📚 Exact Trigonometric Ratios of Special Angles | 特殊角三角函数比的精确值
Exact trigonometric ratios for the special angles 30°, 45° and 60° are a fundamental tool in Edexcel A-Level Mathematics. You will meet them in calculus, coordinate geometry, parametric equations and solved problems in mechanics. Examiners frequently ask for answers in exact surd form, so memorising these values is not enough: you need to understand where they come from and how to apply them confidently.
在 Edexcel A-Level 数学中，30°、45°、60° 等特殊角的三角函数精确值是基础中的基础。它们会出现在微积分、解析几何、参数方程以及力学相关题目中。考官经常要求以精确根式形式作答，因此仅仅记住这些数值还不够：你需要理解它们的来源，并能够自信地运用。
1. Why Exact Values Matter | 为什么需要精确值
When a question says “without a calculator” or “give your answer in exact form”, a rounded decimal will lose marks. Exact values such as √2/2 and √3/2 appear inside trigonometric identities, integration answers and vector calculations. Using exact values also helps you spot simplifications and cancellations that are invisible in decimal approximations.
当题目要求“不使用计算器”或“以精确形式作答”时，写成近似小数会被扣分。像 √2/2、√3/2 这样的精确值经常出现在三角恒等式、积分结果和向量运算中。使用精确值还能帮助你发现小数近似中难以察觉的化简与约分机会。
In addition, many A-Level trigonometry questions require solving equations such as sin θ = 1/2 or cos θ = √3/2 over a given interval. Without the exact values recorded in this article, you will have to pause and use a calculator, which wastes time and can introduce errors.
此外，许多 A-Level 三角方程题要求在给定区间内求解，例如 sin θ = 1/2 或 cos θ = √3/2。如果不掌握本文总结的精确值，你就不得不停下来使用计算器，既浪费时间又容易出错。
2. Deriving 45° from an Isosceles Right Triangle | 由等腰直角三角形推导 45°
Start with a right-angled isosceles triangle whose two legs are both of length 1. By Pythagoras’ theorem, the hypotenuse is √(1² + 1²) = √2. Since the two acute angles are equal, each is 45°.
从一个两条直角边都为 1 的等腰直角三角形出发。根据勾股定理，斜边为 √(1² + 1²) = √2。由于两个锐角相等，每个角都是 45°。
sin 45° = opposite/hypotenuse = 1/√2 = √2/2
cos 45° = adjacent/hypotenuse = 1/√2 = √2/2
tan 45° = opposite/adjacent = 1/1 = 1
Notice that sin 45° and cos 45° are equal because the triangle is isosceles. The value 1/√2 is often written as √2/2 after rationalising the denominator. Both forms are correct, but the rationalised form is preferred in final answers.
注意因为三角形等腰，sin 45° 与 cos 45° 相等。1/√2 通常通分成 √2/2，即对分母有理化。两种写法都正确，但最终答案更推荐使用有理化形式。
3. Deriving 30° and 60° from an Equilateral Triangle | 由等边三角形推导 30° 与 60°
Draw an equilateral triangle of side length 2 and drop a perpendicular from one vertex to the midpoint of the opposite side. The perpendicular bisects the apex angle, so the upper angle becomes 30°, while the base angle remains 60°. The vertical height is √3 by Pythagoras’ theorem.
画一个边长为 2 的等边三角形，并从顶点向对边中点作垂线。垂线平分顶角，因此上方角为 30°，底角仍为 60°。由勾股定理可得高为 √3。
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3
sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
Comparing the two angles, sin 30° equals cos 60°, and cos 30° equals sin 60°. This is not a coincidence: in general, sin(90° – θ) = cos θ. The tangent values show a clear contrast: tan 30° = 1/√3, while tan 60° = √3.
对比两个角可以发现，sin 30° 等于 cos 60°，而 cos 30° 等于 sin 60°。这不是偶然：一般地，sin(90° – θ) = cos θ。正切值则形成鲜明对比：tan 30° = 1/√3，而 tan 60° = √3。
4. The Exact Values Table | 特殊角精确值速查表
The six angles 0°, 30°, 45°, 60° and 90° are collectively referred to as “special angles”. Their exact trigonometric ratios are summarised in the table below.
