📚 Exact Values of Trigonometric Ratios | 三角函数比的精确值
The exact values of sine, cosine, and tangent for special angles are fundamental to A-Level Mathematics. They allow you to evaluate expressions without a calculator, solve trigonometric equations exactly, and simplify results in calculus and geometry.
正弦、余弦和正切在特殊角处的精确值是A-Level数学的基础。它们使你无需计算器就能求值，精确地解三角方程，并在微积分和几何中简化结果。
1. The Special Angles | 特殊角
In this topic, the special angles are 0°, 30°, 45°, 60°, and 90°. These angles appear repeatedly in problems, and their exact trigonometric values must be memorised.
本节中的特殊角为0°、30°、45°、60°和90°。这些角在题目中反复出现，其精确三角函数值必须牢记。
sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1
cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½, cos 90° = 0
tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° is undefined
Some textbooks also include 180°, 270°, and 360° because they are directly related to 0° and 90° through symmetry.
部分教材也包含180°、270°和360°，因为它们可通过对称性与0°和90°直接联系。
2. Derivation from a Right Triangle | 从直角三角形推导
For 30° and 60°, consider an equilateral triangle with side length 2. When it is cut in half, the resulting right triangle has angles 30°, 60°, and 90°, with sides 1, √3, and 2.
对于30°和60°，考虑边长为2的等边三角形。从中间对半切开，得到的直角三角形角度为30°、60°和90°，边长分别为1、√3和2。
= adjacent/hypotenuse, and tan θ = opposite/adjacent, we obtain the exact ratios directly.
利用定义sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边，我们可直接得到精确比值。
For 45°, take a right isosceles triangle with legs 1 and 1. Its hypotenuse is √2, so sin 45° = 1/√2 = √2/2 and similarly for cos 45°. The tangent is 1.
对于45°，取直角边为1和1的等腰直角三角形。其斜边为√2，因此sin 45° = 1/√2 = √2/2，cos 45°同理。正切值为1。
For 30°: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3
For 60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
3. The Unit Circle Definition | 单位圆定义
measured anticlockwise from the positive x-axis gives the coordinates (cos θ, sin θ) of the point on the circle.
在单位圆x² + y² = 1上，从正x轴逆时针测量的角θ给出圆上点的坐标(cos θ, sin θ)。
This definition extends the trigonometric ratios beyond 90° and explains why sin θ and cos θ have ranges between -1 and 1.
该定义将三角函数比扩展到90°以外，并解释了sin θ和cos θ的取值范围在-1和1之间。
(0, 1). These give the exact values cos 0° = 1, sin 0° = 0, cos 90° = 0, and sin 90° = 1.
当θ = 0°时，点为(1, 0)；当θ = 90°时，点为(0, 1)。由此得到cos 0° = 1，sin 0° = 0，cos 90° = 0，sin 90° = 1。
4. Quadrant Signs | 象限符号
The ASTC rule (All, Sine, Tangent, Cosine) tells which ratios are positive in each quadrant: all in Quadrant I, sine in Quadrant II, tangent in Quadrant III, and cosine in Quadrant IV.
ASTC规则（全、正弦、正切、余弦）表明每个象限中哪些比值为正：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。
e, while sin 120° is positive. Using the reference angle 60°, we have sin 120° = sin 60° = √3/2 and cos 120° = -cos 60° = -½.
例如，cos 120°为负，而sin 120°为正。利用参考角60°，可得sin 120° = sin 60° = √3/2，cos 120° = -cos 60° = -½。
Knowing the sign together with the exact value of the reference angle lets you write exact values for angles such as 150°, 210°, and 330°.
知道符号并结合参考角的精确值，你就能写出150°、210°和330°等角的精确值。
5. Related Angles | 关联角
For any angle θ, the reference angle is the acute angle between the terminal side and the x-axis. For 150°, the reference angle is 30°.
对于任意角θ，参考角是终边与x轴之间的锐角。对于150°，参考角是30°。
Therefore, sin 150° = sin 30° = ½, but cos 150° = -cos 30° = -√3/2, and tan 150° = -tan 30° = -√3/3.
因此，sin 150° = sin 30° = ½，但cos 150° = -cos 30° = -√3/2，tan 150° = -tan 30° = -√3/3。
y, 210° has reference angle 30° but lies in Quadrant III. Thus sin 210° = -½, cos 210° = -√3/2, and tan 210° = √3/3.
类似地，210°的参考角为30°，但位于第三象限。因此sin 210° = -½，cos 210° = -√3/2，tan 210° = √3/3。
sin(180° - θ) = sin θ, cos(180° - θ) = -cos θ, tan(180° - θ) = -tan θ
sin(180° + θ) = -sin θ, cos(180° + θ) = -cos θ, tan(180° + θ) = tan θ
6. Exact Values of Reciprocal Ratios | 倒数三角函数比的精确值
sin θ, and cot θ = 1/tan θ. Their exact values follow immediately from the special angles.
倒数函数正割、余割和余切定义为sec θ = 1/cos θ，csc θ = 1/sin θ，cot θ = 1/tan θ。它们的精确值直接从特殊角得出。
For example, sec 45° = 1/(√2/2) = √2, csc 30° = 1/(½) = 2, and cot 60° = 1/√3 = √3/3.
