Unit Circle and Trigonometric Ratios | 单位圆与三角比的关系

📚 Unit Circle and Trigonometric Ratios | 单位圆与三角比的关系

The unit circle is one of the most powerful tools in trigonometry. It connects angles, coordinates, and the six trigonometric ratios in a single, elegant picture. For IB Mathematics students, mastering the unit circle is essential for solving problems involving periodic functions, identities, and equations.

单位圆是三角学中最强大的工具之一。它将角度、坐标和六个三角比统一在一幅简洁而优美的图中。对于IB数学学生来说，掌握单位圆是解决周期函数、恒等式和方程问题的关键。

1. What Is the Unit Circle? | 什么是单位圆？

The unit circle is a circle of radius 1 centered at the origin of a Cartesian coordinate plane. Its equation is:

单位圆是笛卡尔坐标平面中，以原点为圆心、半径为1的圆。其方程为：

x² + y² = 1

Every point on the unit circle satisfies this equation. Because the radius is 1, distances along the circle correspond directly to angle measures in radians.

单位圆上的每一点都满足这个方程。由于半径为1，圆上的弧长可以直接对应角度的弧度制度量。

2. Coordinates as Cosine and Sine | 坐标与余弦、正弦的关系

Take an angle θ measured counter-clockwise from the positive x-axis. The point where the terminal side of θ intersects the unit circle has coordinates (cos θ, sin θ).

从正x轴开始逆时针旋转角度θ，其终边与单位圆的交点坐标为(cos θ, sin θ)。

This definition is far more general than the right-triangle definition. It works for any angle, including negative angles and angles greater than 90°.

这个定义比直角三角形中的定义更为一般化。它适用于任何角度，包括负角和大于90°的角。

3. The Six Trigonometric Ratios | 六个三角比

From the coordinates (x, y) on the unit circle, we define the six trigonometric ratios as follows:

根据单位圆上的坐标(x, y)，六个三角比定义如下：

sin θ = y	csc θ = 1/y (y ≠ 0)
cos θ = x	sec θ = 1/x (x ≠ 0)
tan θ = y/x (x ≠ 0)	cot θ = x/y (y ≠ 0)

Because the radius is 1, these ratios have a clear geometric meaning. The tangent can also be seen as the length of a vertical segment tangent to the circle at (1, 0).

由于半径为1，这些比值具有清晰的几何意义。正切还可以看作过点(1, 0)的圆切线上的一条垂直线段。

4. Special Angles on the Unit Circle | 单位圆上的特殊角

The angles π/6 (30°), π/4 (45°), and π/3 (60°) appear frequently in IB exams. Their coordinates are derived from simple right triangles.

角π/6（30°）、π/4（45°）和π/3（60°）在IB考试中经常出现。它们的坐标可以由简单直角三角形导出。

Angle θ	sin θ	cos θ	tan θ
0	0	1	0
π/6	1/2	√3/2	1/√3
π/4	√2/2	√2/2	1
π/3	√3/2	1/2	√3
π/2	1	0	undefined

Notice the pattern of increasing values for sine and decreasing values for cosine as θ moves from 0 to π/2. These exact values must be memorized or quickly derived.

注意当θ从0到π/2时，正弦值递增、余弦值递减的规律。这些精确值必须牢记或快速推导。

5. Signs in the Four Quadrants | 四个象限中的符号

The unit circle makes it easy to determine whether each trigonometric ratio is positive or negative. The signs depend only on the signs of x and y in each quadrant.

单位圆使我们能够轻松判断每个三角比的正负。符号仅取决于每个象限中x和y的符号。

Quadrant I (0 to π/2): x > 0, y > 0 → all ratios positive.

第一象限（0到π/2）：x > 0，y > 0 → 所有三角比为正。
Quadrant II (π/2 to π): x < 0, y > 0 → only sine and cosecant are positive.

第二象限（π/2到π）：x < 0，y > 0 → 只有正弦和余割为正。
Quadrant III (π to 3π/2): x < 0, y < 0 → only tangent and cotangent are positive.

第三象限（π到3π/2）：x < 0，y < 0 → 只有正切和余切为正。
Quadrant IV (3π/2 to 2π): x > 0, y < 0 → only cosine and secant are positive.

第四象限（3π/2到2π）：x > 0，y < 0 → 只有余弦和正割为正。

A common mnemonic is “All Students Take Calculus” to remember the positive ratios in quadrants I, II, III, IV respectively.

常见的助记口诀是“ASTC”（All Students Take Calculus），分别表示第一、二、三、四象限中为正的三角比。

6. Reference Angles | 参考角

A reference angle is the acute angle between the terminal side of θ and the x-axis. It is always between 0 and π/2.

参考角是角θ的终边与x轴之间的锐角。它总是在0到π/2之间。

To find the reference angle α for an angle θ in standard position:

求标准位置角θ的参考角α的方法：

Quadrant	Reference angle α
II	α = π − θ
III	α = θ − π
IV	α = 2π − θ

Once the reference angle is known, the absolute values of the trigonometric ratios are the same as those for α. Only the signs differ, according to the quadrant.

求出参考角后，各三角比的绝对值与α的对应值相同，只需根据象限调整符号。

7. Periodicity of Trigonometric Functions | 三角函数的周期性

Because one full revolution around the unit circle corresponds to an angle of 2π, all trigonometric functions are periodic:

由于单位圆上旋转一整圈对应角度2π，所有三角函数都是周期函数：

sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ, tan(θ + π) = tan θ

Sine and cosine have period 2π, while tangent and cotangent have period π. This is because tangent repeats after half a revolution, since y/x and (−y)/(−x) are equal.