0°、30°、45°、60° 和 90° 这六个角统称为“特殊角”。它们的三角函数精确值汇总如下表。
| Angle θ / 角度 θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined（无定义） |
You also need to know these values in radians when studying trigonometry later. The conversions are 30° = π/6, 45° = π/4 and 60° = π/3. The exact ratios are identical.
后续学习三角学时，你还需要知道这些角的弧度制形式：30° = π/6，45° = π/4，60° = π/3。对应的精确比值完全相同。
5. The Unit Circle and Extension to 0° and 90° | 单位圆与 0°、90° 的延拓
The unit circle provides a powerful way to define trigonometric functions for all angles. On a circle of radius 1 centred at the origin, the coordinates of a point at angle θ measured anticlockwise from the positive x-axis are (cos θ, sin θ).
单位圆为定义任意角的三角函数提供了有力工具。在圆心位于原点、半径为 1 的圆上，从 x 轴正方向逆时针旋转角度 θ 所对应的点坐标为 (cos θ, sin θ)。
At 0°, the point is (1, 0), so cos 0° = 1 and sin 0° = 0. At 90°, the point is (0, 1), so cos 90° = 0 and sin 90° = 1. At 180° the point is (-1, 0), giving cos 180° = -1 and sin 180° = 0. These values complete the special angle table.
在 0° 时，点坐标为 (1, 0)，因此 cos 0° = 1，sin 0° = 0。在 90° 时，点坐标为 (0, 1)，因此 cos 90° = 0，sin 90° = 1。在 180° 时，点坐标为 (-1, 0)，所以 cos 180° = -1，sin 180° = 0。这些数值补全了特殊角表格。
The unit circle also makes the signs of the ratios clear. In the first quadrant, both sin and cos are positive; in the second, sin is positive but cos is negative; in the third, both are negative; in the fourth, cos is positive but sin is negative.
单位圆还清楚说明了三角函数的符号规律。第一象限中，sin 和 cos 均为正；第二象限中，sin 为正而 cos 为负；第三象限中，两者均为负；第四象限中，cos 为正而 sin 为负。
6. Reference Angles in Other Quadrants | 其他象限中的参考角
Exact values can be extended to angles outside the range 0° to 90° using reference angles. The reference angle is the acute angle that a given angle makes with the x-axis. For example, 120° has a reference angle of 180° – 120° = 60°.
借助参考角，精确值可以推广到 0° 到 90° 之外的角。参考角是指给定角与 x 轴之间所夹的锐角。例如，120° 的参考角为 180° – 120° = 60°。
sin 120° = + sin 60° = √3/2
cos 120° = – cos 60° = -1/2
tan 120° = – tan 60° = -√3
Similarly, 210° has a reference angle of 30° and lies in the third quadrant, so its sine and cosine are both negative: sin 210° = -1/2 and cos 210° = -√3/2. An angle of 315° has a reference angle of 45° and lies in the fourth quadrant, so tan 315° = -1.
同理，210° 的参考角为 30°，且位于第三象限，因此其正弦和余弦均为负：sin 210° = -1/2，cos 210° = -√3/2。315° 的参考角为 45°，位于第四象限，因此 tan 315° = -1。
7. Memory Aids for the Table | 速记技巧
One reliable memory aid is to write the sine values for 0°, 30°, 45°, 60° and 90° as square roots over 2:
一个可靠的速记技巧是：将 0°、30°、45°、60°、90° 的正弦值写成带根号的分数，分母为 2：
sin θ: √0/2, √1/2, √2/2, √3/2, √4/2
This simplifies to 0, 1/2, √2/2, √3/2 and 1. For cosine, simply reverse the order. For tangent, use tan θ = sin θ / cos θ.
这依次化简为 0、1/2、√2/2、√3/2 和 1。对于余弦，只需将顺序倒过来。对于正切，使用 tan θ = sin θ / cos θ 计算即可。
Many students also use the CAST diagram or the phrase “All Silly Turtles Crawl” to remember which functions are positive in each quadrant: All positive in Quadrant I, Sin positive in Quadrant II, Tan positive in Quadrant III, Cos positive in Quadrant IV.
