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

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

Trigonometry forms a cornerstone of A-Level Mathematics, and within this field, the ability to recall and apply exact values for sin, cos, and tan of key angles is a fundamental skill. Edexcel exam papers consistently test students on these values, not only in pure trigonometry questions but also within calculus, vectors, and complex number problems. Memorising and working fluently with the surd forms of trigonometric ratios for 30°, 45°, 60°, and related angles saves time, reduces rounding errors, and demonstrates true mathematical understanding.

三角函数是A-Level数学的基石，而在这门学科中，记住并应用关键角度下sin、cos和tan的精确值是一项基本技能。Edexcel历年试卷不仅会在纯三角函数题中考查这些值，也会在微积分、向量和复数问题中涉及。熟练记忆并运用30°、45°、60°及其相关角的根式形式的三角比，不仅能节省时间、减少舍入误差，更能体现真正的数学理解能力。

1. What Are Exact Values? | 什么是精确值？

In mathematics, an exact value is a representation that uses integers, fractions, and surds (such as √2, √3) rather than decimal approximations. For example, the exact value of sin 45° is 1/√2 or √2/2, not 0.7071... Edexcel mark schemes explicitly require answers to be given in exact form unless the question states otherwise. Leaving an answer as √3/2 rather than 0.866 is preferred and often essential for full marks.

在数学中，精确值是指用整数、分数和根式（如√2、√3）而非小数近似来表示的数值。例如，sin 45°的精确值是1/√2或√2/2，而不是0.7071...。Edexcel的评分标准明确要求在未特别说明的情况下，答案应以精确形式给出。将结果保留为√3/2而非0.866，往往是获得满分的必要条件。

2. Key Angles in Degrees and Radians | 关键角度：度与弧度

A-Level mathematics uses both degrees and radians. The exact values you need to know are based on the angles 0°, 30°, 45°, 60°, and 90°, which correspond to 0, π/6, π/4, π/3, and π/2 radians respectively. Being comfortable converting between these two measures is vital because many calculus and trigonometric equation questions require radian mode. The symmetrical properties of the unit circle then allow you to find exact values for angles such as 120°, 135°, 150°, and beyond.

A-Level数学同时使用角度制和弧度制。你需要掌握的精确值基于0°、30°、45°、60°和90°这些角度，它们分别对应0、π/6、π/4、π/3和π/2弧度。熟练地在两种度量之间转换至关重要，因为许多微积分和三角方程问题都要求在弧度模式下求解。利用单位圆的对称性质，你还能求出120°、135°、150°乃至更大角度的精确值。

3. Exact Values for 45° (π/4) | 45°(π/4)的精确值

Consider an isosceles right-angled triangle with equal sides of length 1. The hypotenuse is √1²+1² = √2. Using the definitions: sin 45° = opposite/hypotenuse = 1/√2, cos 45° = adjacent/hypotenuse = 1/√2 n 45° = opposite/adjacent = 1. These results are often rationalised as sin 45° = cos 45° = √2/2. Remember that 1/√2 and √2/2 are equivalent exact forms; both are acceptable unless the mark scheme specifies a preference.

考虑一个等腰直角三角形，两腰长度均为1，斜边长为√1²+1² = √2。根据定义：sin 45° = 对边/斜边 = 1/√2，cos 45° = 邻边/斜边 = 1/√2，tan 45° = 对边/邻边 = 1。这些结果通常有理化写作sin 45° = cos 45° = √2/2。请记住1/√2与√2/2是等价的精确形式，除非评分标准特别指明，两者均可接受。

4. Exact Values for 30° and 60° (π/6 and π/3) | 30°和60°(π/6和π/3)的精确值

0°, and 90°. The sides are: hypotenuse = 2, side opposite 30° = 1, and side opposite 60° = √3 (by Pythagoras). From this triangle: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 (or √3/3). For 60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3 sin 30° = cos 60° and sin 60° = cos 30°, reflecting the complementary angle relationship.

