📚 IB Mathematics Core: Geometry and Trigonometry | IB数学核心：几何与三角学
Geometry and trigonometry form a fundamental part of the IB Mathematics curriculum, equipping students with tools to analyze shapes, angles, and spatial relationships. This core topic covers right-angled triangle trigonometry, sine and cosine rules, radian measure, the unit circle, trigonometric identities, coordinate geometry, and three-dimensional applications. Mastering these concepts is essential for success in both Standard Level and Higher Level examinations.
几何与三角学是IB数学课程的基础组成部分，为学生提供分析形状、角度和空间关系的工具。该核心主题涵盖直角三角形三角学、正弦与余弦定理、弧度制、单位圆、三角恒等式、坐标几何以及立体几何应用。掌握这些概念对于在标准水平和高级水平考试中取得成功至关重要。
1. Right-Angled Triangle Trigonometry | 直角三角形三角学
e, tan θ = opposite / adjacent. The mnemonic SOH CAH TOA helps memorise these definitions.
在直角三角形中，锐角 θ 的正弦、余弦和正切定义为边长之比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记忆口诀 SOH CAH TOA 有助于记住这些定义。
When solving for an unknown side or angle, you can rearrange these ratios and use the inverse trigonometric functions, sin⁻¹, cos⁻¹, tan⁻¹. Always ensure your calculator is in the correct degree or radian mode.
在求解未知边或角时，可以重新排列这些比，并使用反三角函数 sin⁻¹、cos⁻¹、tan⁻¹。务必确保计算器处于正确的度数或弧度模式。
Exact values for selected special angles must be learned. They often appear in non-calculator papers.
必须记住特殊角的精确值，它们经常出现在非计算器卷中。
| θ (degrees) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3 / 2 | 1 / √3 |
| 45° | √2 / 2 | √2 / 2 | 1 |
| 60° | √3 / 2 | 1/2 | √3 |
2. Sine Rule and Cosine Rule | 正弦定理与余弦定理
useful when given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
对于任意非直角三角形 ABC，边 a 对应角 A，b 对应角 B，c 对应角 C，正弦定理为：a / sin A = b / sin B = c / sin C。当已知两角一边（AAS 或 ASA）或两边及一个非夹角（SSA）时使用。
es cos A = (b² + c² - a²) / (2bc), which allows finding an angle when all sides are known.
余弦定理将三边与一个角联系起来：a² = b² + c² - 2bc cos A。当已知三边（SSS）或两边及其夹角（SAS）时使用。变形后得 cos A = (b² + c² - a²) / (2bc)，可在已知三边时求角。
ound using Area = (1/2) ab sin C, where a and b are two sides and C is the included angle.
任何三角形的面积可用 Area = (1/2) ab sin C 计算，其中 a 和 b 是两边，C 是它们的夹角。
3. Radian Measure, Arc Length, and Sector Area | 弧度、弧长与扇形面积
t degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π.
角度可以用弧度度量。一弧度是弧长等于半径时所对应的圆心角。转换关系为 π rad = 180°。度转弧度乘以 π/180；弧度转度乘以 180/π。
of the sector is A = (1/2) r²θ. The segment area is (1/2) r²(θ - sin θ) for θ in radians.
在半径为 r 的圆中，对于圆心角 θ（弧度），弧长 s = rθ，扇形面积 A = (1/2) r²θ。弓形面积为 (1/2) r²(θ - sin θ)，其中 θ 为弧度。
4. The Unit Circle and Trigonometric Ratios | 单位圆与三角比
The unit circle has radius 1 centred at the origin. For any angle θ measured from the positive x-axis, the coordinates of the point on the circle are (cos θ, sin θ). This definition extends trigonometric ratios to all real angles, including those greater than 90° and negative angles.
单位圆是半径为 1、中心在原点的圆。对于从正 x 轴测量的任意角 θ，圆上点的坐标为 (cos θ, sin θ)。这一定义将三角比扩展到所有实数角，包括大于 90° 的角和负角。
Signs of sin, cos, and tan in the four quadrants are remembered by 'All Students Take Calculus' (ASTC): All positive in Quadrant I, Sine positive in II, Tangent positive in III, Cosine positive in IV.
各象限中 sin、cos、tan 的符号可通过 'All Students Take Calculus' (ASTC) 记忆：第一象限全正，第二象限正弦正，第三象限正切正，第四象限余弦正。
trig functions become clear. sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ, tan(θ + π) = tan θ.
根据单位圆，基本恒等式如 sin²θ + cos²θ = 1 以及三角函数的周期性质显而易见：sin(θ + 2π) = sin θ，cos(θ + 2π) = cos θ，tan(θ + π) = tan θ。
5. Graphs of Trigonometric Functions | 三角函数的图像
arting at the origin, with maximum 1, minimum -1, and period 2π. It is an odd function, symmetric about the origin.
y = sin x 的图像是一条光滑的波浪线，从原点开始，最大值为 1，最小值为 -1，周期为 2π。它是奇函数，关于原点对称。
at its maximum (0,1), also period 2π, and is an even function, symmetric about the y-axis.
y = cos x 的图像类似，但从最大值 (0,1) 开始，周期也为 2π，是偶函数，关于 y 轴对称。
symptotes at x = π/2 + kπ, period π, and passes through the origin. It is an odd function.
y = tan x 在 x = π/2 + kπ 处有渐近线，周期为 π，并经过原点。它是奇函数。
Transformations of the form y = a sin(bx + c) + d introduce amplitude |a|, period 2π/|b|, phase shift -c/b, and vertical shift d. Similar rules apply for cosine and tangent.
