Exact Values of Trigonometric Ratios | 三角函数比的精确值

📚 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。

Using the definitions sin θ = opposite/hypotenuse, cos θ = 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

On the unit circle x² + y² = 1, an angle θ 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。

At θ = 0°, the point is (1, 0); at θ = 90°, it is (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。

For example, cos 120° is negative, 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。

Similarly, 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

The reciprocals secant, cosecant, and cotangent are defined as sec θ = 1/cos θ, csc θ = 1/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

The most important identity is sin² θ + cos² θ = 1. It holds for 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。

Dividing 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

To solve sin θ = ½ for 0° ≤ θ < 360°, identify 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°。

For tan θ = 1, the reference 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. Remember 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。

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Exact Values of Trigonometric Ratios | 三角函数比的精确值