很多同学还会借助 CAST 图或口诀“All Silly Turtles Crawl”来记忆每个象限中哪些函数为正：第一象限“All”全部为正，第二象限“Sin”正弦为正，第三象限“Tan”正切为正，第四象限“Cos”余弦为正。
8. Worked Examples | 典型例题精讲
Worked Example 1: Evaluate sin²45° + cos 60° without a calculator.
例题 1：不使用计算器，求 sin²45° + cos 60° 的值。
From the table, sin 45° = √2/2, so sin²45° = (√2/2)² = 1/2. Also cos 60° = 1/2. Therefore the sum is 1/2 + 1/2 = 1.
由表格可知 sin 45° = √2/2，所以 sin²45° = (√2/2)² = 1/2。又 cos 60° = 1/2。因此和为 1/2 + 1/2 = 1。
sin²45° + cos 60° = 1
Worked Example 2: Solve 2cos θ + √3 = 0 for 0° ≤ θ < 360°.
例题 2：在 0° ≤ θ < 360° 内解方程 2cos θ + √3 = 0。
Rearrange to cos θ = -√3/2. The reference angle is 30° because cos 30° = √3/2. Since cosine is negative in the second and third quadrants, the solutions are θ = 180° – 30° = 150° and θ = 180° + 30° = 210°.
整理得 cos θ = -√3/2。由于 cos 30° = √3/2，参考角为 30°。余弦在第二、第三象限为负，因此解为 θ = 180° – 30° = 150° 和 θ = 180° + 30° = 210°。
θ = 150° 或 θ = 210°
9. Common Mistakes and How to Avoid Them | 常见错误与规避方法
A very common error is writing sin 30° = √3/2 and sin 60° = 1/2. To avoid this, remember the symmetry: sin 30° = cos 60° = 1/2, and sin 60° = cos 30° = √3/2.
一个非常常见的错误是把 sin 30° 写成 √3/2，把 sin 60° 写成 1/2。为避免混淆，请记住对称关系：sin 30° = cos 60° = 1/2，sin 60° = cos 30° = √3/2。
Another common mistake is leaving 1/√3 instead of rationalising to √3/3. Many examiners accept both, but the rationalised form is safer. Also, when solving equations, forgetting the negative sign in the second or third quadrant leads to missed solutions.
另一个常见问题是写作 1/√3 而未有理化为 √3/3。多数考官两种形式都会接受，但有理化形式更稳妥。另外，在解方程时，忘记第二或第三象限中的负号会导致漏解。
Finally, be careful with 45°: tan 45° = 1, not √2/2. The square root value √2/2 belongs to both sin 45° and cos 45°. Keep the three functions for 45° clearly separated in your mind.
最后，请特别注意 45°：tan 45° = 1，而不是 √2/2。√2/2 属于 sin 45° 和 cos 45°。请将 45° 的三个函数值清晰地分开记忆。
10. Final Revision Checklist | 考前复习清单
Before your exam, make sure you can complete all of the following: derive sin, cos and tan for 45° from an isosceles right triangle; derive the 30° and 60° values from an equilateral triangle; recall the exact value table without hesitation; use the unit circle to extend values to 0°, 90°, 180°, 270°; solve simple equations using reference angles; and rationalise surd denominators in final answers.
考试之前，请确保自己能完成以下全部任务：从等腰直角三角形推导 45° 的正弦、余弦、正切；从等边三角形推导 30° 和 60° 的精确值；毫不迟疑地默写出特殊角精确值表；利用单位圆将精确值拓展到 0°、90°、180°、270°；使用参考角解简单三角方程；在最终答案中对根式分母进行有理化。
With regular practice, these values become automatic. Because they appear in almost every trigonometry, calculus and geometry topic at A-Level, investing time in mastering them will pay off across your whole examination.
通过定期练习，这些数值会逐渐变成条件反射。由于特殊角精确值几乎出现在 A-Level 的三角学、微积分和几何等所有相关专题中，投入时间掌握它们，将为你整场考试带来丰厚回报。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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