📚 Investigation 1 - The Trigonometric Ratios | 探究1 – 三角比
Trigonometry is a branch of mathematics that explores the relationships between angles and sides of triangles. "Investigation 1" focuses on the foundational trigonometric ratios: sine, cosine, and tangent. By understanding these ratios, you can solve right-angled triangles, analyse periodic functions, and model real-world phenomena such as sound waves and tides. This investigation will guide you through definitions, unit circle extensions, exact values, identities, graphs, and applications, providing a thorough grounding for IB Mathematics.
三角学是数学的一个分支，探索三角形中角度与边之间的关系。“探究1”聚焦于基础的三角比：正弦、余弦和正切。通过理解这些比率，你可以解直角三角形，分析周期函数，并对声波和潮汐等现实世界现象建模。本探究将引导你了解定义、单位圆扩展、精确值、恒等式、图像和应用，为IB数学打下坚实基础。
1. What are Trigonometric Ratios? | 什么是三角比？
The three primary trigonometric ratios relate an acute angle of a right triangle to the ratios of two of its sides. They are named sine (sin), cosine (cos), and tangent (tan). These ratios are universal for a given angle, regardless of the triangle's size, because similar triangles have proportional sides. This property makes trigonometry a powerful tool for indirect measurement.
三个主要三角比将直角三角形的一个锐角与它的两条边的比值联系起来。它们分别称为正弦（sin）、余弦（cos）和正切（tan）。这些比值对于给定的角度是通用的，与三角形的大小无关，因为相似三角形的边是成比例的。这一性质使得三角学成为间接测量的强大工具。
2. Defining Sine, Cosine & Tangent in Right Triangles | 直角三角形中的正弦、余弦和正切定义
In a right triangle with an acute angle θ, the sides are labelled relative to θ: the hypotenuse is the longest side opposite the right angle; the opposite side is the side facing θ; the adjacent side is the other side that forms the angle θ. Then the trigonometric ratios are defined as follows:
在一个含有锐角 θ 的直角三角形中，各边相对于 θ 进行标记：斜边是直角所对的最长边；对边是正对着 θ 的边；邻边是构成角 θ 的另一条边。那么三角比定义如下：
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
3/5, cos θ = 4/5, and tan θ = 3/4. Note that tan θ can also be expressed as sin θ / cos θ.
例如，在一个3-4-5的三角形中，如果角 θ 对着长度为3的边，那么 sin θ = 3/5，cos θ = 4/5，tan θ = 3/4。注意 tan θ 也可以表示为 sin θ / cos θ。
3. The Unit Circle & Extension to Any Angle | 单位圆与任意角的扩展
While right-triangle definitions work only for acute angles (0° < θ < 90°), the unit circle extends trigonometric ratios to any real angle. The unit circle is a circle of radius 1 centered at the origin. For an angle θ measured from the positive x-axis, the coordinates of the point where the terminal side intersects the circle are (cos θ, sin θ). This holds for all angles, including those greater than 90° or negative angles.
虽然直角三角形的定义只适用于锐角（0° < θ < 90°），但单位圆将三角比扩展到任意实数角。单位圆是以原点为中心、半径为1的圆。对于一个从正x轴开始测量的角 θ，其终边与单位圆交点的坐标为 (cos θ, sin θ)。这对于所有角都成立，包括大于90°的角或负角。
The unit circle also explains the periodic nature of sine and cosine, with a period of 360° or 2π radians. It also clarifies why sine is positive in the first and second quadrants, cosine positive in the first and fourth, and tangent positive in the first and third.
单位圆也解释了正弦和余弦的周期性，周期为360°或2π弧度。它还说明了为什么正弦在第一和第二象限为正，余弦在第一和第四象限为正，正切在第一和第三象限为正。
4. Special Angles & Exact Values | 特殊角及其精确值
Certain angles have exact trigonometric values that you should memorise for IB Mathematics. These angles are usually 0°, 30°, 45°, 60°, and 90° (0, π/6, π/4, π/3, π/2 radians). The exact values are derived from special right triangles: the 45°-45°-90° triangle (isosceles right triangle) and the 30°-60°-90° triangle. Knowing these values allows you to solve problems without a calculator.
一些特殊角有精确的三角值，你应该为IB数学记住它们。这些角通常是0°、30°、45°、60°和90°（0、π/6、π/4、π/3、π/2弧度）。精确值源自特殊的直角三角形：45°-45°-90°三角形（等腰直角三角形）和30°-60°-90°三角形。了解这些值可以使你在不借助计算器的情况下解决问题。
| θ (degrees) | θ (radians) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | ½ | √3/2 | 1/√3 |
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