A-Level Maths: Signs of Trigonometric Functions in the Four Quadrants | A-Level 数学：四个象限内角的三角函数符号

📚 A-Level Maths: Signs of Trigonometric Functions in the Four Quadrants | A-Level 数学：四个象限内角的三角函数符号

Before you can solve trigonometric equations, sketch graphs, or apply identities, you must know whether a sine, cosine or tangent value is positive or negative for a given angle. In this article you will learn a reliable way to determine these signs using the unit circle and the ASTC rule, with worked examples and exam-style practice.

在解三角方程、画函数图像或使用恒等式之前，你必须先判断某个角的正弦、余弦或正切值是正还是负。在本文中，你将学习如何利用单位圆和 ASTC 法则可靠地判断这些符号，并配有例题与考试风格练习。

1. Introduction to the Unit Circle and Quadrants | 单位圆与四个象限简介

In A-Level mathematics, trigonometric functions are defined using a unit circle: a circle of radius 1 centred at the origin of the coordinate plane. For an angle θ measured anticlockwise from the positive x-axis, the point where the circle meets the terminal side has coordinates (cos θ, sin θ).

在 A-Level 数学中，三角函数通过单位圆定义：单位圆是圆心在坐标原点、半径为 1 的圆。对于从 x 轴正方向开始逆时针旋转的角 θ，单位圆与角的终边交点的坐标为 (cos θ, sin θ)。

The coordinate axes divide the plane into four quadrants, labelled I, II, III and IV in anticlockwise order.

坐标轴把平面分为四个象限，按逆时针方向依次标为 I、II、III、IV。
Angles between 0° and 90° lie in QI; between 90° and 180° in QII; between 180° and 270° in QIII; between 270° and 360° in QIV.

0° 到 90° 的角位于第一象限；90° 到 180° 位于第二象限；180° 到 270° 位于第三象限；270° 到 360° 位于第四象限。
In Edexcel A-Level both degrees and radians are used; 180° = π radians, so the quadrant boundaries are 0, π/2, π, 3π/2 and 2π.

在 Edexcel A-Level 中，角度制和弧度制都会使用；180° = π 弧度，因此象限边界为 0、π/2、π、3π/2 和 2π。

2. Defining the Three Basic Trigonometric Ratios | 三个基本三角比的定义

On the unit circle, the x-coordinate of a point equals cos θ and the y-coordinate equals sin θ. The tangent ratio is defined as the quotient of these two coordinates.

在单位圆上，一点的 x 坐标等于 cos θ，y 坐标等于 sin θ。正切比定义为这两个坐标的商。

sin θ = y, cos θ = x, tan θ = sin θ / cos θ

Since tan θ is the quotient of sin θ and cos θ, its sign depends on whether sin θ and cos θ have the same sign or opposite signs.

因为 tan θ 是 sin θ 与 cos θ 的商，所以它的符号取决于 sin θ 和 cos θ 是同号还是异号。

sin θ is positive when y > 0 and negative when y < 0.

当 y > 0 时 sin θ 为正，当 y < 0 时 sin θ 为负。
cos θ is positive when x > 0 and negative when x < 0.

当 x > 0 时 cos θ 为正，当 x < 0 时 cos θ 为负。
The reciprocal functions sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ have the same sign as their corresponding basic function.

倒数函数 sec θ = 1/cos θ、cosec θ = 1/sin θ 和 cot θ = 1/tan θ 与它们对应基本函数的符号相同。

3. How Signs Are Determined: Coordinates in Each Quadrant | 符号如何确定：各象限的坐标

The sign of every trigonometric ratio is ultimately determined by the signs of the x- and y-coordinates in that quadrant. The following table summarises the sign pattern.

每个三角函数值的符号最终都由该象限中 x 坐标和 y 坐标的符号决定。下表总结了符号规律。

Quadrant	x-coordinate	y-coordinate	sin θ	cos θ	tan θ
I	+	+	+	+	+
II	–	+	+	–	–
III	–	–	–	–	+
IV	+	–	–	+	–

Notice that in QIII, both x and y are negative, so y / x is positive. This is why tan θ is positive in QIII even though sin θ and cos θ are both negative.

