Angles in All Four Quadrants | 四个象限中的角度

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

Understanding how angles behave in all four quadrants is a core skill in A-Level Mathematics. This article explains quadrant signs, reference angles, the CAST diagram, exact values and simple trigonometric equations, with examples aligned to the Edexcel specification.

理解角在四个象限中的行为是 A-Level 数学的核心技能。本文讲解象限符号、参考角、CAST 图、精确值以及简单三角方程，并配以符合 Edexcel 考试大纲的例题。

1. The Coordinate Plane and Quadrant Signs | 坐标平面与象限符号

The xy-plane is divided by the coordinate axes into four regions called quadrants. Quadrant I is the top-right region where x > 0 and y > 0. Quadrant II is the top-left region where x < 0 and y > 0. Quadrant III is the bottom-left region where x < 0 and y < 0. Quadrant IV is the bottom-right region where x > 0 and y < 0.

xy 平面被坐标轴分成四个区域，称为象限。第一象限是右上区域，x > 0 且 y > 0。第二象限是左上区域，x < 0 且 y > 0。第三象限是左下区域，x < 0 且 y < 0。第四象限是右下区域，x > 0 且 y < 0。

For an angle measured anticlockwise from the positive x-axis, Quadrant I contains angles from 0° to 90°, Quadrant II from 90° to 180°, Quadrant III from 180° to 270°, and Quadrant IV from 270° to 360°.

对于从正 x 轴逆时针测量的角，第一象限包含 0° 到 90° 的角，第二象限包含 90° 到 180°，第三象限包含 180° 到 270°，第四象限包含 270° 到 360°。

2. Standard Position and Directed Angles | 标准位置与有向角

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is obtained by rotating the initial side through the angle. Positive angles are measured anticlockwise, while negative angles are measured clockwise.

当角的顶点位于原点且始边位于正 x 轴时，该角处于标准位置。终边由始边旋转该角度得到。正角按逆时针方向测量，负角按顺时针方向测量。

For example, 135° has its terminal side in Quadrant II, and −225° also has the same terminal side because −225° clockwise is equivalent to 135° anticlockwise. Angles that share a terminal side are coterminal.

例如，135° 的终边在第二象限，而 −225° 也具有相同的终边，因为顺时针 −225° 等同于逆时针 135°。终边相同的角称为终边相同角。

3. The Unit Circle Definition | 单位圆定义

The unit circle is a circle of radius 1 centred at the origin. If a point P(x, y) lies on the unit circle at angle θ, then the definitions are sin θ = y, cos θ = x and tan θ = y / x, provided x ≠ 0.

单位圆是以原点为圆心、半径为 1 的圆。如果点 P(x, y) 位于单位圆上且对应角 θ，则定义 sin θ = y，cos θ = x，tan θ = y / x，其中 x ≠ 0。

Because the radius is 1, the coordinates of P are simply (cos θ, sin θ). This makes the signs of sine and cosine identical to the signs of y and x in the corresponding quadrant.

由于半径为 1，点 P 的坐标就是 (cos θ, sin θ)。这使得正弦和余弦的符号与相应象限中 y 和 x 的符号完全一致。

4. Sine, Cosine and Tangent in Each Quadrant | 各象限中的正弦、余弦与正切

The sign of sin θ depends only on y, so sine is positive in Quadrants I and II where y > 0, and negative in Quadrants III and IV where y < 0. The sign of cos θ depends only on x, so cosine is positive in Quadrants I and IV where x > 0, and negative in Quadrants II and III where x < 0.

sin θ 的符号只取决于 y，因此正弦在第一和第二象限（y > 0）为正，在第三和第四象限（y < 0）为负。cos θ 的符号只取决于 x，因此余弦在第一和第四象限（x > 0）为正，在第二和第三象限（x < 0）为负。

Since tan θ = y / x, tangent is positive when x and y have the same sign, which occurs in Quadrants I and III. Tangent is negative when x and y have opposite signs, which occurs in Quadrants II and IV.

由于 tan θ = y / x，当 x 和 y 同号时正切为正，这发生在第一和第三象限。当 x 和 y 异号时正切为负，这发生在第二和第四象限。

5. The CAST Diagram | CAST 图

The CAST diagram is a memory aid for the signs of sine, cosine and tangent. Starting in Quadrant I and moving anticlockwise, the letters are A, S, T, C. A means all three functions are positive

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