正弦和余弦的周期为2π，而正切和余切的周期为π。这是因为正切在半圈后重复，因为y/x与(−y)/(−x)相等。

The unit circle also gives rise to important symmetry identities:

单位圆还给出了重要的对称恒等式：

sin(−θ) = −sin θ (odd function)

sin(−θ) = −sin θ（奇函数）
cos(−θ) = cos θ (even function)

cos(−θ) = cos θ（偶函数）
sin(π − θ) = sin θ

sin(π − θ) = sin θ
cos(π − θ) = −cos θ

cos(π − θ) = −cos θ

8. The Pythagorean Identity | 毕达哥拉斯恒等式

Since (cos θ, sin θ) lies on the unit circle, its coordinates must satisfy x² + y² = 1. Substituting gives the most important trigonometric identity:

由于点(cos θ, sin θ)位于单位圆上，其坐标必须满足x² + y² = 1。代入可得最重要的三角恒等式：

sin²θ + cos²θ = 1

Dividing by cos²θ or sin²θ yields two related identities:

分别除以cos²θ或sin²θ，可得两个相关恒等式：

1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ

These identities are invaluable for simplifying expressions and proving other identities in IB exams.

这些恒等式在化简表达式和证明其他恒等式时非常有用，是IB考试中的必备工具。

9. Parametric Representation of the Circle | 圆的参数表示

The unit circle can be parameterized by the angle θ as:

单位圆可以用角度θ参数化表示为：

(x, y) = (cos θ, sin θ), 0 ≤ θ < 2π

This representation is used extensively in calculus, physics, and computer graphics. For example, circular motion is described by r(t) = (R cos ωt, R sin ωt).

这种表示在微积分、物理学和计算机图形学中广泛使用。例如，圆周运动可以描述为r(t) = (R cos ωt, R sin ωt)。

In IB Mathematics: Analysis and Approaches, this idea connects to vectors and complex numbers, where the unit circle appears as the set of complex numbers with modulus 1.

在IB数学分析与方法中，这一思想与向量和复数相联系，单位圆表现为模为1的复数集合。

10. Solving Trigonometric Equations | 解三角方程

The unit circle provides a visual method for solving equations such as sin θ = ½. First, identify the reference angle α = π/6. Then, determine which quadrants have positive sine: QI and QII.

单位圆为解方程提供了直观方法，例如sin θ = ½。首先确定参考角α = π/6。然后判断哪些象限的正弦为正：第一象限和第二象限。

Thus the solutions in the interval [0, 2π) are:

因此，在区间[0, 2π)内的解为：

θ = π/6 and θ = 5π/6

Adding the period 2πn gives the general solution. This method works for any standard equation.

加上周期2πn即可得到通解。这个方法适用于任何标准三角方程。

For example, cos θ = −√2/2: the reference angle is π/4, cosine is negative in QII and QIII, so θ = 3π/4 and 5π/4.

例如，cos θ = −√2/2：参考角为π/4，余弦在第二、三象限为负，所以θ = 3π/4和5π/4。

11. Common Mistakes and Pitfalls | 常见错误与易错点

Many students make sign errors because they forget the quadrant. Always draw the unit circle or use the ASTC rule.

许多学生因为忘记象限而出现符号错误。务必画出单位圆或使用ASTC规则。

Using degrees instead of radians on exact-value problems. Always check the angle unit required.

在精确值题目中混用角度制和弧度制。务必确认题目要求的度量单位。
Confusing the order of coordinates. The point is (cos θ, sin θ), not (sin θ, cos θ).

混淆坐标顺序。点的坐标是(cos θ, sin θ)，不是(sin θ, cos θ)。
Forgetting that tan θ is undefined when x = 0, i.e., at θ = π/2 and 3π/2.

忘记当x = 0时，即θ = π/2和3π/2时，tan θ无定义。
Using the reference angle as the final answer without adjusting the quadrant sign.

只求参考角而不根据象限调整符号，导致答案错误。

12. Practice Example: Full IB-Style Problem | 练习例题：完整IB风格题目

Let us work through a typical IB-style question:

我们来看一道典型的IB风格题目：

Given that cos θ = −3/5 and π/2 < θ < π, find exact values of sin θ and tan θ.

已知cos θ = −3/5，且π/2 < θ < π，求sin θ和tan θ的精确值。

Using the identity sin²θ + cos²θ = 1:

使用恒等式sin²θ + cos²θ = 1：

sin²θ = 1 − (−3/5)² = 1 − 9/25 = 16/25

Since θ is in Quadrant II, sin θ is positive. Therefore sin θ = 4/5.

由于θ在第二象限，sin θ为正。因此sin θ = 4/5。

Then tan θ = sin θ / cos θ = (4/5) / (−3/5) = −4/3.

于是tan θ = sin θ / cos θ = (4/5) / (−3/5) = −4/3。

Notice that the unit circle picture would show the same result: a point in QII with x = −3/5 and y = 4/5, lying on the circle of radius 1.

注意单位圆图像会显示同样的结果：第二象限中的点，x = −3/5，y = 4/5，位于半径为1的圆上。

The unit circle is not just a diagram; it is the unifying definition of trigonometric ratios. Once you internalize its geometry, you will be able to derive exact values, signs, identities, and equation solutions quickly and accurately.

单位圆不仅是图形，更是三角比的统一定义。一旦你内化了它的几何结构，你就能快速而准确地推导出精确值、符号、恒等式和方程的解。

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

更多咨询请联系16621398022（同微信）

Unit Circle and Trigonometric Ratios | 单位圆与三角比的关系