Exact Values of Trigonometric Ratios | 三角比精确值

📚 Exact Values of Trigonometric Ratios | 三角比精确值

In Edexcel A-Level Mathematics, you are expected to quote exact values for sin, cos and tan of 30°, 45°, 60° and related angles without using a calculator. These values come from two special triangles and are essential for trigonometry, calculus and vector problems.

在爱德思 A-Level 数学中，你需要不借助计算器直接写出 30°、45°、60° 及相关角的正弦、余弦和正切的精确值。这些值来自两个特殊三角形，是三角函数、微积分和向量问题的基础。

1. Why Exact Values Matter | 为什么精确值重要

Exact values allow you to give answers such as √3/2 rather than a rounded decimal like 0.866. In A-Level exams, leaving a trig ratio as a decimal often loses accuracy marks, especially when the question asks for an exact answer.

精确值让你能够写出 √3/2 这样的答案，而不是 0.866 这样的四舍五入小数。在 A-Level 考试中，如果题目要求精确答案，把三角比写成小数通常会丢失准确性分数。

These values also appear when solving trigonometric equations, differentiating trigonometric functions, and working with complex numbers. Memorising them correctly saves time and reduces sign errors.

这些值还会在解三角方程、对三角函数求导以及处理复数时出现。正确记住它们可以节省时间，减少符号错误。

2. The Two Special Triangles | 两个特殊三角形

The exact trig values are derived from two right-angled triangles. The first is half of an equilateral triangle, which gives the angles 30° and 60°. The second is a right-angled isosceles triangle, which gives the angle 45°.

精确三角值来自两个直角三角形。第一个是等边三角形的一半，得到 30° 和 60° 角。第二个是等腰直角三角形，得到 45° 角。

For the half-equilateral triangle, start with an equilateral triangle of side length 2. Splitting it down an altitude creates a right-angled triangle with hypotenuse 2, short side 1, and remaining side √3.

对于半个等边三角形，从边长为 2 的等边三角形开始。沿一条高将其分成两个直角三角形，得到斜边为 2、短直角边为 1、另一条直角边为 √3 的三角形。

For the isosceles right-angled triangle, take the two shorter sides as 1 and 1. By Pythagoras’ theorem, the hypotenuse is √2.

对于等腰直角三角形，设两条较短的边为 1 和 1。根据勾股定理，斜边为 √2。

3. Deriving sin, cos and tan for 30° and 60° | 推导 30° 和 60° 的正弦、余弦和正切

Using the half-equilateral triangle, place the 30° angle between the hypotenuse and the side of length √3. Applying SOH CAH TOA gives the exact ratios.

使用半个等边三角形，将 30° 角放在斜边和长度为 √3 的边之间。应用 SOH CAH TOA 可以得到精确比值。

sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3

For 60°, the adjacent and opposite sides swap. Therefore sin 60° is √3/2, cos 60° is 1/2, and tan 60° is √3.

对于 60° 角，邻边和对边互换。因此 sin 60° = √3/2，cos 60° = 1/2，tan 60° = √3。

sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Notice that sin 30° equals cos 60°, and sin 60° equals cos 30°. This is because 30° and 60° are complementary angles.

注意 sin 30° 等于 cos 60°，sin 60° 等于 cos 30°。这是因为 30° 和 60° 互为余角。

4. Deriving sin, cos and tan for 45° | 推导 45° 的正弦、余弦和正切

In the right-angled isosceles triangle, both acute angles are 45°, and the two shorter sides are equal. With sides 1, 1 and √2, the ratios are straightforward.

在等腰直角三角形中，两个锐角都是 45°，两条较短的边相等。边长为 1、1 和 √2 时，比值很容易得出。

sin 45° = 1/√2 = √2/2, cos 45° = 1/√2 = √2/2, tan 45° = 1

Both sin 45° and cos 45° are equal because the opposite and adjacent sides are equal. The tangent is simply 1 divided by 1, so tan 45° = 1.

sin 45° 和 cos 45° 相等，因为对边和邻边相等。正切就是 1 除以 1，所以 tan 45° = 1。

You should be comfortable writing these values in both forms. The rationalised form √2/2 is usually preferred in Edexcel mark schemes.

你应该能够熟练写出这两种形式。在爱德思评分标准中，通常更倾向于使用有理化形式 √2/2。

5. Standard Exact Value Table | 标准精确值表

The table below summarises the exact values you must know for the first quadrant. It is worth memorising the whole table rather than deriving it every time.