从一个边长为2的等边三角形入手。将其对半切开，得到一个角度为30°、60°和90°的直角三角形。各边长为：斜边=2，30°对边=1，60°对边=√3（根据勾股定理）。由此可得：sin 30° = 1/2，cos 30° = √3/2，tan 30° = 1/√3（或√3/3）。对于60°：sin 60° = √3/2，cos 60° = 1/2，tan 60° = √3。注意sin 30° = cos 60°而sin 60° = cos 30°，这体现了互余角的关系。

5. Using Right-Angled Triangles to Derive | 利用直角三角形推导

Rather than rote memorisation, many students find it helpful to visualise the two standard triangles: the 45°-45°-90° triangle (sides 1, 1, √2) and the 30°-60°-90° triangle (sides 1, √3, 2). By recalling the side lengths and applying SOH CAH TOA, you can quickly reconstruct any exact ratio. This method reduces mistakes under exam pressure and deepens your conceptual grasp. Practice quickly sketching these triangles on the side of your exam paper whenever you need an exact value.

与其死记硬背，许多学生发现想象两个标准三角形更有帮助：45°-45°-90°三角形（边长1、1、√2）和30°-60°-90°三角形（边长1、√3、2）。通过回忆边长并运用SOH CAH TOA口诀，你就能迅速重建任何精确比值。这种方法能减轻考试压力下的错误，并加深你的概念理解。建议你练习在试卷边缘快速画出这些三角形，以备随时查阅。

6. Summary Table of Exact Values | 精确值汇总表

The table below consolidates the essential exact trigonometric ratios. This format mirrors what you might encounter in Edexcel formula booklets or revision resources, though the values are not provided in the exam booklet — you must know them by heart.

下表汇总了必备的精确三角比。这种格式与你在Edexcel公式手册或复习资料中可能见到的类似，但这些数值并不会在考场提供的小册子中给出——你必须熟记于心。

Angle (deg)	Angle (rad)	sin	cos	tan
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	1/√2	1/√2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

ision by zero. Some students also find it useful to remember the pattern in the sine column: √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90° respectively; the cosine column is the same sequence reversed.

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注意tan 90°是无定义的，因为cos 90° = 0导致分母为零。有些学生觉得记住正弦列的规律很有用：0°, 30°, 45°, 60°, 90°分别对应√0/2, √1/2, √2/2, √3/2, √4/2；余弦列则是这一序列的倒序。

7. Exact Values for 0° and 90° | 0°和90°的精确值

se equals the adjacent side. Hence sin 0° = 0, cos 0° = 1, tan 0° = 0. At 90°, the adjacent side shrinks to zero, so cos 90° = 0 and sin 90° = 1; tan 90° is undefined because we cannot divide by zero. Although these are often obvious, Edexcel questions can test them in the context of trigonometric identities or graph transformations.

这些值可以通过想象直角三角形的极端情况来理解。在0°时，对边长度趋近于0，而斜边等于邻边，因此sin 0° = 0，cos 0° = 1，tan 0° = 0。在90°时，邻边收缩为零，故cos 90° = 0，sin 90° = 1；tan 90°由于分母为零而无定义。尽管这些往往显而易见，Edexcel试题仍可能在三角恒等式或图像变换的背景下考查它们。

8. The Unit Circle and Symmetry | 单位圆与对称性

xis, the coordinates of the point on the circle are (cos θ, sin θ). Because the circle has radius 1, symmetry about the axes and the line y = x allows you to determine sin and cos for angles in the second, third, and fourth quadrants. For example, sin 150° = sin(180°−30°) = sin 30° = 1/2, and cos 150° = −cos 30° = −√3/2.

