📚 Coordinates of Multiples of π/6 and π/4 on the Unit Circle | 单位圆上π/6与π/4倍数角的坐标
The unit circle is one of the most important visual tools in IB Mathematics. By drawing an angle θ in standard position and marking the point where its terminal side meets the circle, we obtain the coordinates (cos θ, sin θ).
单位圆是IB数学中最重要的可视化工具之一。在标准位置画出角θ，并标出其终边与圆的交点，就能得到坐标 (cos θ, sin θ)。
1. The Unit Circle and Radian Measure | 单位圆与弧度制
circle is the circle with centre at the origin and radius 1. Its equation is x² + y² = 1.
单位圆是圆心在原点、半径为1的圆，其方程为 x² + y² = 1。
°. A full revolution around the circle is 2π radians, and the circumference is 2π(1) = 2π.
在弧度制中，π 弧度等于 180°。绕圆一整圈为 2π 弧度，圆周长为 2π(1) = 2π。
s, then P = (cos θ, sin θ). This definition is the foundation of all trigonometric values.
对于单位圆上的任意点 P，若半径 OP 与 x 轴正方向成角 θ，则 P = (cos θ, sin θ)。这个定义是所有三角函数值的基础。
2. Why Multiples of π/6 and π/4 Matter | 为什么 π/6 与 π/4 的倍数角如此重要
Angles that are multiples of π/6 and π/4 correspond to the familiar 30°-60°-90° and 45°-45°-90° triangles.
π/6 与 π/4 的倍数角对应我们熟悉的 30°-60°-90° 和 45°-45°-90° 三角形。
Multiples of π/6 include 0, π/6, π/3, π/2, 2π/3, 5π/6, π, and their reflections in all quadrants.
π/6 的倍数角包括 0, π/6, π/3, π/2, 2π/3, 5π/6, π 以及它们在各个象限中的反射角。
Multiples of π/4 include 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, and 7π/4.
π/4 的倍数角包括 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2 和 7π/4。
Because these angles appear constantly in IB exam questions, knowing their exact coordinates saves time and prevents errors.
因为这些角度在IB考试中反复出现，熟记它们对应的精确坐标可以节省时间并避免错误。
3. Deriving the π/4 Family Using the 45°-45°-90° Triangle | 用 45°-45°-90° 三角形推导 π/4 族坐标
ay between the positive x-axis and the positive y-axis. The right triangle formed has two equal legs.
当 θ = π/4 时，终边位于 x 轴正方向与 y 轴正方向的中间，所形成的直角三角形两条直角边相等。
If the hypotenuse is 1, then by the Pythagorean theorem:
x² + x² = 1 → 2x² = 1 → x = √2/2
因此 cos(π/4) = √2/2，sin(π/4) = √2/2，坐标为 (√2/2, √2/2)。
This same triangle gives all π/4 family coordinates when reflected across the axes.
当这个三角形关于坐标轴反射时，就能得到所有 π/4 族的坐标。
4. Deriving the π/6 and π/3 Coordinates Using the 30°-60°-90° Triangle | 用 30°-60°-90° 三角形推导 π/6 与 π/3 坐标
For θ = π/6, the triangle has a shortest side of length 1/2 and a longer leg of length √3/2 when the hypotenuse is 1.
当 θ = π/6 时，斜边为1的三角形中，最短边长度为 1/2，较长直角边长度为 √3/2。
Therefore:
cos(π/6) = √3/2, sin(π/6) = 1/2
因此 cos(π/6) = √3/2，sin(π/6) = 1/2，坐标为 (√3/2, 1/2)。
For θ = π/3, the roles of the legs are swapped:
当 θ = π/3 时，两条直角边的作用互换：
cos(π/3) = 1/2, sin(π/3) = √3/2
所以 cos(π/3) = 1/2，sin(π/3) = √3/2，坐标为 (1/2, √3/2)。
5. Complete Table for Multiples of π/6 | π/6 倍数角完整坐标表
The table below lists the coordinates of all multiples of π/6 from 0 to 2π.
