IB Math: Multiples of π/6 and π/4 | IB数学：π/6与π/4的倍数角

📚 IB Math: Multiples of π/6 and π/4 | IB数学：π/6与π/4的倍数角

In IB Mathematics, the exact values of trigonometric functions at multiples of π/6 and π/4 are essential for solving problems in algebra, geometry, calculus, and vectors. These angles appear in the unit circle, in solving equations, in differentiation and integration, and in real-world modelling. Mastering them means you can work quickly and accurately without a calculator, and you will understand the underlying symmetry of periodic functions.

在IB数学中，三角函数在π/6与π/4的倍数角上的精确值，是代数、几何、微积分和向量中解题的基石。这些角出现在单位圆、方程求解、微分积分以及实际建模问题中。掌握它们，意味着你无需计算器也能快速准确作答，并且能更深入地理解周期函数的对称性。

1. Radians and the Unit Circle | 弧度与单位圆

A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full revolution is 2π radians, so π/6 radians equals 30°, π/4 radians equals 45°, and π/2 radians equals 90°. On the unit circle, the coordinates of a point at angle θ are (cos θ, sin θ). For multiples of π/6 and π/4, these coordinates take simple exact forms such as (√3/2, 1/2) and (√2/2, √2/2).

弧度是长度等于半径的弧所对应的圆心角。整圈为2π弧度，所以π/6弧度等于30°，π/4弧度等于45°，π/2弧度等于90°。在单位圆上，角度θ对应的点坐标为(cos θ, sin θ)。对π/6和π/4的倍数角，这些坐标具有简单精确的形式，如(√3/2, 1/2)和(√2/2, √2/2)。

Visualising these angles on the unit circle helps you understand why sin and cos values change sign in different quadrants. The radius of the unit circle is 1, so the hypotenuse of any right triangle formed with the x-axis is always 1. This is why the exact values are just the ratios of the side lengths of the associated right triangles.

在单位圆上观察这些角，有助于理解为什么sin和cos在不同象限会变号。单位圆的半径为1，因此与x轴构成的任何直角三角形斜边都恒为1。这正是精确值等于相应直角三角形各边比例的原因。

2. Exact Values for Key Angles | 关键角的精确值

The table below gives the exact values of sine, cosine, and tangent for multiples of π/6 and π/4 from 0 to 2π radians. You should be able to reproduce this table from memory by using symmetry and the special triangles.

下表给出了0到2π弧度之间所有π/6和π/4倍数角的sin、cos和tan精确值。你应该能够利用对称性和特殊三角形，在脑海中再现这张表。