📚 Angles in All Four Quadrants | 四个象限中的角
Understanding how angles behave in all four quadrants is a cornerstone of A-Level trigonometry. It enables you to extend sine, cosine, and tangent beyond acute angles, solve equations in a full 360° range, and work confidently with radian measure. This article unpacks the unit circle, CAST diagram, related angle formulas, and practical techniques for finding all solutions.
理解角度在四个象限中的行为是 A-Level 三角学的基石。它让你能够将正弦、余弦和正切扩展到锐角之外，在 360° 范围内解方程，并能熟练使用弧度制。本文讲解单位圆、CAST 图、相关角公式以及寻找所有解的实用技巧。
1. The Unit Circle and Quadrants | 单位圆与象限
The unit circle is a circle of radius 1 centred at the origin (0,0). An angle θ is measured from the positive x-axis, moving anticlockwise for positive angles and clockwise for negative angles. The circle is divided into four quadrants: Quadrant I (0° to 90°), Quadrant II (90° to 180°), Quadrant III (180° to 270°), and Quadrant IV (270° to 360°).
单位圆是以原点 (0,0) 为圆心、半径为 1 的圆。角 θ 从正 x 轴开始测量，逆时针为正角，顺时针为负角。圆被分为四个象限：第一象限（0° 到 90°）、第二象限（90° 到 180°）、第三象限（180° 到 270°）和第四象限（270° 到 360°）。
Any point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle formed with the positive x-axis. This simple fact connects the geometry of the circle directly to trigonometric ratios, allowing us to determine the sign of sine and cosine in each quadrant.
单位圆上任意一点的坐标是 (cos θ, sin θ)，其中 θ 是与正 x 轴形成的角。这一简单事实将圆的几何性质与三角比直接联系起来，使我们能够确定每个象限中正弦和余弦的符号。
2. The CAST Diagram | CAST 图
The CAST diagram is a mnemonic that shows which trigonometric functions are positive in each quadrant. Moving anticlockwise from Quadrant I: C (Cos positive), A (All positive), S (Sin positive), T (Tan positive). Many students remember this order as 'All Students Take Calculus' or 'CAST'.
CAST 图是一种助记符，显示每个象限中哪些三角函数为正。从第一象限逆时针移动：C（余弦正）、A（全部正）、S（正弦正）、T（正切正）。许多学生通过 'All Students Take Calculus' 或 'CAST' 来记住这个顺序。
| Quadrant | Range | Positive functions | Negative functions |
|---|---|---|---|
| I | 0° – 90° | All (sin, cos, tan) | None |
| II | 90° – 180° | Sin | Cos, Tan |
| III | 180° – 270° | Tan | Sin, Cos |
| IV | 270° – 360° | Cos | Sin, Tan |
Knowing which function is positive in a given quadrant is essential when solving trigonometric equations, because it tells you the sign of the solution before you even calculate the related acute angle.
在解三角方程时，知道哪个函数在给定象限为正至关重要，因为它在计算相关锐角之前就告诉你解的符号。
3. Related Acute Angle (Reference Angle) | 相关锐角（参考角）
For any angle θ, the related acute angle, α, is the smallest positive angle between the terminal side of θ and the x-axis. It is always between 0° and 90° and is found differently depending on the quadrant.
对于任意角 θ，相关锐角 α 是 θ 的终边与 x 轴之间最小的正角。它始终在 0° 到 90° 之间，并根据象限的不同以不同的方式求得。
- Quadrant I: α = θ
- Quadrant II: α = 180° – θ (or π – θ in radians)
- Quadrant III: α = θ – 180° (or θ – π)
- Quadrant IV: α = 360° – θ (or 2π – θ)
Once you have found α, the magnitude of the trigonometric ratio for θ equals the value of that ratio for α. The sign is then determined by the quadrant using CAST.
一旦找到 α，θ 的三角比的绝对值就等于 α 的那个比值。然后根据象限用 CAST 确定符号。
4. Exact Values for Common Angles | 常见角度的精确值
and 90°, and extend these to the four quadrants. For example, sin 30° = 1/2, cos 45° = 1/√2, tan 60° = √3. These values come from special right-angled triangles: the isosceles right triangle (45°-45°-90°) and the equilateral triangle halved (30°-60°-90°).
你必须能够记住 0°、30°、45°、60° 和 90° 的正弦、余弦和正切的精确值，并将它们扩展到四个象限。例如，sin 30° = 1/2，cos 45° = 1/√2，tan 60° = √3。这些值来自特殊的直角三角形：等腰直角三角形（45°-45°-90°）和对半分的等边三角形（30°-60°-90°）。
To find, say, sin 210°, note that 210° is in Quadrant III with α = 210° – 210° = -sin 30° = -1/2. Practising this process quickly builds fluency for exam questions.
