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

📚 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

The unit 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。

In radian measure, π radians equals 180°. 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。

For any point P on the unit circle, if the radius OP makes an angle θ with the positive x-axis, 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

For θ = π/4, the terminal side is halfway 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,

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）