下表列出从 0 到 2π 之间所有 π/6 倍数角的坐标。
| θ | cos θ | sin θ | Point (x, y) |
|---|---|---|---|
| 0 | 1 | 0 | (1, 0) |
| π/6 | √3/2 | 1/2 | (√3/2, 1/2) |
| π/3 | 1/2 | √3/2 | (1/2, √3/2) |
| π/2 | 0 | 1 | (0, 1) |
| 2π/3 | -1/2 | √3/2 | (-1/2, √3/2) |
| 5π/6 | -√3/2 | 1/2 | (-√3/2, 1/2) |
| π | -1 | 0 | (-1, 0) |
| 7π/6 | -√3/2 | -1/2 | (-√3/2, -1/2) |
| 4π/3 | -1/2 | -√3/2 | (-1/2, -√3/2) |
| 3π/2 | 0 | -1 | (0, -1) |
| 5π/3 | 1/2 | -√3/2 | (1/2, -√3/2) |
| 11π/6 | √3/2 | -1/2 | (√3/2, -1/2) |
| 2π | 1 | 0 | (1, 0) |
6. Complete Table for Multiples of π/4 | π/4 倍数角完整坐标表
The table below lists the coordinates of all multiples of π/4 from 0 to 2π.
下表列出从 0 到 2π 之间所有 π/4 倍数角的坐标。
| θ | cos θ | sin θ | Point (x, y) |
|---|---|---|---|
| 0 | 1 | 0 | (1, 0) |
| π/4 | √2/2 | √2/2 | (√2/2, √2/2) |
| π/2 | 0 | 1 | (0, 1) |
| 3π/4 | -√2/2 | √2/2 | (-√2/2, √2/2) |
| π | -1 | 0 | (-1, 0) |
| 5π/4 | -√2/2 | -√2/2 | (-√2/2, -√2/2) |
| 3π/2 | 0 | -1 | (0, -1) |
| 7π/4 | √2/2 | -√2/2 | (√2/2, -√2/2) |
| 2π | 1 | 0 | (1, 0) |
7. Symmetry and Quadrant Signs | 对称性与象限符号
Once the first-quadrant coordinates are known, reflection across the axes gives all other coordinates.
一旦知道了第一象限的坐标，通过关于坐标轴的反射就能得到所有其他象限的坐标。
Quadrant I: x > 0, y > 0, so both cos θ and sin θ are positive.
第一象限：x > 0，y > 0，所以 cos θ 和 sin θ 均为正。
Quadrant II: x < 0, y > 0, so cos θ is negative and sin θ is positive.
第二象限：x < 0，y > 0，所以 cos θ 为负，sin θ 为正。
Quadrant III: x < 0, y < 0, so both cos θ and sin θ are negative.
第三象限：x < 0，y < 0，所以 cos θ 和 sin θ 均为负。
Quadrant IV: x > 0, y < 0, so cos θ is positive and sin θ is negative.
第四象限：x > 0，y < 0，所以 cos θ 为正，sin θ 为负。
Memorising the mnemonic "All School Teachers Cheat" or "ASTC" can help: in Quadrants I, II, III, IV, the positive functions are All, Sine, Tangent, Cosine respectively.
可以借助口诀“ASTC”来记忆：在第一、二、三、四象限中，分别为全正、sin 正、tan 正、cos 正。
8. Tangent Values at These Angles | 这些角的正切值
Since tan θ = sin θ ÷ cos θ, the unit circle coordinates give tangent values immediately.
因为 tan θ = sin θ ÷ cos θ，所以单位圆坐标可以直接给出正切值。
tan θ = sin θ ÷ cos θ
For example:
例如：
tan(π/6) = (1/2) ÷ (√3/2) = 1/√3 = √3/3
tan(π/6) = (1/2) ÷ (√3/2) = 1/√3 = √3/3
tan(π/4) = (√2/2) ÷ (√2/2) = 1
tan(π/4) = (√2/2) ÷ (√2/2) = 1
tan(2π/3) = (√3/2) ÷ (-1/2) = -√3
tan(2π/3) = (√3/2) ÷ (-1/2) = -√3
At θ = π/2 and θ = 3π/2, cos θ = 0, so tan θ is undefined.
当 θ = π/2 和 θ = 3π/2 时，cos θ = 0，因此 tan θ 无定义。
9. Reference-Angle Method | 参考角法
For any angle θ in standard position, the reference angle α is the acute angle between the terminal side and the x-axis.
对于标准位置上的任意角 θ，参考角 α 是终边与 x 轴之间的锐角。
Once α is known, we use the first-quadrant value and adjust the signs according to the quadrant.
一旦求出 α，就可以使用第一象限的值，再根据象限调整符号。
(cos θ, sin θ) = (s₁ cos α, s₂ sin α)
Here s₁ is the sign of cosine in the quadrant, and s₂ is the sign of sine in the quadrant.
其中 s₁ 是余弦在该象限的符号，s₂ 是正弦在该象限的符号。
For θ = 5π/6, the reference angle is π/6; because it is in Quadrant II,
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