📚 Trigonometry | 三角学
Trigonometry is a fundamental branch of mathematics that explores the relationships between angles and sides of triangles, extending to periodic functions and wave phenomena. It is essential for IB Mathematics, underpinning topics in geometry, calculus, and real-world applications such as physics and engineering.
三角学是数学的一个基础分支，研究三角形中角度与边长之间的关系，并扩展到周期函数和波动现象。它是 IB 数学的核心内容，为几何、微积分以及物理和工程等实际应用奠定基础。
1. Trigonometric Ratios in Right-Angled Triangles | 直角三角形中的三角比
se, tanθ = opposite/adjacent. The mnemonic SOH CAH TOA helps remember these relationships.
在直角三角形中，锐角 θ 的正弦（sin）、余弦（cos）和正切（tan）定义为边长之比：sinθ = 对边/斜边，cosθ = 邻边/斜边，tanθ = 对边/邻边。助记口诀 SOH CAH TOA 有助于记住这些关系。
These ratios allow us to find unknown sides or angles in right triangles, forming the basis for solving many practical problems.
利用这些比值可以求出直角三角形中未知的边或角，为解决许多实际问题奠定了基础。
2. The Unit Circle and Radian Measure | 单位圆与弧度制
The unit circle (radius 1) extends trigonometric definitions to all real numbers. For any angle θ measured from the positive x-axis, the coordinates of the point on the circle are (cosθ, sinθ). This definition makes the functions periodic and valid beyond 0° to 90°.
单位圆（半径为 1）将三角函数的定义扩展到所有实数。对于从 x 轴正方向开始量度的任意角 θ，圆上点的坐标为 (cosθ, sinθ)。这一定义使三角函数具有周期性，且在 0°–90° 之外也有效。
it circle. 2π radians = 360°, and common conversions are: π/6 = 30°, π/4 = 45°, π/3 = 60°.
引入弧度制后，角 θ 等于单位圆上所截弧的长度。2π 弧度 = 360°，常用换算：π/6 = 30°，π/4 = 45°，π/3 = 60°。
3. Exact Values for Special Angles | 特殊角的精确值
0°, 30°, 45°, 60°, 90° (and their radian equivalents). For example, sin30° = 1/2, cos45° = √2/2, tan60° = √3. These values arise from equilateral and isosceles right triangles.
IB 考生需要熟记特殊角（如 0°、30°、45°、60°、90° 及其弧度相等值）的精确三角值。例如，sin30° = 1/2，cos45° = √2/2，tan60° = √3。这些值来源于等边三角形和等腰直角三角形。
A table can be used for quick reference:
可使用表格快速查阅：
| θ (degrees) | θ (radians) | sinθ | cosθ | tanθ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 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 |
| 90° | π/2 | 1 | 0 | undefined |
4. Graphs of Trigonometric Functions | 三角函数图像
s such as y = a sin(bx + c) + d affect amplitude, period, phase shift, and vertical shift.
正弦和余弦函数的图像呈波浪形，周期为 2π，振幅为 1，值域为 [−1, 1]。正弦曲线过原点 (0,0)，而余弦曲线从 (0,1) 开始。变换形式 y = a sin(bx + c) + d 会影响振幅、周期、相移和垂直平移。
l asymptotes at x = π/2 + kπ. Its graph is discontinuous and ranges over all real numbers.
正切函数的周期为 π，并在 x = π/2 + kπ 处存在垂直渐近线。其图像不连续，值域为全体实数。
5. Trigonometric Identities | 三角恒等式
θ = csc²θ. These are derived from the unit circle and allow simplification of expressions.
基本勾股恒等式包括 sin²θ + cos²θ = 1、1 + tan²θ = sec²θ 以及 1 + cot²θ = csc²θ。它们源于单位圆，可用于化简表达式。
sin²θ + cos²θ = 1
Compound angle formulas such as sin(A±B) = sinA cosB ± cosA sinB and cos(A±B) = cosA cosB ∓ d proving identities. Double-angle formulas (e.g., sin2θ = 2 sinθ cosθ) are special cases.
和差角公式例如 sin(A±B) = sinA cosB ± cosA sinB 和 cos(A±B) = cosA cosB ∓ sinA sinB 对于解方程和证明恒等式至关重要。倍角公式（如 sin2θ = 2 sinθ cosθ）是其特例。
6. Solving Trigonometric Equations | 解三角方程
dding multiples of the period. For sinx = k, x = arcsin(k) + 2πn or π − arcsin(k) + 2πn, n ∈ ℤ.
由于周期性，诸如 sinx = 0.5 的三角方程有无穷多个解。通过考虑主值并加上周期的整数倍，可得到通解。对于 sinx = k，解为 x = arcsin(k) + 2πn 或 π − arcsin(k) + 2πn，n ∈ ℤ。
Equations may require use of identities, factoring, or substitution. It is important to check solutions within the specified interval, often [0, 2π).
解方程可能需要利用恒等式、因式分解或代换。重要的一点是在给定区间（通常为 [0, 2π)）内检验解。
7. Sine and Cosine Rules | 正弦定理与余弦定理
/sinC. It is used when given two angles and a side (AAS) or two sides and a non-included angle (SSA), though the ambiguous case must be considered.
对于非直角三角形，正弦定理指出：a/sinA = b/sinB = c/sinC。当已知两角一边（AAS）或两边及一对角（SSA）时使用，但须注意多解情况。
a/sinA = b/sinB = c/sinC
useful for SAS (two sides and included angle) or SSS (three sides) scenarios. It generalises the Pythagorean theorem.
余弦定理：a² = b² + c² − 2bc cosA，适用于 SAS（两边及其夹角）或 SSS（三边）的情况。它是勾股定理的推广。
a² = b² + c² − 2bc cosA
8. Area of a Triangle | 三角形面积
ight. This formula is particularly useful when two sides and the included angle are known.
任意三角形的面积可用 ½ ab sinC 计算，其中 C 为两边 a 和 b 的夹角。对于直角三角形，该公式简化为 ½ × 底 × 高。当已知两边及其夹角时，此公式尤其实用。
Heron' −a)(s−b)(s−c)] with s = (a+b+c)/2, provides an alternative when all three sides are given.
海伦公式，面积 = √[s(s−a)(s−b)(s−c)]，其中 s = (a+b+c)/2，为已知三边时提供了另一种方法。
9. Inverse Trigonometric Functions | 反三角函数
The inverse sine function arcsin or sin⁻¹ has domain [−1,1] and range [−π/2, π/2]; arccos has domain [−1,1] and range [0,π]; arctan has domain ℝ and range (−π/2, π/2). These functions return an angle whose sine, cosine, or tangent is the given value.
反正弦函数 arcsin 或 sin⁻¹ 的定义域为 [−1,1]，值域为 [−π/2, π/2]；反余弦 arccos 定义域为 [−1,1]，值域为 [0,π]；反正切 arctan 定义域为 ℝ，值域为 (−π/2, π/2)。这些函数返回给定三角值所对应的角。
They are used when solving equations like
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