📚 IB & CIE Maths: Trigonometry – Key Concepts & Exam Tips | IB & CIE 数学：三角函数考点精讲
Trigonometry is a cornerstone of both IB and CIE mathematics syllabi. From solving right triangles with basic sine, cosine, and tangent ratios to manipulating complex identities and analysing periodic functions, a firm grasp of trigonometry is vital. This article consolidates key concepts, formulas, and exam-focused strategies to help you excel in your assessments.
三角学是IB和CIE数学教学大纲的基石。从使用基本正弦、余弦和正切比解直角三角形，到操作复杂的恒等式和分析周期函数，牢固掌握三角函数至关重要。本文将整合关键概念、公式和应试策略，助你在考试中脱颖而出。
1. Radian Measure & Angle Conversions | 弧度制与角度转换
an arc equal in length to the radius. Since the full circumference is 2πr, a complete turn is 2π radians. The conversion linking degrees and radians is 180° = π rad. To convert degrees to radians, multiply by π/180. Conversely, multiply radians by 180/π to obtain degrees. Mastering radian mode is essential for calculus applications in trigonometric functions.
弧度制提供了一种基于圆半径来描述角度的自然方式。1弧度定义为弧长等于半径的圆心角。由于整个圆周长为 2πr，因此一整圈为 2π 弧度。度与弧度的换算关系为 180° = π 弧度。将度数转换为弧度，乘以 π/180。反之，将弧度乘以 180/π 可得到度数。掌握弧度模式对于三角函数中的微积分应用至关重要。
5° = π/4, 60° = π/3, and 90° = π/2. Being fluent in these equivalents saves time in exams.
常见角度转换包括 30° = π/6，45° = π/4，60° = π/3，90° = π/2。熟记这些等价关系能在考试中节省时间。
| Degrees | Radians | Degrees | Radians |
|---|---|---|---|
| 0° | 0 | 180° | π |
| 30° | π/6 | 270° | 3π/2 |
| 45° | π/4 | 360° | 2π |
| 60° | π/3 | 720° | 4π |
| 90° | π/2 | 1080° | 6π |
2. Trigonometric Ratios & the Unit Circle | 三角比与单位圆
the coordinates are (cos θ, sin θ). The tangent function is defined as tan θ = sin θ / cos θ, provided cos θ ≠ 0. The reciprocal functions are cosec θ = 1 / sin θ, sec θ = 1 / cos θ, and cot θ = 1 / tan θ. Because the circle has radius 1, the sine and cosine values always lie between -1 and 1, and the signs of the functions depend on the quadrant.
单位圆以 (0,0) 为圆心、半径为 1，为任意角定义三角函数。对于圆上对应角 θ 的点 P，其坐标为 (cos θ, sin θ)。正切函数定义为 tan θ = sin θ / cos θ，前提是 cos θ ≠ 0。倒数函数为 cosec θ = 1 / sin θ，sec θ = 1 / cos θ，cot θ = 1 / tan θ。由于圆的半径为 1，正弦和余弦值始终介于 -1 和 1 之间，函数的符号取决于所在象限。
In the first quadrant, all six functions are positive. In the second quadrant, only sin θ and cosec θ are positive. In the third quadrant, only tan θ and cot θ are positive. In the fourth quadrant, only cos θ and sec θ are positive. The mnemonic "All Students Take Calculus" helps recall this pattern.
在第一象限，六个函数均为正。在第二象限，只有 sin θ 和 cosec θ 为正。在第三象限，只有 tan θ 和 cot θ 为正。在第四象限，只有 cos θ 和 sec θ 为正。口诀 "All Students Take Calculus" 有助于记忆这一规律。
180°n) = tan θ. These periodic properties are fundamental when finding general solutions.
对于任意角 θ，以下关系成立：sin(θ + 360°n) = sin θ，cos(θ + 360°n) = cos θ，tan(θ + 180°n) = tan θ。这些周期性质是求通解的基础。
3. Exact Values & Special Angles | 特殊角的精确值
Exact values for 30°, 45°, and 60° are frequently required in both IB and CIE examinations. These values come from the two special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. Knowing them by heart allows you to avoid calculator errors and solve problems quickly.
30°、45° 和 60° 的精确值在 IB 和 CIE 考试中经常出现。这些值来自两个特殊直角三角形：45-45-90 三角形和 30-60-90 三角形。熟记它们可以避免计算器错误并快速解题。
| Angle θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | style="border:1
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