Euler’s Formula | 欧拉公式

📚 Euler’s Formula | 欧拉公式

Euler’s formula is one of the most important bridges between algebra and trigonometry in A-level Mathematics. It states that for any real number θ, the complex exponential e^iθ can be written as cos θ + i sin θ. This result is central to the AQA A-level Further Mathematics syllabus and underpins work on complex numbers, de Moivre’s theorem, roots of equations and trigonometric identities.

欧拉公式是 A-level 数学中连接代数与三角学最重要的桥梁之一。它表明对于任意实数 θ，复数指数 e^iθ 可以写成 cos θ + i sin θ。这一结果是 AQA A-level 进阶数学课程的核心，支撑着复数、棣莫弗定理、方程求根以及三角恒等式等内容。

Understanding Euler’s formula allows students to move fluently between polar form and exponential form, making multiplication, division and powers of complex numbers much easier. It also gives a compact way to express all roots of unity and to derive many standard trigonometric identities.

理解欧拉公式能让学生熟练地在极式和指数式之间转换，从而使复数的乘法、除法和幂运算变得更加简单。它还提供了一种简洁的方式来表示所有的单位根，并推导出许多标准的三角恒等式。

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1. Introduction to Euler’s Formula | 欧拉公式简介

Euler’s formula states that for any real angle θ measured in radians, e^iθ = cos θ + i sin θ. Here e is the base of natural logarithms, i is the imaginary unit with i² = −1, and cos θ and sin θ are the usual trigonometric functions.

欧拉公式指出，对于任何以弧度表示的实角 θ，都有 e^iθ = cos θ + i sin θ。其中 e 是自然对数的底数，i 是虚数单位且 i² = −1，cos θ 和 sin θ 是通常的三角函数。

The formula is remarkable because it links exponential growth, which is normally associated with real powers, with periodic trigonometric behaviour. It gives a precise meaning to raising e to an imaginary power.

该公式之所以引人注目，是因为它将通常与实数幂相关的指数增长与周期性的三角行为联系了起来。它为 e 的虚数幂赋予了精确的含义。

e^iθ = cos θ + i sin θ

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2. Deriving Euler’s Formula from Maclaurin Series | 由麦克劳林级数推导欧拉公式

One standard A-level derivation uses Maclaurin series. The exponential, cosine and sine functions have the following expansions for all real x and θ:

一种标准的 A-level 推导使用麦克劳林级数。指数函数、余弦函数和正弦函数对所有实数 x 和 θ 有以下展开式：

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + …

cos θ = 1 − θ²/2! + θ⁴/4! − θ⁶/6! + …

sin θ = θ − θ³/3! + θ⁵/5! − θ⁷/7! + …

If we replace x by iθ in the series for e^x and use i² = −1, i³ = −i and i⁴ = 1, we get:

如果我们在 e^x 的级数中用 iθ 替换 x，并利用 i² = −1、i³ = −i 和 i⁴ = 1，就得到：

e^iθ = 1 + iθ − θ²/2! − iθ³/3! + θ⁴/4! + iθ⁵/5! − …

Collecting real and imaginary parts shows that the real part is exactly the series for cos θ and the imaginary part is exactly the series for sin θ. This gives Euler’s formula.

分别收集实部和虚部可以发现，实部正是 cos θ 的级数，虚部正是 sin θ 的级数。这就得到了欧拉公式。

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3. The Exponential Form of Complex Numbers | 复数的指数形式

Any non-zero complex number z can be written in polar form as z = r(cos θ + i sin θ), where r = |z| is the modulus and θ = arg z is the argument. Using Euler’s formula, this becomes z = re^iθ.

任何非零复数 z 都可以写成极式 z = r(cos θ + i sin θ)，其中 r = |z| 是模，θ = arg z 是辐角。利用欧拉公式，它就可以写成 z = re^iθ。

z = r(cos θ + i sin θ) = re^iθ

This is called the exponential form of a complex number. It is extremely compact and makes many operations easier. In AQA questions, students are often asked to convert between Cartesian form x + iy, polar form and exponential form.

