📚 Exact Values of Trigonometric Ratios | 三角比的精确值
In A-level mathematics, exact values of trigonometric ratios for special angles are essential. They appear in equations, calculus, and proofs. Rather than entering sin(30°) into a calculator and obtaining 0.5, you are expected to know that sin30° = 1/2 exactly. This skill saves time and ensures exactness in final answers.
在 A-level 数学中，特殊角三角比的精确值是必备基础。它们出现在方程、微积分和证明中。你不应依赖计算器得到 sin(30°) ≈ 0.5，而应熟记 sin30° = 1/2。这项技能能节省时间并保证最终答案的精确性。
1. Why Exact Values Matter | 为什么精确值很重要
Using exact values avoids rounding errors. In multi-step problems, an approximate value can lead to incorrect final results. Exact values also link trigonometry with algebra and allow cancellation, factorisation, and simplification of expressions.
使用精确值可避免舍入误差。在多步骤问题中，近似值可能导致最终结果出错。精确值还将三角学与代数联系起来，便于约分、因式分解和化简表达式。
Examination mark schemes often award marks specifically for exact answers, not decimal approximations. For example, leaving an answer as √3/2 is preferred over 0.866. A calculator may be allowed, but exact knowledge demonstrates deeper understanding.
考试评分标准通常明确给“精确值”得分，而不接受小数近似。例如，保留 √3/2 优于 0.866。虽然考试可能允许使用计算器，但精确值更能体现对概念的理解。
2. The Special Angles | 特殊角
The special angles you must know are 0°, 30°, 45°, 60° and 90°. In radians, these are 0, π/6, π/4, π/3 and π/2. Their sine, cosine and tangent values can be summarised in one exact-value table.
需要熟记的特殊角为 0°、30°、45°、60° 和 90°。用弧度表示分别为 0、π/6、π/4、π/3 和 π/2。它们的正弦、余弦和正切值可用一张精确值表总结。
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
Notice the pattern in sine values: 0, 1/2, √2/2, √3/2, 1. Cosine is the reverse. This pattern helps you recall the table quickly.
注意正弦值的规律：0、1/2、√2/2、√3/2、1。余弦值正好相反。借助这个规律可以快速记住整个表。
3. Derivation from Special Triangles | 从特殊三角形推导
The 45-45-90 triangle is an isosceles right triangle. If both legs are 1, the hypotenuse is √2. Therefore sin45° = hypotenuse / opposite = 1/√2 = √2/2, and cos45° = 1/√2 = √2/2. The tangent is 1 because opposite and adjacent are equal.
45-45-90 三角形是等腰直角三角形。若两条直角边均为 1，则斜边为 √2。因此 sin45° = 对边/斜边 = 1/√2 = √2/2，cos45° = 1/√2 = √2/2。正切值为 1，因为对边等于邻边。
For 30° and 60°, consider an equilateral triangle with side length 2. Split it in half along an altitude. The shorter leg is 1, the longer leg is √3, and the hypotenuse is 2. Hence sin30° = 1/2, cos30° = √3/2, sin60° = √3/2 and cos60° = 1/2.
对于 30° 和 60°，考虑边长为 2 的等边三角形，沿高线对半切开。较短直角边为 1，较长直角边为 √3，斜边为 2。因此 sin30° = 1/2，cos30° = √3/2，sin60° = √3/2，cos60° = 1/2。
4. The Unit Circle | 单位圆
On a unit circle with radius 1, a point at angle θ from the positive x-axis has coordinates (cosθ, sinθ). This definition extends the trigonometric ratios to all real angles, including negative angles and angles greater than 360°.
在半径为 1 的单位圆上，从 x 轴正方向旋转角度 θ 所对应的点坐标为 (cosθ, sinθ)。这一定义将三角比推广到所有实数角度，包括负角和大于 360° 的角。
For example, at θ = π/2, the point is (0,1), so cosπ/2 = 0 and sinπ/2 = 1. At θ = π, the point is (-1,0), giving cosπ = -1 and sinπ = 0. These quadrantal angles are exact and must be memorised.
例如，在 θ = π/2 时，点为 (0,1)，所以 cosπ/2 = 0，sinπ/2 = 1。在 θ = π 时，点为 (-1,0)，因此 cosπ = -1，sinπ = 0。这些象限角也是精确值，必须熟记。
5. Reference Angles and the CAST Rule | 参考角与 CAST 规则
A reference angle is the acute angle between the terminal arm of an angle and the x-axis. For 150°, the reference angle is 30°. The CAST rule tells which trigonometric ratios are positive in each quadrant: All positive in QI, Sin positive in QII, Tan positive in QIII, Cos positive in QIV.
参考角是角的终边与 x 轴之间夹的锐角。例如 150° 的参考角是 30°。CAST 规则用于判断各象限中哪些三角比为正：第一象限全为正，第二象限正弦为正，第三象限正切为正，第四象限余弦为正。
Using this rule, sin150° = sin30° = 1/2, because sine is positive in QII. But cos150° = -cos30° = -√3/2, because cosine is negative in QII. The absolute value comes from the reference angle; the sign comes from the quadrant.
根据此规则，sin150° = sin30° = 1/2，因为正弦在第二象限为正；而 cos150° = -cos30° = -√3/2，因为余弦在第二象限为负。绝对值来自参考角，正负号来自象限。
6. Exact Values in All Quadrants | 所有象限中的精确值
By combining the reference angle with the CAST rule, you can find exact values for angles such as 120°, 135°, 150°, 210°, 225°, 240°, 300°, 315° and 330°. The following table shows some frequently tested examples.
将参考角与 CAST 规则结合，可以求出 120°、135°、150°、210°、225°、240°、300°、315°、330° 等角的精确值。下表列出一些常见的考点。
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 |
| 210° | 7π/6 | -1/2 | -√3/2 | √3/3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -√3/3 |
Always ask: what is the reference angle, and what sign does the quadrant impose? For example, sin240° has reference 60° and lies in QIII where sine is negative, so sin240° = -√3/2.
计算时永远问自己：参考角是多少？该象限的符号是什么？例如 sin240° 的参考角为 60°，且处于正弦为负的第三象限，因此 sin240° = -√3/2。
7. Exact Values for Tangent | 正切的精确值
Since tanθ = sinθ / cosθ, exact values of tangent follow directly from the sine and cosine table. For 30°, tan30° = (1/2) ÷ (√3/2) = 1/√3 = √3/3. For 45°, tan45° = (√2/2) ÷ (√2/2) = 1. For 60°, tan60° = (√3/2) ÷ (1/2) = √3.
因为 tanθ = sinθ / cosθ，正切的精确值可直接由正弦和余弦表求出。对于 30°，tan30° = (1/2) ÷ (√3/2) = 1/√3 = √3/3。对于 45°，tan45° = (√2/2) ÷ (√2/2) = 1。对于 60°，tan60° = (√3/2) ÷ (1/2) = √3。
At 0°, tan0° = 0/1 = 0. At 90°, cos90° = 0, so tan90° is undefined; the graph has a vertical asymptote there. Also note that tanθ has period 180° (π), so tan210° = tan30° = √3/3.
在 0°，tan0° = 0/1 = 0。在 90°，cos90° = 0，所以 tan90° 无定义，图像在该处有垂直渐近线。另外注意正切函数的周期为 180°（π），所以 tan210° = tan30° = √3/3。
8. Reciprocal Exact Values | 倒数三角比的精确值
The reciprocal ratios cosecant (csc), secant (sec) and cotangent (cot) also have
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