例如，要求 sin 210°，注意到 210° 在第三象限，α = 210° – 180° = 30°。第三象限正弦为负，所以 sin 210° = -sin 30° = -1/2。练习这个过程能让你在考试中快速建立流利度。
5. Trigonometric Identities and Quadrants | 三角恒等式与象限
The two fundamental identities, sin²θ + cos²θ ≡ 1 and tan θ ≡ wn. For instance, if sin θ = 3/5 and θ is in Quadrant II, then cos θ must be negative, and using the identity gives cos θ = -√(1 – (3/5)²) = -4/5.
两个基本恒等式 sin²θ + cos²θ ≡ 1 和 tan θ ≡ sin θ / cos θ 对所有角度都成立，而不仅仅是锐角。解题时，如果已知其中一个，你可以利用象限确定 sin θ 或 cos θ 的符号。例如，如果 sin θ = 3/5 且 θ 在第二象限，那么 cos θ 必须为负，利用恒等式得到 cos θ = -√(1 – (3/5)²) = -4/5。
ities, such as proving sin(θ) / cos(θ) = tan(θ) or simplifying expressions like cos²θ / (1 – sin θ). Always consider the quadrant to handle absolute values correctly.
这种方法也适用于更复杂的恒等式，例如证明 sin(θ) / cos(θ) = tan(θ) 或简化 cos²θ / (1 – sin θ) 等表达式。始终考虑象限以正确处理绝对值。
6. Negative Angles and Symmetry | 负角与对称性
tive angles using symmetry: sin(-θ) = -sin θ (odd function), cos(-θ) = cos θ (even function), tan(-θ) = -tan θ. These identities are consistent with the unit circle: a negative angle places the point in the corresponding quadrant below the x-axis.
负角是从正 x 轴开始顺时针测量的。三角函数的负角可以用正角通过对称性表达：sin(-θ) = -sin θ（奇函数），cos(-θ) = cos θ（偶函数），tan(-θ) = -tan θ。这些恒等式与单位圆一致：负角将点置于 x 轴下方的相应象限。
may be asked to evaluate something like sin(-60°). You can write sin(-60°) = -sin 60° = -√3/2. This skill helps when solving equations with negative domains or transforming graphs.
在考试中，你可能需要计算类似 sin(-60°) 的值。你可以写 sin(-60°) = -sin 60° = -√3/2。这个技能在求解负定义域方程或变换图像时很有帮助。
7. General Solutions for Trigonometric Equations | 三角方程的通解
Trigonometric equations often have infinitely many solutions due to the periodic nature of the functions. The general solutions for sine, cosine, and tangent are:
由于函数的周期性，三角方程通常有无穷多个解。正弦、余弦和正切的通解为：
- sin θ = k → θ = nπ + (-1)ⁿ α (in radians) or θ = 180°n + (-1)ⁿ α (in degrees), where n ∈ Z.
- cos θ = k → θ = 2nπ ± α (rad) or θ = 360°n ± α (deg).
- tan θ = k → θ = nπ + α (rad) or θ = 180°n + α (deg), because tan has a period of π (180°).
Here α is the related acute angle (or principal value for the inverse function). These formulas generate all possible solutions by adding multiples of the period. Understanding their derivation from the unit circle is more important than memorising them.
这里 α 是相关锐角（或反函数的主值）。这些公式通过加上周期的整数倍产生所有可能的解。理解它们从单位圆中的推导比记忆更重要。
8. Solving Equations in a Given Interval | 在给定区间内解方程
Typical A-Level problems ask: 'Solve 2 sin θ = 1 for 0° ≤ θ ≤ 360°.' d acute angle α = 30°. Sin is positive in Quadrants I and II. So θ = 30° (QI) and θ = 180° – xtended to -180° ≤ θ ≤ 180°, you would include negative coterminal angles or use symmetry.
典型的 A-Level 题目要求：'解 2 sin θ = 1，0° ≤ θ ≤ 360°。' 首先，整理得 sin θ = 1/2。相关锐角 α = 30°。正弦在第一象限和第二象限为正。所以 θ = 30°（第一象限）和 θ = 180° – 30° = 150°（第二象限）。如果区间扩展到 -180° ≤ θ ≤ 180°，你需要包括负的同终边角或使用对称性。
e bracket first, then adjust for θ. Let the bracket equal the appropriate general solution before isolating θ.