这被称为复数的指数形式。它非常简洁，使许多运算更加容易。在 AQA 考题中，学生经常被要求在笛卡尔形式 x + iy、极式和指数式之间进行转换。

The argument θ must be chosen in radians. The principal argument is usually taken in the interval (−π, π], but some questions may allow 0 ≤ θ < 2π. Always check the convention used in the question.

辐角 θ 必须用弧度表示。主辐角通常取在区间 (−π, π] 内，但有些题目可能允许 0 ≤ θ < 2π。一定要检查题目中采用的约定。

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4. From Polar Form to Exponential Form | 从极式到指数式

To convert a complex number from Cartesian form z = x + iy into exponential form, first calculate the modulus r = √(x² + y²). Then find the argument θ using tan θ = y/x, but be careful to place θ in the correct quadrant.

要将复数从笛卡尔形式 z = x + iy 转换为指数形式，首先计算模 r = √(x² + y²)。然后利用 tan θ = y/x 求出辐角 θ，但要注意把 θ 放在正确的象限。

The sign of x and y determines the quadrant of θ. A common AQA technique is to first find the acute angle α = tan⁻¹(|y/x|) and then adjust it depending on the signs of x and y.

x 和 y 的符号决定 θ 的象限。AQA 中常用的方法是先求锐角 α = tan⁻¹(|y/x|)，然后根据 x 和 y 的符号进行调整。

Quadrant	Signs of x, y	Argument θ
1	x > 0, y > 0	θ = α
2	x < 0, y > 0	θ = π − α
3	x < 0, y < 0	θ = −π + α or θ = π + α depending on range
4	x > 0, y < 0	θ = −α

Once r and θ are known, write z = re^iθ. In exam answers, you should always give the modulus and argument clearly before stating the exponential form.

一旦知道了 r 和 θ，就可以写出 z = re^iθ。在考试答案中，你应该先清楚地给出模和辐角，再写出指数形式。

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5. Euler’s Identity | 欧拉恒等式

When θ = π, Euler’s formula gives e^iπ = cos π + i sin π. Since cos π = −1 and sin π = 0, this becomes e^iπ = −1, or equivalently:

当 θ = π 时，欧拉公式给出 e^iπ = cos π + i sin π。因为 cos π = −1 且 sin π = 0，所以得到 e^iπ = −1，或者等价地：

e^iπ + 1 = 0

This is known as Euler’s identity. It is often described as one of the most beautiful equations in mathematics because it connects five fundamental constants: 0, 1, e, i and π.

这就是著名的欧拉恒等式。它常被称为数学中最美的方程之一，因为它将五个基本常数 0、1、e、i 和 π 联系在了一起。

In AQA exams, Euler’s identity is usually not tested on its own, but it is a useful check when evaluating e^iθ at special angles such as θ = π/2, π, 3π/2 and 2π.

在 AQA 考试中，欧拉恒等式通常不会单独考查，但在计算特殊角如 θ = π/2、π、3π/2 和 2π 处的 e^iθ 时，它是一个有用的检验方法。

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6. Multiplication and Division in Exponential Form | 指数形式的乘除运算

One of the biggest advantages of Euler’s formula is how it simplifies multiplication and division. If z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, then by the laws of exponents:

欧拉公式最大的优点之一在于它简化了乘法和除法。如果 z₁ = r₁e^iθ₁ 且 z₂ = r₂e^iθ₂，那么根据指数法则：

z₁z₂ = r₁r₂ e^{i(θ₁ + θ₂)}

z₁ / z₂ = (r₁ / r₂) e^{i(θ₁ − θ₂)}

This matches the polar form rules: multiply the moduli and add the arguments; divide the moduli and subtract the arguments. The exponential form makes the algebra concise and less prone to sign errors.