例如，sec 45° = 1/(√2/2) = √2，csc 30° = 1/(½) = 2，cot 60° = 1/√3 = √3/3。
| Angle | sec | csc | cot |
| 30° | 2√3/3 | 2 | √3 |
| 45° | √2 | √2 | 1 |
| 60° | 2 | 2√3/3 | √3/3 |
7. Fundamental Identities | 基本恒等式
r all angles and provides a way to find one ratio from another when the quadrant is known.
最重要的恒等式是sin² θ + cos² θ = 1。它对所有角成立，并在已知象限时提供从一种比值求另一种比值的方法。
ing by cos² θ gives 1 + tan² θ = sec² θ, and dividing by sin² θ gives 1 + cot² θ = csc² θ.
除以cos² θ得到1 + tan² θ = sec² θ，除以sin² θ得到1 + cot² θ = csc² θ。
For example, given sin θ = ½ and θ in Quadrant I, cos θ = √1 - ¼ = √3/2. This matches the exact value for 30°.
例如，已知sin θ = ½且θ在第一象限，则cos θ = √1 - ¼ = √3/2。这与30°的精确值一致。
8. Exact Values of 15° and 75° | 15°和75°的精确值
Using the sum and difference formulas, we can derive exact values for 15° and 75°, which are not in the basic special-angle table.
利用和差公式，我们可以推导出15°和75°的精确值，它们不在基本特殊角表中。
sin(45° - 30°) = sin45°cos30° - cos45°sin30° = (√2/2)(√3/2) - (√2/2)(½) = (√6 - √2)/4
cos(45° - 30°) = cos45°cos30° + sin45°sin30° = (√2/2)(√3/2) + (√2/2)(½) = (√6 + √2)/4
Thus sin 15° = (√6 - √2)/4 and cos 15° = (√6 + √2)/4. For 75° = 45° + 30°, sin 75° = (√6 + √2)/4 and cos 75° = (√6 - √2)/4.
因此sin 15° = (√6 - √2)/4，cos 15° = (√6 + √2)/4。对于75° = 45° + 30°，sin 75° = (√6 + √2)/4，cos 75° = (√6 - √2)/4。
These values are occasionally requested in A-Level papers, particularly in questions involving compound angle identities.
这些值偶尔出现在A-Level试卷中，尤其是在涉及复合角恒等式的题目中。
9. Solving Equations with Exact Values | 用精确值解方程
y the reference angle 30° and apply the ASTC rule. The solutions are θ = 30° and θ = 150°.
在0° ≤ θ < 360°内解sin θ = ½，先确定参考角30°，再应用ASTC规则。解为θ = 30°和θ = 150°。
angle is 45°, and tangent is positive in Quadrants I and III. Hence θ = 45° and θ = 225°.
对于tan θ = 1，参考角为45°，正切在第一和第三象限为正。因此θ = 45°和θ = 225°。
When exact values are available, you can write solution sets without a calculator, which is often required in non-calculator papers.
当精确值可用时，你可以在不使用计算器的情况下写出解集，这在不允许使用计算器的试卷中通常是必需的。
10. Common Mistakes | 常见错误
A frequent error is writing tan 60° as 1/√3 instead of √3 emember that tan is increasing from 0° to 90°, so tan 60° must be larger than tan 45° = 1.
常见错误是把tan 60°写成1/√3而不是√3。记住正切在0°到90°之间递增，因此tan 60°必须大于tan 45° = 1。
Another mistake is forgetting signs in non-first-quadrant angles. For example, cos 120° is -½, not ½. Always check the quadrant before writing the final answer.
另一个错误是忘记非第一象限角的符号。例如，cos 120°是-½而不是½。在写最终答案前务必检查象限。
Students also confuse sin and cos values: sin 30° = ½ but cos 30° = √3/2. One useful mnemonic is that sine starts at 0 and increases, while cosine starts at 1 and decreases.
学生还会混淆sin和cos的值：sin 30° = ½但cos 30° = √3/2。一个有用的记忆法是正弦从0开始增大，而余弦从1开始减小。
11. Memory Table and Tips | 记忆表与技巧
The following table lists the sine and cosine exact values for the five basic angles. The pattern 0, 1, √2, √3, 2 divided by 2 appears in sine and can be remembered correctly.
下表列出五个基本角的正弦和余弦精确值。正弦值中0、1、√2、√3、2除以2的规律可以用来记忆。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | ½ | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | ½ | 0 |
For tangent, divide sine by cosine: tan 30° = (½)/(√3/2) = 1/√3 = √3/3.
对于正切，用正弦除以余弦：tan 30° = (½)/(√3/2) = 1/√3 = √3/3。
Practising these values through flashcards and sketching the unit circle will greatly improve your speed and accuracy in exams.
通过闪卡和画单位圆来练习这些值，将极大提高你在考试中的速度和准确性。
12. Summary | 总结
Exact values of trigonometric ratios for 0°, 30°, 45°, 60°, and 90° are essential. They arise from simple right triangles and the unit circle, and they extend to all angles using reference angles and the ASTC sign rule.
0°、30°、45°、60°和90°的三角函数精确值是必需的。它们来自简单的直角三角形和单位圆，并可借助参考角和ASTC符号规则扩展到所有角度。
Master the basic table, understand the derivation, and practise the identities and equation-solving techniques. These exact values will serve you throughout A-Level Mathematics and beyond.
掌握基本表格，理解推导过程，并练习恒等式和解方程技巧。这些精确值将在整个A-Level数学及以后的学习中为你服务。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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