Angles in the Four Quadrants | 四个象限中的角

📚 Angles in the Four Quadrants | 四个象限中的角

Angles are more than just turns between two lines. In coordinate geometry and trigonometry, an angle is often placed on a coordinate plane, and the quadrant it lies in tells us valuable information about the sign and size of its trigonometric ratios. Understanding angles in the four quadrants is a fundamental skill for solving problems in A-level mathematics.

角不仅仅是两条线之间的旋转。在坐标几何和三角学中，角经常被放置在坐标平面上，而它所在的象限会告诉我们关于其三角函数值的符号和大小的宝贵信息。理解四个象限中的角是解决 A-level 数学问题的基本技能。

1. The Coordinate Plane and Quadrants | 坐标平面与象限

The coordinate plane is divided by two perpendicular axes: the horizontal x-axis and the vertical y-axis. These two axes intersect at the origin O and divide the plane into four regions called quadrants. The quadrants are numbered anticlockwise, starting from the upper-right region.

坐标平面由两条互相垂直的轴分割：水平的 x 轴和竖直的 y 轴。这两条轴在原点 O 相交，并将平面分成四个称为象限的区域。象限按逆时针方向编号，从右上方的区域开始。

Quadrant I: x > 0, y > 0
Quadrant II: x < 0, y > 0
Quadrant III: x < 0, y < 0
Quadrant IV: x > 0, y < 0

Quadrant I: x > 0, y > 0

第一象限：x > 0，y > 0

Quadrant II: x < 0, y > 0

第二象限：x < 0，y > 0

Quadrant III: x < 0, y < 0

第三象限：x < 0，y < 0

Quadrant IV: x > 0, y < 0

第四象限：x > 0，y < 0

2. Standard Position of an Angle | 角的标准位置

An angle is said to be in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The angle is then formed by rotating the initial side to a terminal side. A positive angle is generated by an anticlockwise rotation, while a negative angle is generated by a clockwise rotation.

当一个角的顶点位于原点，且始边沿 x 轴正方向时，这个角被称为处于标准位置。然后通过将始边旋转到终边来形成该角。正角由逆时针旋转产生，而负角由顺时针旋转产生。

For example, an angle of 135° in standard position has its terminal side in Quadrant II. An angle of −60° has its terminal side in Quadrant IV, because the clockwise rotation places it below the positive x-axis.

例如，标准位置下的 135° 角，其终边位于第二象限。−60° 角的终边位于第四象限，因为顺时针旋转使其落在 x 轴正方向的下方。

3. Measuring Angles: Degrees and Radians | 角的度量：度与弧度

Angles can be measured in degrees or radians. A full revolution around the origin is 360°, which is equal to 2π radians. Therefore, 180° = π radians and 90° = π/2 radians. When working with the four quadrants, it is useful to remember the quadrant boundaries in both units.

角可以用度或弧度来度量。绕原点旋转一整圈是 360°，等于 2π 弧度。因此，180° = π 弧度，90° = π/2 弧度。在处理四个象限时，记住两种单位下的象限分界线是很有用的。

180° = π radians, 90° = π/2 radians, 360° = 2π radians

180° = π 弧度，90° = π/2 弧度，360° = 2π 弧度

The quadrant boundaries are:

象限分界线为：

Quadrant I: 0° to 90° (0 to π/2)
Quadrant II: 90° to 180° (π/2 to π)
Quadrant III: 180° to 270° (π to 3π/2)
Quadrant IV: 270° to 360° (3π/2 to 2π)

Quadrant I: 0° to 90° (0 to π/2)

第一象限：0° 到 90°（0 到 π/2）

Quadrant II: 90° to 180° (π/2 to π)

第二象限：90° 到 180°（π/2 到 π）

Quadrant III: 180° to 270° (π to 3π/2)

第三象限：180° 到 270°（π 到 3π/2）

Quadrant IV: 270° to 360° (3π/2 to 2π)

第四象限：270° 到 360°（3π/2 到 2π）

4. Reference Angles | 参考角

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. For any angle in standard position, the reference angle α is always between 0° and 90° (0 and π/2). It is a powerful tool because the absolute values of all trigonometric functions of the original angle are the same as those of its reference angle.