始终绘制单位圆或 CAST 图以避免遗漏解。当处理变换角如 sin(2θ + 30°) = 0.5 时，先对括号求解，然后调整得到 θ。在分离 θ 之前，让括号等于相应的通解。
9. Radians and Four Quadrants | 弧度与四个象限
V (3π/2 to 2π). The CAST diagram works identically. Related acute angle α is found using π – θ, θ – π, 2π – θ. Exact values in radians are essential: sin(π/6)=1/2, cos(π/4)=1/√2, tan(π/3)=√3.
在 A-Level 阶段，你必须熟练使用弧度。弧度的象限为：第一象限（0 到 π/2），第二象限（π/2 到 π），第三象限（π 到 3π/2），第四象限（3π/2 到 2π）。CAST 图同样适用。相关锐角 α 使用 π – θ、θ – π、2π – θ 来计算。记住弧度下的精确值至关重要：sin(π/6)=1/2，cos(π/4)=1/√2，tan(π/3)=√3。
tions like cos θ = -1/2 for 0 ≤ θ < 2π, α = π/3. Cos is negative in QII and QIII. So θ = π – π/3 = 4π/3. Radians often simplify calculus work later, so practising fluency now is wise.
当解像 cos θ = -1/2 且 0 ≤ θ < 2π 的方程时，α = π/3。余弦在第二、三象限为负。所以 θ = π – π/3 = 2π/3，以及 θ = π + π/3 = 4π/3。弧度通常能简化之后的微积分计算，所以尽早练习熟练度是明智的。
10. Trigonometric Functions Beyond 360° and Periodic Nature | 超过 360° 的三角函数与周期性
80°) = tan θ. Understanding periodicity helps you to simplify angles greater than 360° by subtracting multiples of the period until the angle lies in the primary range 0°–360°.
正弦和余弦的周期为 360° (2π)，而正切的周期为 180° (π)。这意味着 sin(θ + 360°) = sin θ，cos(θ + 360°) = cos θ，tan(θ + 180°) = tan θ。理解周期性有助于通过减去周期的倍数简化大于 360° 的角，直到角落在 0°–360° 的主范围内。
is idea is crucial when handling general solutions or real-world contexts like rotating machinery.
例如，求 cos 480°，减去 360° 得到 120°。120° 在第二象限，α = 60°，所以 cos 120° = -cos 60° = -1/2。这个思路在处理通解或现实情境（如旋转机械）时至关重要。
11. Graphs of Sine, Cosine, and Tangent Across Quadrants | 各象限的正弦、余弦、正切图像
ction values change sign and magnitude across the four quadrants. The sine graph starts at 0, rises to 1 at 90°, falls to 0 at 180°, becomes -1 at 270°, and returns to 0 at 360°. The cosine graph starts at 1, drops to 0 at 90°, -1 at 180°, 0 at 270°, and 1 at 360°. The tangent graph has asymptotes at 90° and 270°, with a period of 180°.
绘制 y = sin θ、y = cos θ 和 y = tan θ 的图像可以突出显示函数值在四个象限中如何改变符号和大小。正弦图像从 0 开始，在 90° 上升到 1，在 180° 下降到 0，在 270° 变为 -1，在 360° 回到 0。余弦图像从 1 开始，在 90° 降到 0，在 180° 到 -1，在 270° 到 0，在 360° 回到 1。正切图像在 90° 和 270° 有渐近线，周期为 180°。
These graphs provide a visual confirmation of CAST. For instance, sin θ is positive where the graph is above the θ-axis (QI and QII), and negative below (QIII, QIV). Being able to sketch these quickly will support your equation-solving and transformation work.
这些图像为 CAST 提供了视觉确认。例如，sin θ 在图像位于 θ 轴上方时为正（第一、二象限），在下方的第三、四象限为负。能够快速绘制这些图像将有助于你解方程和进行图像变换。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
sin 2θ, the interval becomes 0° ≤ 2θ ≤ 720°, so you must find all solutions for 2θ in that extended domain before halving. Another mistake is misapplying CAST signs, especially for negative angles or when using calculators that give only principal values.
一个典型错误是在解涉及 2θ 或 (θ + 30°) 函数的方程时忘记调整定义域。例如，如果 0° ≤ θ ≤ 360°，那么对于 sin 2θ，区间变为 0° ≤ 2θ ≤ 720°，因此必须先在该扩展定义域内找到所有的 2θ 的解，然后再除以 2。另一个错误是误用 CAST 符号，尤其是对于负角或只给出主值的计算器。
Always draw a quick sketch, label the quadrants, and write the CAST letters. Use exact values wherever possible, and rationalise denominators if required. Check your solutions by plugging them back into the original equation; this catches extraneous solutions from squaring or algebraic manipulation.
始终画个简图，标注象限，写下 CAST 字母。尽可能使用精确值，需要时分母有理化。将解代回原方程检查；这能捕捉到因平方或代数操作产生的外来解。
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