下表总结了你必须掌握的第一象限精确值。建议记住整张表，而不是每次都重新推导。

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3

A common memory pattern is that the sine values under 30°, 45° and 60° are √1/2, √2/2 and √3/2, while the cosine values reverse that order.

一个常见的记忆规律是：30°、45°、60° 的正弦值分别为 √1/2、√2/2、√3/2，而余弦值的顺序与之相反。

6. Including 0°, 90° and Related Angles | 包括 0°、90° 及相关角

For completeness, you should also know the boundary angles. At 0°, sin 0° = 0, cos 0° = 1 and tan 0° = 0. At 90°, sin 90° = 1, cos 90° = 0 and tan 90° is undefined.

为了完整掌握，你还应该知道边界角。在 0° 时，sin 0° = 0，cos 0° = 1，tan 0° = 0。在 90° 时，sin 90° = 1，cos 90° = 0，tan 90° 无定义。

Using the CAST diagram or the unit circle, you can extend these exact values to angles in the other quadrants. For example, sin 120° = sin 60° = √3/2, while cos 150° = -cos 30° = -√3/2.

利用 CAST 图或单位圆，你可以把这些精确值推广到其他象限的角。例如，sin 120° = sin 60° = √3/2，而 cos 150° = -cos 30° = -√3/2。

When dealing with negative angles, remember that cos is even and sin is odd: cos(-θ) = cos θ and sin(-θ) = -sin θ.

处理负角时，记住余弦是偶函数，正弦是奇函数：cos(-θ) = cos θ，sin(-θ) = -sin θ。

7. Radians and Exact Values | 弧度制与精确值

In A-Level Maths, angles are often given in radians. The exact values are the same, but the angles are written as π/6, π/4 and π/3 instead of 30°, 45° and 60°.

在 A-Level 数学中，角通常以弧度制给出。精确值相同，但角度写作 π/6、π/4 和 π/3，而不是 30°、45° 和 60°。

sin(π/6) = 1/2, sin(π/4) = √2/2, sin(π/3) = √3/2

cos(π/6) = √3/2, cos(π/4) = √2/2, cos(π/3) = 1/2

You must be able to switch between degrees and radians fluently. In Edexcel papers, questions involving calculus almost always use radians, so exact trig values in radian form are particularly important.

你必须能够在角度制和弧度制之间熟练转换。在爱德思试卷中，涉及微积分的题目几乎总是使用弧度制，因此弧度形式的精确三角值尤为重要。

8. Reciprocal Trigonometric Ratios | 倒数三角比

Edexcel also expects you to know the reciprocal ratios: secant, cosecant and cotangent. They are defined as sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ.

爱德思还要求你掌握倒数三角比：正割、余割和余切。它们定义为 sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ。

Using the standard exact values, you can find results such as sec 60° = 2, cosec 30° = 2 and cot 45° = 1. These often appear when simplifying trigonometric identities.

利用标准精确值，你可以得到 sec 60° = 2、cosec 30° = 2、cot 45° = 1 等结果。这些在化简三角恒等式时经常出现。

sec 60° = 2, cosec 30° = 2, cot 45° = 1

Be careful with rationalisation: sec 30° = 2/√3 = 2√3/3, and cot 60° = 1/√3 = √3/3.

注意有理化：sec 30° = 2/√3 = 2√3/3，cot 60° = 1/√3 = √3/3。

9. Using Exact Values in Algebraic Expressions | 在代数式中使用精确值

Exact trig values are frequently combined with algebra. You might be asked to simplify an expression such as sin² 60° + cos² 60°, which equals 1 by the Pythagorean identity.

精确三角值经常与代数结合。你可能会被要求化简 sin² 60° + cos² 60° 这样的表达式，根据勾股恒等式，它等于 1。

sin² 60° + cos² 60° = (√3/2)² + (1/2)² = 3/4 + 1/4 = 1

Another common type is evaluating a product or sum of exact values, such as sin 60° tan 30° + cos 45°. Substituting the exact forms gives (√3/2)(√3/3) + √2/2 = 1/2 + √2/2 = (1 + √2)/2.

另一种常见题型是计算精确值的乘积或和，例如 sin 60° tan 30° + cos 45°。代入精确形式得到 (√3/2)(√3/3) + √2/2 = 1/2 + √2/2 = (1 + √2)/2。

When solving equations, exact values help you state answers precisely. For example, if 2 cos x = √3, then cos x = √3/2, giving x = 30° or x = 330° in the interval 0° ≤ x ≤ 360°.