参考角是指给定角的终边与 x 轴之间形成的锐角。对于标准位置下的任意角，参考角 α 总是在 0° 到 90°（0 到 π/2）之间。它非常有用，因为原角的所有三角函数值的绝对值都等于其参考角的三角函数值的绝对值。

To find the reference angle, use these rules based on the quadrant of the terminal side:

要根据终边所在象限使用以下规则来求参考角：

Quadrant I: reference angle = θ
Quadrant II: reference angle = 180° − θ (π − θ)
Quadrant III: reference angle = θ − 180° (θ − π)
Quadrant IV: reference angle = 360° − θ (2π − θ)

Quadrant I: reference angle = θ

第一象限：参考角 = θ

Quadrant II: reference angle = 180° − θ (π − θ)

第二象限：参考角 = 180° − θ（π − θ）

Quadrant III: reference angle = θ − 180° (θ − π)

第三象限：参考角 = θ − 180°（θ − π）

Quadrant IV: reference angle = 360° − θ (2π − θ)

第四象限：参考角 = 360° − θ（2π − θ）

5. Signs of Trigonometric Functions in Each Quadrant | 各象限中三角函数的符号

The signs of sin θ, cos θ, and tan θ depend on the quadrant in which the terminal side of θ lies. This is determined by the signs of x and y coordinates of a point on the terminal side. Recall that sin θ = y/r, cos θ = x/r, and tan θ = y/x, where r > 0 is the distance from the origin to the point.

sin θ、cos θ 和 tan θ 的符号取决于 θ 的终边所在的象限。这是由终边上一点的 x 和 y 坐标的符号决定的。回顾一下：sin θ = y/r，cos θ = x/r，tan θ = y/x，其中 r > 0 是原点到该点的距离。

Quadrant	sin θ	cos θ	tan θ
I	+	+	+
II	+	−	−
III	−	−	+
IV	−	+	−

Note that cosecant (csc θ) has the same sign as sin θ, secant (sec θ) has the same sign as cos θ, and cotangent (cot θ) has the same sign as tan θ.

注意：余割（csc θ）与 sin θ 同号，正割（sec θ）与 cos θ 同号，余切（cot θ）与 tan θ 同号。

6. The ASTC Rule | ASTC 法则

The ASTC rule is a quick memory aid for which trigonometric functions are positive in each quadrant. The letters stand for:

ASTC 法则是快速记忆各象限中哪些三角函数为正的技巧。这些字母代表：

All (Quadrant I): All functions are positive.
Sin (Quadrant II): Only sin and cosecant are positive.
Tan (Quadrant III): Only tan and cotangent are positive.
Cos (Quadrant IV): Only cos and secant are positive.

All（第一象限）：所有函数均为正。

Sin（第二象限）：只有 sin 和 cosecant 为正。

Tan（第三象限）：只有 tan 和 cotangent 为正。

Cos（第四象限）：只有 cos 和 secant 为正。

Many students remember the phrase ‘All Students Take Calculus’ to recall the order ASTC starting from Quadrant I and moving anticlockwise.

许多学生用短语“All Students Take Calculus”来记住从第一象限开始逆时针排列的 ASTC 顺序。

7. Worked Example: Finding the Reference Angle | 实例：求参考角

Find the reference angle for each of the following angles: 135°, 210°, 330°, and 5π/4 radians.

求以下各角的参考角：135°、210°、330° 和 5π/4 弧度。

For 135°, the terminal side is in Quadrant II. Therefore the reference angle is 180° − 135° = 45°.

对于 135°，终边在第二象限。因此参考角为 180° − 135° = 45°。

For 210°, the terminal side is in Quadrant III. Therefore the reference angle is 210° − 180° = 30°.

对于 210°，终边在第三象限。因此参考角为 210° − 180° = 30°。

For 330°, the terminal side is in Quadrant IV. Therefore the reference angle is 360° − 330° = 30°.

对于 330°，终边在第四象限。因此参考角为 360° − 330° = 30°。

For 5π/4 radians, note that 5π/4 is in Quadrant III because π < 5π/4 < 3π/2. The reference angle is 5π/4 − π = π/4.

对于 5π/4 弧度，注意 5π/4 在第三象限，因为 π < 5π/4 < 3π/2。参考角为 5π/4 − π = π/4。

8. Worked Example: Determining the Sign | 实例：判断符号

Determine the sign of sin 240°, cos 240°, and tan 240°.