单位圆为将精确值推广到90°以上角度提供了一种强有力的直观方法。对于从正x轴开始逆时针测量的任意角度θ，单位圆上点的坐标为(cos θ, sin θ)。由于圆的半径为1，利用关于坐标轴和直线y = x的对称性，你可以确定第二、第三和第四象限角度的sin和cos值。例如，sin 150° = sin(180°−30°) = sin 30° = 1/2，而cos 150° = −cos 30° = −√3/2。

9. Extending Beyond 90° | 扩展到90°以上角度

in quadrant II, sin is positive while cos and tan are negative. The related angle is 180°−120° = 60°, hence sin 120° = sin 60° = √3/2, cos 120° = −cos 60° = −1/2, tan 120° = −tan 60° = −√3 h is 180°+30°, sin is negative and cos is negative, giving sin(7π/6) = −1/2, cos(7π/6) = −√3/2.

要求诸如120°、225°或300°等角度的精确值，先找出其相关锐角，再应用适当的CAST图符号规则。120°位于第二象限，sin为正，cos和tan为负。其相关角为180°−120°=60°，因此sin 120° = sin 60° = √3/2，cos 120° = −cos 60° = −1/2，tan 120° = −tan 60° = −√3。在弧度制下，同样的原则适用：对于7π/6，即180°+30°，sin和cos均为负，得sin(7π/6) = −1/2，cos(7π/6) = −√3/2。

10. Tan, Cosec, Sec, Cot Values | 正切、余割、正割、余切值

obtain: cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. For example, sec 45° = 1/(1/√2) = √2. cot 60° = 1/√3, which is often rationalised as √3 with angles where the denominator is zero, such as sec 90° and cot 0°, which are undefined. Presenting these reciprocal values in exact surd form is frequently required in integration and proof questions.

Edexcel同样要求你掌握反三角函数。一旦知道了sin、cos和tan，你就能求得：cosec θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。例如，sec 45° = 1/(1/√2) = √2。cot 60° = 1/√3，通常有理化为√3/3。cosec 30° = 1/(1/2) = 2。但需注意分母为零的情况，如sec 90°和cot 0°没有定义。在积分和证明题中，经常需要将这些倒数值表示为精确的根式形式。

11. Applications in Solving Equations | 解方程中的应用

given interval, a staple of Edexcel P2 and P3 papers. Suppose you need to solve 2 sin x = √3 for 0 ≤ x ≤ 2π. Dividing gives sin x = √3 sin is positive in the first and second quadrants). Without knowing exact values, you would be forced to use inverse trig functions on a calculator, which often yields a decimal — losing marks if an exact answer is specified.

在给定区间内解三角方程时，精确值至关重要，这也是Edexcel P2和P3试卷的常见题型。假设要求解2 sin x = √3，0 ≤ x ≤ 2π。两边同除以2得sin x = √3/2，你能识别出这是sin 60°的值。在给定区间内的主解为x = π/3和x = 2π/3（因为sin在第一和第二象限为正）。若不了解精确值，你将不得不用计算器的反三角函数功能，结果往往是小数——如果题目要求精确答案，就会失分。

12. Common Mistakes | 常见错误

Even well-prepared students make errors with exact values. Confusing sin 30° with sin 60° values is common; remember that 30° has the smaller sine value (1/2, not √3/2). Forgetting to adjust the sign for different quadrants is another pitfall — always check CAST. Many write √3/2 for tan 60° when it should be √3. Also, when rationalising, students sometimes mishandle the surd, writing 1/√3 as √3 without the denominator. Finally, be meticulous about writing 'undefined' rather than 'infinity' for tan 90° or sec 90° in the context of exact values.

即便是准备充分的学生也常在精确值上犯错。混淆sin 30°和sin 60°的值很常见：记住30°的正弦值较小（是1/2，而非√3/2）。忘记根据不同象限调整符号是另一个陷阱——务必检查CAST图。许多人将tan 60°误写为√3/2，而正确值是√3。此外，在进行分母有理化时，学生有时会错误处理根式，将1/√3写成没有分母的√3。最后，在精确值语境下，要严谨地将tan 90°或sec 90°记为'undefined'，而非'infinity'。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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