形如 y = a sin(bx + c) + d 的变换引入了振幅 |a|、周期 2π/|b|、相移 -c/b 及垂直位移 d。对余弦和正切函数也有类似的规则。
6. Trigonometric Identities and Equations | 三角恒等式与解三角方程
ngle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ.
重要的恒等式包括：sin²θ + cos²θ = 1，tan θ = sin θ / cos θ。双角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ - sin²θ = 2 cos²θ - 1 = 1 - 2 sin²θ。
When solving trigonometric equations, first simplify using identities, then find the principal solution and add the appropriate period to obtain all solutions within the given interval.
在解三角方程时，先用恒等式化简，然后求出主解并加上适当的周期，得到给定区间内的所有解。
, so x = π/6 or 5π/6. Always consider the unit circle or graph to avoid missing solutions.
例如：求解 2 sin x - 1 = 0，0 ≤ x ≤ 2π。得 sin x = 1/2，因此 x = π/6 或 5π/6。始终结合单位圆或图像以防漏解。
7. Coordinate Geometry: Lines and Distances | 坐标几何：直线与距离
etween them is d = √[(x₂ - x₁)² + (y₂ - y₁)²]. The midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
对于两点 A(x₁, y₁) 和 B(x₂, y₂)，距离为 d = √[(x₂ - x₁)² + (y₂ - y₁)²]。中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。
rough point (x₁, y₁) has equation y - y₁ = m(x - x₁). The general form is ax + by + c = 0.
直线 AB 的梯度（斜率）为 m = (y₂ - y₁) / (x₂ - x₁)。过点 (x₁, y₁) 且斜率为 m 的直线方程为 y - y₁ = m(x - x₁)。一般形式为 ax + by + c = 0。
The perpendicular bisector of a segment combines both midpoint and perpendicular gradient.
平行线斜率相等：m₁ = m₂。垂直线的斜率满足 m₁ m₂ = -1。线段的垂直平分线结合了中点与垂直斜率。
8. Three-Dimensional Geometry Problems | 立体几何问题
In three dimensions, distances and angles often require right-angled triangles extracted from the solid. The diagonal of a rectangular prism with edges l, w, h is √l² + w² + h².
在三维空间中，距离和角度的计算通常需要从立体中提取直角三角形。棱长分别为 l、w、h 的长方体的对角线长为 √l² + w² + h²。
Pyramid problems involve slant height, vertical height, and base dimensions. Use Pythagoras' theorem and basic trigonometry to find face angles, edge lengths, and the angle between a face and the base.
棱锥问题涉及斜高、垂直高度和底面尺寸。利用勾股定理和基础三角比可求出面角、棱长以及面与底面的夹角。
When dealing with spheres, cones, and cylinders, the surface area and volume formulas are given in the IB formula booklet. Applying trigonometry helps find unknown lengths from central angles or cross-sections.
在处理球体、圆锥和圆柱时，表面积和体积公式可在 IB 公式手册中找到。应用三角学可从圆心角或截面中求出未知长度。
9. Applications of Trigonometry | 三角学的应用
IB problems often present real-world contexts: heights of buildings, angles of elevation and depression, bearings in navigation, and inclined planes. Draw a clear diagram and label all given information.
IB 题目常给出实际情境：建筑物高度、仰角和俯角、航海中的方位角以及斜面。画一张清晰的图并标注所有已知信息。
For bearings, measure clockwise from north. When using the sine and cosine rules, check whether you have the ambiguous case (SSA), which may yield two possible triangles.
对于方位角，从正北顺时针测量。使用正弦和余弦定理时，需检查是否存在多义情况（SSA），该情况可能产生两个可能的三角形。
Periodic phenomena like tides, temperature, and sound waves can be modelled using sine and cosine functions, with parameters representing amplitude, period, and phase shift.
周期现象如潮汐、温度和声波，可用正弦和余弦函数建模，其参数分别表示振幅、周期和相移。
10. Summary of Key Formulas | 关键公式一览
The table below collects the most important formulas from this core topic:
下表汇总了该核心主题中最重要的公式：
| Formula | 中文公式 | Use |
|---|---|---|
| a / sin A = b / sin B = c / sin C | 正弦定理 | Non-right triangles, AAS, ASA, SSA |
| a² = b² + c² - 2bc cos A | 余弦定理 | SSS, SAS |
| Area = (1/2) ab sin C | 三角形面积 | Two sides and included angle |
| s = rθ, A = (1/2) r²θ | 弧长与扇形面积 | θ in radians |
| sin²θ + cos²θ = 1 | 三角恒等式 | Proofs, solving equations |
| sin 2θ = 2 sin θ cos θ | 正弦双角公式 | Integrals, equation solving |
| cos 2θ = cos²θ - sin²θ | 余弦双角公式 | Multiple angle problems |
| d = √[(x₂ - x₁)² + (y₂ - y₁)²] | 两点距离 | Coordinate geometry |
| m = (y₂ - y₁) / (x₂ - x₁) | 斜率公式 | Gradient, parallel/perpendicular |
Regular practice with these concepts and formulas, combined with past paper problems, will build confidence for the IB Mathematics examination.
定期练习这些概念和公式，并结合历年真题进行训练，将为 IB 数学考试建立信心。
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