ = A/H, tan θ = O/A
A common way to remember these is “SOH CAH TOA”: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent.
一个常用的记忆法是“SOH CAH TOA”：正弦是对边比斜边，余弦是邻边比斜边，正切是对边比邻边。
2. Exact Values for Special Angles | 特殊角的精确值
The following table gives the exact values for the angles you must know for IGCSE.
下表列出了IGCSE考试中必须掌握的特殊角的精确值。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 or √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Not defined / 未定义 |
3. Finding Missing Sides and Angles | 求未知边和角
To find a missing side, choose the ratio that contains the given angle, the given side and the unknown side, then rearrange.
要求未知边，先选择包含已知角、已知边和未知边的三角比，然后进行变形。
Example: In a right triangle, angle θ is 40°, hypotenuse is 10 cm. Find the opposite side. We use sin θ = Opposite/Hypotenuse, so Opposite = 10 × sin 40° ≈ 6.43 cm.
例如：在直角三角形中，角θ为40°，斜边为10 cm。求对边。使用 sin θ = 对边/斜边，因此对边 = 10 × sin 40° ≈ 6.43 cm。
To find a missing angle, apply the inverse function, for example θ = tan⁻¹(Opposite/Adjacent).
求未知角时，需要运用反三角函数，例如 θ = tan⁻¹(对边/邻边)。
4. The Sine Rule | 正弦定理
For any triangle (not just right-angled) with angles A, B, C and opposite sides a, b, c, the sine rule states:
对于任意三角形（不仅是直角三角形）来说，若三个角分别为A、B、C，其对边分别为a、b、c，则正弦定理为：
a / sin A = b / sin B = c / sin C
Use the sine rule when you know two angles and one side (AAS) or two sides and one non-included angle (SSA).
当已知两个角和一个边（AAS），或已知两个边和一个非夹角（SSA）时，使用正弦定理。
Example: In triangle ABC, A=35°, B=60°, a=8 cm. Find b. Since b/sin B = a/sin A, b = 8 sin 60° / sin 35° ≈ 12.1 cm.
例如：在三角形ABC中，A=35°，B=60°，a=8 cm。求b。因为 b/sin B = a/sin A，所以 b = 8 × sin 60° / sin 35° ≈ 12.1 cm。
5. The Cosine Rule | 余弦定理
The cosine rule is used to find a side when you know two sides and the included angle (SAS), or to find an angle when you know all three sides (SSS).
余弦定理用于已知两边及其夹角（SAS）时求第三边，或已知三边（SSS）时求角。
a² = b² + c² – 2bc cos A
For an angle, rearrange to: cos A = (b² + c² – a²) / (2bc).
求角时，可将公式变为：cos A = (b² + c² – a²) / (2bc)。
Example: If b = 5 cm, c = 7 cm and A = 40°, then a² = 5² + 7² – 2 × 5 × 7 × cos 40° ≈ 20.4, so a ≈ 4.51 cm.
例如：若b = 5 cm，c = 7 cm，A = 40°，则 a² = 5² + 7² – 2 × 5 × 7 × cos 40° ≈ 20.4，因此 a ≈ 4.51 cm。
6. Area of a Triangle Using Trigonometry | 用三角函数求三角形面积
The area of a triangle can be found using the formula Area = ½ ab sin C, where a and b are two sides and C is the included angle between them.
三角形的面积可以使用公式 面积 = ½ ab sin C 来求，其中a和b是两条边，C是它们之间的夹角。
Area = ½ ab sin C
Example: If a = 8 cm, b = 11 cm and C = 30°, then Area = ½ × 8 × 11 × sin 30° = 22 cm².
例如：若a = 8 cm，b = 11 cm，C = 30°，则面积 = ½ × 8 × 11 × sin 30° = 22 cm²。
7. Graphs of Trigonometric Functions | 三角函数的图像
You should know the shape and key features of y = sin θ, y = cos θ and y = tan θ for 0° ≤ θ ≤ 360°.
你需要了解 y = sin θ、y = cos θ 和 y = tan θ 在 0° ≤ θ ≤ 360° 范围内的图像形状和关键特征。
-
The sine graph starts at 0, rises to 1 at 90°, returns to 0
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply