📚 Mastering IGCSE Mathematics: Exact Values of Trigonometric Ratios | IGCSE数学精讲：三角比精确值
In IGCSE Mathematics, you will often evaluate the sine, cosine and tangent of key angles such as 30°, 45° and 60°. Instead of giving decimal answers, you are expected to write exact values using square roots. This skill is essential for solving many trigonometry problems without a calculator.
在 IGCSE 数学中，你经常需要计算 30°、45°、60° 等重要角度的正弦、余弦和正切值。与给出小数答案不同，你要学会用根号表示精确值。这项技能对于在不使用计算器的情况下解决许多三角学问题至关重要。
1. What Are Exact Values? | 什么是精确值？
An exact value is a number written in its precise mathematical form, such as √2/2 or √3/2, rather than an approximate decimal like 0.7071 or 0.8660.
精确值是以精确数学形式表示的数，例如 √2/2 或 √3/2，而不是如 0.7071 或 0.8660 这样的近似小数。
You must be comfortable converting between these forms because examiners award full marks only for exact answers in certain questions.
你必须熟练掌握这些形式之间的转换，因为在某些题目中，只有写出精确答案才能获得满分。
2. The Key Angles | 关键角度
The angles you must memorise are 0°, 30°, 45°, 60° and 90°. These appear constantly in triangles, graphs and identities.
你必须牢记的角度是 0°、30°、45°、60° 和 90°。它们频繁出现在三角形、图像和恒等式中。
30° and 60° come from cutting an equilateral triangle in half, giving a right-angled triangle with side ratio 1 : √3 : 2.
30° 和 60° 来自将一个等边三角形对半切开，得到一个边长比为 1 : √3 : 2 的直角三角形。
45° comes from an isosceles right triangle, with side ratio 1 : 1 : √2.
45° 来自一个等腰直角三角形，其边长比为 1 : 1 : √2。
0° and 90° are the boundary angles that you must remember separately.
0° 和 90° 是必须单独记住的边界角度。
3. Special Triangles | 特殊三角形
All exact values for 30°, 45° and 60° can be derived from two special right-angled triangles.
30°、45° 和 60° 的所有精确值都可以从两个特殊的直角三角形推导出来。
| Triangle | Side opposite smaller angle | Side opposite larger angle | Hypotenuse |
|---|---|---|---|
| 30° - 60° - 90° | 1 (opposite 30°) | √3 (opposite 60°) | 2 |
| 45° - 45° - 90° | 1 | 1 | √2 |
in θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse gives all the key exact values.
利用 sin θ = 对边/斜边 和 cos θ = 邻边/斜边 即可得到所有关键精确值。
4. Exact Values Table | 精确值表
The following table summarises the exact values you must know for IGCSE.
下表总结了 IGCSE 中你必须掌握的精确值。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Notice that √3/3 is the same as 1/√3, but √3/3 is the rationalised form preferred in exams.
注意 √3/3 与 1/√3 相同，但 √3/3 是考试中更受欢迎的有理化形式。
5. Unit Circle and Quadrants | 单位圆与象限
The unit circle helps you determine the sign of each trigonometric ratio in different quadrants. The mnemonic ASTC tells you which ratios are positive.
单位圆帮助你判断不同象限中各三角比的正负号。口诀 ASTC 告诉你哪些比值是正的。
Quadrant I (0° to 90°): All ratios are positive.
第一象限（0° 到 90°）：所有比值均为正。
Quadrant II (90° to 180°): Only sin is positive.
第二象限（90° 到 180°）：只有 sin 为正。
Quadrant III (180° to 270°): Only tan is positive.
第三象限（180° 到 270°）：只有 tan 为正。
Quadrant IV (270° to 360°): Only cos is positive.
第四象限（270° 到 360°）：只有 cos 为正。
positive, while cos 120° is negative. The reference angle is 60°, so sin 120° = sin 60° = √3/2 and cos 120° = -cos 60° = -1/2.
例如，sin 120° 为正，而 cos 120° 为负。参考角为 60°，因此 sin 120° = sin 60° = √3/2，cos 120° = -cos 60° = -1/2。
6. Related Angles Beyond 90° | 90° 以上的相关角
By using reference angles and the ASTC rule, you can find exact values for many angles between 0° and 360°.
通过使用参考角和 ASTC 规则，你可以求出 0° 到 360° 之间许多角度的精确值。
| Angle | Quadrant | sin | cos | tan |
|---|---|---|---|---|
| 120° | II | √3/2 | -1/2 | -√3 |
| 135° | II | √2/2 | -√2/2 | -1 |
| 150° | II | 1/2 | -√3/2 | -√3/3 |
| 180° | on x-axis | 0 | -1 | 0 |
| 210° | III | -1/2 | -√3/2 | √3/3 |
| 225° | III | -√2/2 | -√2/2 | 1 |
| 240° | III | -√3/2 | -1/2 | √3 |
| 270° | on y-axis | -1 | 0 | undefined |
| 300° | IV | -√3/2 | 1/2 | -√3 |
| 315° | IV | -√2/2 | √2/2 | -1 |
| 330° | IV | -1/2 | √3/2 | -√3/3 |
| 360° | on x-axis | 0 | 1 | 0 |
You are not expected to memorise every row, but you must know how to find any row using the reference angle and the correct sign.
你不一定需要记住每一行，但必须知道如何通过参考角和正确的符号求出任意一行。
7. Using Exact Values in Equations | 在方程中使用精确值
When solving trigonometric equations, exact values help you identify all solutions in a given interval.
在解三角方程时，精确值帮助你找出给定区间内的所有解。
Example: Solve sin θ = √2/2 for 0° ≤ θ < 360°.
示例：解方程 sin θ = √2/2，其中 0° ≤ θ < 360°。
two solutions share the reference angle 45°. Therefore θ = 45° and θ = 180° - 45° = 135°.
由于 sin 在第一象限和第二象限为正，两个解共享参考角 45°。因此 θ = 45°，θ = 180° - 45° = 135°。
θ = 45°, 135°
8. Identities and Verifications | 恒等式与验证
Exact values allow you to verify fundamental trigonometric identities.
精确值使你能够验证基本的三角恒等式。
For θ = 45°, verify that sin²θ + cos²θ = 1:
对于 θ = 45°，验证 sin²θ + cos²θ = 1：
(√2/2)² + (√2/2)² = 2/4 + 2/4 = 1/2 + 1/2 = 1
For θ = 30°, verify that tan θ = sin θ / cos θ:
对于 θ = 30°，验证 tan θ = sin θ / cos θ：
(1/2) ÷ (√3/2) = 1/√3 = √3/3
This matches the table value for tan 30°.
这与表中 tan 30° 的值一致。
9. Worked Examples | 例题精讲
Now let us apply exact values to composite expressions and triangle problems.
下面我们将精确值应用于复合表达式和三角形问题。
Example 1: Evaluate 3 sin 30° + 2 cos 60°.
示例 1：计算 3 sin 30° + 2 cos 60°。
3 × (1/2) + 2 × (1/2) = 3/2 + 1 = 5/2
Example 2: A right-angled triangle has an angle of 60° and a hypotenuse of 10 cm. Find the length of the side opposite the 60° angle.
示例 2：一个直角三角形有一个 60° 角，斜边长为 10 cm。求 60° 角所对边的长度。
x = 10 × sin 60° = 10 × (√3/2) = 5√3 cm
Always include the units and leave the answer in surd form unless told otherwise.
除非另有说明，否则始终写上单位并以根号形式保留答案。
10. Common Mistakes | 常见错误
Avoid these frequent errors to gain full marks.
避免这些常见错误，才能获得满分。
Writing 1/√2 instead of √2/2. Always rationalise the denominator.
把 1/√2 写成 √2/2。一定要对分母进行有理化。
Forgetting the sign in Quadrants II, III and IV. Use ASTC.
忘记第二、三、四象限中的正负号。请使用 ASTC。
Saying tan 90° is infinity. The correct answer is that it is undefined.
说 tan 90° 是无穷大。正确的答案是没有定义。
Confusing sin 30° = 1/2 with sin 60° = √3/2. Draw the special triangles.
混淆 sin 30° = 1/2 和 sin 60° = √3/2。画出特殊三角形。
11. Practice Questions | 练习
Try these questions and check your answers below.
试试以下问题，并在下方核对答案。
1. Write down the exact value of sin 60°, cos 45° and tan 30°.
1. 写出 sin 60°、cos 45° 和 tan 30° 的精确值。
2. Solve sin θ = √3/2 for 0° ≤ θ < 360°.
2. 解方程 sin θ = √3/2，其中 0° ≤ θ < 360°。
3. Evaluate 4 tan 45° - 2 sin 90°.
3. 计算 4 tan 45° - 2 sin 90°。
Answers: 1. √3/2, √2/2, √3/3; 2. 60° and 120°; 3. 4(1) - 2(1) = 2.
答案：1. √3/2、√2/2、√3/3；2. 60° 和 120°；3. 4(1) - 2(1) = 2。
12. Summary | 总结
Mastering exact values means memorising the table for 0°, 30°, 45°, 60° and 90°, understanding the two special triangles, and applying the ASTC rule to extend these values to all four quadrants.
掌握精确值意味着熟记 0°、30°、45°、60° 和 90° 的表格，理解两个特殊三角形，并运用 ASTC 规则将这些值推广到四个象限。
With regular practice, these values will become automatic, and you will solve trigonometry questions faster and more accurately.
通过定期练习，这些值将成为你的本能反应，你将更快、更准确地解决三角学问题。
Published by TutorHao | Mathematics Revision Series | alevel
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