判断 sin 240°、cos 240° 和 tan 240° 的符号。

First, identify the quadrant. Since 180° < 240° < 270°, the terminal side lies in Quadrant III. According to the ASTC rule, only tan is positive in Quadrant III. Therefore:

首先，确定象限。因为 180° < 240° < 270°，终边位于第三象限。根据 ASTC 法则，第三象限中只有 tan 为正。因此：

sin 240° is negative.

sin 240° 为负。

cos 240° is negative.

cos 240° 为负。

tan 240° is positive.

tan 240° 为正。

We can verify using reference angles. The reference angle is 240° − 180° = 60°. We know sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. Applying the signs from Quadrant III gives:

我们可以用参考角来验证。参考角为 240° − 180° = 60°。我们知道 sin 60° = √3/2，cos 60° = 1/2，tan 60° = √3。加上第三象限的符号后得到：

sin 240° = −√3/2, cos 240° = −1/2, tan 240° = √3

sin 240° = −√3/2，cos 240° = −1/2，tan 240° = √3

9. Angles Greater Than 360° and Negative Angles | 大于 360° 的角和负角

Angles greater than 360° or less than 0° can be simplified by finding their coterminal angle — an angle in the range 0° to 360° that shares the same terminal side. To find a coterminal angle, add or subtract multiples of 360° (or 2π radians).

大于 360° 或小于 0° 的角可以通过求其同终边角来简化——即在 0° 到 360° 范围内、具有相同终边的角。要求同终边角，可以加上或减去 360°（或 2π 弧度）的整数倍。

For example, to work with 780°, subtract 360° twice: 780° − 360° − 360° = 60°. Thus 780° lies in Quadrant I. Similarly, −120° can be converted by adding 360°: −120° + 360° = 240°, so −120° lies in Quadrant III.

例如，要处理 780°，减去两次 360°：780° − 360° − 360° = 60°。因此 780° 位于第一象限。类似地，−120° 可以通过加上 360° 来转换：−120° + 360° = 240°，所以 −120° 位于第三象限。

For radians, add or subtract 2π. For example, 9π/4 − 2π = π/4, which is in Quadrant I.

对于弧度，加上或减去 2π。例如，9π/4 − 2π = π/4，位于第一象限。

10. Common Mistakes and Exam Tips | 常见错误与考试提示

One common mistake is confusing the quadrant of negative angles. Remember that a negative angle is measured clockwise from the positive x-axis, so −30° is in Quadrant IV, not Quadrant I.

一个常见错误是混淆负角所在的象限。记住负角是从 x 轴正方向顺时针测量的，所以 −30° 在第四象限，而不是第一象限。

Another mistake is using the wrong formula for the reference angle. Always check the quadrant first, then apply the correct subtraction. For example, in Quadrant III, the reference angle is θ − 180°, not 180° − θ.

另一个错误是使用错误的参考角公式。一定要先判断象限，再应用正确的减法。例如，在第三象限中，参考角是 θ − 180°，而不是 180° − θ。

When solving trigonometric equations, it is essential to find all solutions in the required interval. For example, if sin θ = 1/2 and 0° ≤ θ < 360°, the solutions are θ = 30° and θ = 150°. The first is in Quadrant I, and the second is in Quadrant II where sin is still positive.

在解三角方程时，必须求出给定区间内的所有解。例如，如果 sin θ = 1/2 且 0° ≤ θ < 360°，解为 θ = 30° 和 θ = 150°。第一个在第一象限，第二个在第二象限，而 sin 在第二象限仍为正。

Exam tip: When you evaluate a trigonometric function for any angle, first write down the reference angle and the quadrant sign. Then combine them. This systematic method prevents sign errors.

考试提示：当你求任意角的三角函数值时，先写出参考角和象限符号，然后将两者结合。这种系统方法可以避免符号错误。

11. Conclusion | 总结

Angles in the four quadrants are a central concept in trigonometry. By understanding standard position, reference angles, and the signs of trigonometric functions through the ASTC rule, you can confidently evaluate trigonometric expressions, solve equations, and sketch graphs. Mastery of these ideas is essential for success in A-level mathematics.

四个象限中的角是三角学的核心概念。通过理解标准位置、参考角以及借助 ASTC 法则判断三角函数的符号，你可以自信地计算三角表达式、解方程和绘制图像。掌握这些思想对在 A-level 数学中取得成功至关重要。

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Angles in the Four Quadrants | 四个象限中的角