解方程时，精确值帮助你准确写出答案。例如，若 2 cos x = √3，则 cos x = √3/2，在 0° ≤ x ≤ 360° 内得到 x = 30° 或 x = 330°。

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common error is mixing up sin 30° and sin 60°. Remember that sin 30° is the smaller value 1/2, while sin 60° is the larger value √3/2.

一个常见错误是混淆 sin 30° 和 sin 60°。记住 sin 30° 是较小的值 1/2，而 sin 60° 是较大的值 √3/2。

Another mistake is forgetting to rationalise denominators. Leaving tan 30° as 1/√3 may be accepted sometimes, but Edexcel mark schemes usually prefer √3/3.

另一个错误是忘记有理化分母。把 tan 30° 写成 1/√3 有时可以接受，但爱德思评分标准通常更喜欢 √3/3。

Students also forget that tan 90° is undefined, not infinite as a number. In exact value questions, do not write tan 90° = ∞; write that it is undefined.

学生还会忘记 tan 90° 是无定义的，而不是一个等于无穷大的数。在精确值题目中，不要写 tan 90° = ∞，应写为无定义。

Finally, check whether the question uses degrees or radians. Using the wrong angle mode can change the entire solution, especially in calculus problems.

最后，检查题目使用的是角度制还是弧度制。使用错误的角单位会改变整个解法，尤其是在微积分问题中。

11. Exam-Style Worked Examples | 考试风格例题

Worked example 1: Find the exact value of sin 60° + tan 45° – cos 30°.

例题 1：求 sin 60° + tan 45° – cos 30° 的精确值。

Substitute the known values: sin 60° = √3/2, tan 45° = 1, cos 30° = √3/2. Therefore the expression equals √3/2 + 1 – √3/2 = 1.

代入已知值：sin 60° = √3/2，tan 45° = 1，cos 30° = √3/2。因此原式等于 √3/2 + 1 – √3/2 = 1。

Worked example 2: Solve 2 sin x = √2 for 0 ≤ x ≤ 2π, giving answers in radians.

例题 2：解方程 2 sin x = √2，其中 0 ≤ x ≤ 2π，答案用弧度制表示。

Divide both sides by 2 to obtain sin x = √2/2. The exact value √2/2 corresponds to sin(π/4). Since sin is positive in the first and second quadrants, the solutions are x = π/4 and x = 3π/4.

两边同时除以 2，得到 sin x = √2/2。精确值 √2/2 对应 sin(π/4)。由于正弦在第一和第二象限为正，解为 x = π/4 和 x = 3π/4。

Worked example 3: Show that sin 60° cos 30° + cos 60° sin 30° = 1.

例题 3：证明 sin 60° cos 30° + cos 60° sin 30° = 1。

Using exact values, the left-hand side is (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1. This is also an application of sin(60° + 30°) = sin 90° = 1.

使用精确值，左边为 (√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1。这也是 sin(60° + 30°) = sin 90° = 1 的应用。

12. Summary and Memory Aids | 总结与记忆技巧

A reliable memory aid is to write the sine values for 0°, 30°, 45°, 60° and 90° as √0/2, √1/2, √2/2, √3/2 and √4/2. The cosine values run in the opposite direction, and tan is sin divided by cos.

一个可靠的记忆方法是把 0°、30°、45°、60°、90° 的正弦值写成 √0/2、√1/2、√2/2、√3/2、√4/2。余弦值按相反方向排列，正切值等于正弦除以余弦。

Practise writing the full table from memory before exam day. Once you can reproduce the two special triangles quickly, you will always be able to reconstruct any exact trig value if you forget it.

在考试前练习默写整张表。一旦你能够快速画出两个特殊三角形，即使忘记某个精确三角值，也能随时重新推导出来。

Keep a clear distinction between sin 30° = 1/2 and sin 60° = √3/2. For tan, remember that tan 45° = 1 is the midpoint case, with tan 30° less than 1 and tan 60° greater than 1.

清楚区分 sin 30° = 1/2 和 sin 60° = √3/2。对于正切，记住 tan 45° = 1 是中间情况，tan 30° 小于 1，tan 60° 大于 1。

Finally, always check whether your answer is rationalised and whether the angle is in the correct unit. These small checks prevent unnecessary loss of marks in Edexcel exams.

最后，始终检查答案是否已有理化，以及角度单位是否正确。这些小检查可以避免在爱德思考试中不必要地失分。

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Exact Values of Trigonometric Ratios | 三角比精确值