注意在第三象限中，x 和 y 都为负，因此 y / x 为正。这就是为什么在第三象限中 sin θ 和 cos θ 都为负，但 tan θ 为正。

4. The ASTC / CAST Rule | ASTC / CAST 法则

The ASTC rule is a quick way to recall which trigonometric function is positive in each quadrant. Starting from QI and moving anticlockwise, the letters stand for All, Sine, Tangent, Cosine.

ASTC 法则是一种快速记忆各象限中哪个三角函数为正的方法。从第一象限开始逆时针移动，字母依次代表 All、Sine、Tangent、Cosine。

A in QI: all six trigonometric functions are positive.

QI 中的 A：六个三角函数全部为正。
S in QII: only sine and its reciprocal cosec are positive.

QII 中的 S：只有正弦及其倒数 cosec 为正。
T in QIII: only tangent and its reciprocal cot are positive.

QIII 中的 T：只有正切及其倒数 cot 为正。
C in QIV: only cosine and its reciprocal sec are positive.

QIV 中的 C：只有余弦及其倒数 sec 为正。

Many students remember the mnemonic ‘All Silly Trig Classes’ or ‘All Students Take Calculus’. The key idea is that the letters tell you which ratio is positive, not which ratio is negative.

许多学生会用口诀 ‘All Silly Trig Classes’ 或 ‘All Students Take Calculus’ 来记忆。关键思想是字母告诉你哪个比值为正，而不是哪个比值为负。

5. Quadrant I: All Positive | 第一象限：全部为正

In QI, the angle satisfies 0° < θ < 90° or 0 < θ < π/2. Here x > 0 and y > 0, so sin θ, cos θ and tan θ are all positive.

在第一象限中，角满足 0° < θ < 90° 或 0 < θ < π/2。此时 x > 0 且 y > 0，所以 sin θ、cos θ 和 tan θ 都为正。

sin θ > 0, cos θ > 0, tan θ > 0

For example, take θ = 60°. The exact values are sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3, all positive.

例如，取 θ = 60°，精确值为 sin 60° = √3/2、cos 60° = 1/2、tan 60° = √3，全部为正。

6. Quadrant II: Sine Positive, Cosine and Tangent Negative | 第二象限：正弦为正，余弦和正切为负

In QII, the angle satisfies 90° < θ < 180° or π/2 < θ < π. Here x < 0 and y > 0, so sin θ is positive while cos θ and tan θ are negative.

在第二象限中，角满足 90° < θ < 180° 或 π/2 < θ < π。此时 x < 0 且 y > 0，所以 sin θ 为正，而 cos θ 和 tan θ 为负。

sin θ > 0, cos θ < 0, tan θ < 0

For example, θ = 120° gives sin 120° = √3/2, cos 120° = -1/2 and tan 120° = -√3.

例如，θ = 120° 时，sin 120° = √3/2、cos 120° = -1/2、tan 120° = -√3。

7. Quadrant III: Tangent Positive, Sine and Cosine Negative | 第三象限：正切为正，正弦和余弦为负

In QIII, the angle satisfies 180° < θ < 270° or π < θ < 3π/2. Here x < 0 and y < 0, so sin θ and cos θ are both negative, but tan θ is positive because y / x is a positive quotient of two negatives.

在第三象限中，角满足 180° < θ < 270° 或 π < θ < 3π/2。此时 x < 0 且 y < 0，所以 sin θ 和 cos θ 都为负，但 tan θ 为正，因为两个负数相除得到正数。

sin θ < 0, cos θ < 0, tan θ > 0

For example, θ = 225° gives sin 225° = -√2/2, cos 225° = -√2/2 and tan 225° = 1.

例如，θ = 225° 时，sin 225° = -√2/2、cos 225° = -√2/2、tan 225° = 1。

8. Quadrant IV: Cosine Positive, Sine and Tangent Negative | 第四象限：余弦为正，正弦和正切为负

In QIV, the angle satisfies 270° < θ < 360° or 3π/2 < θ < 2π. Here x > 0 and y < 0, so cos θ is positive while sin θ and tan θ are negative.

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A-Level Maths: Signs of Trigonometric Functions in the Four Quadrants | A-Level 数学：四个象限内角的三角函数符号