这与极式规则一致：模相乘，辐角相加；模相除，辐角相减。指数形式使代数运算更加简洁，也减少了符号错误。

For AQA questions, you should be able to multiply and divide complex numbers given in exponential form and then convert the result back to Cartesian form if required.

对于 AQA 考题，你应该能够对以指数形式给出的复数进行乘法和除法，并在需要时将结果转换回笛卡尔形式。

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7. De Moivre’s Theorem and Euler’s Formula | 棣莫弗定理与欧拉公式

De Moivre’s theorem follows directly from Euler’s formula. Since e^iθ = cos θ + i sin θ, raising both sides to the power n gives:

棣莫弗定理直接由欧拉公式推出。因为 e^iθ = cos θ + i sin θ，将两边同时求 n 次幂得到：

(cos θ + i sin θ)ⁿ = e^inθ = cos nθ + i sin nθ

For integer n, this is exactly de Moivre’s theorem. It allows you to find multiple-angle formulae such as cos 3θ and sin 3θ by expanding the left-hand side using the binomial theorem.

对于整数 n，这正是棣莫弗定理。它允许你通过二项式定理展开左边来求多倍角公式，如 cos 3θ 和 sin 3θ。

For non-integer rational powers, de Moivre’s theorem can still be used but it produces multiple possible values. This is important when finding roots of complex numbers.

对于非整数有理数幂，棣莫弗定理仍然可以使用，但会产生多个可能的值。这在求复数的根时非常重要。

Euler’s formula therefore provides a clean justification for de Moivre’s theorem, which is a key tool in the AQA Further Mathematics specification.

因此，欧拉公式为棣莫弗定理提供了一个简洁的证明，而棣莫弗定理是 AQA 进阶数学考试大纲中的关键工具。

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8. Finding Roots of Complex Numbers | 复数求根

To solve zⁿ = w, write w in exponential form as w = re^{i(θ + 2kπ)}, where k is any integer. Then taking the nth root gives:

要求解 zⁿ = w，先将 w 写成指数形式 w = re^{i(θ + 2kπ)}，其中 k 是任意整数。然后取 n 次方根得到：

z = r^1/n e^{i(θ + 2kπ)/n}

Letting k = 0, 1, 2, …, n − 1 produces n distinct roots. These roots are equally spaced around a circle of radius r^1/n in the Argand diagram.

令 k = 0, 1, 2, …, n − 1，可得到 n 个不同的根。这些根在阿尔冈图中均匀分布在半径为 r^1/n 的圆上。

For example, the cube roots of unity are found by writing 1 = e^{i(0 + 2kπ)}, giving z = e^i(2kπ/3) for k = 0, 1, 2. These simplify to 1, e^2πi/3 and e^4πi/3.

例如，单位立方根可以通过写 1 = e^{i(0 + 2kπ)} 来求，得到 z = e^i(2kπ/3)，其中 k = 0, 1, 2。它们化简为 1、e^2πi/3 和 e^4πi/3。

AQA questions often ask for all roots in Cartesian form, so you should be confident converting each exponential root back to the form x + iy using Euler’s formula.

AQA 考题经常要求用笛卡尔形式写出所有根，因此你应该能够熟练地利用欧拉公式将每个指数形式的根转换回 x + iy 的形式。

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9. Proving Trigonometric Identities | 证明三角恒等式

Euler’s formula gives a powerful method for proving trigonometric identities. By writing cos θ = (e^iθ + e^−iθ) / 2 and sin θ = (e^iθ − e^−iθ) / 2i, you can derive many standard results.

欧拉公式为证明三角恒等式提供了一种强有力的方法。通过写出 cos θ = (e^iθ + e^−iθ) / 2 和 sin θ = (e^iθ − e^−iθ) / 2i，你可以推导出许多标准结果。

For example, to express cos³θ in terms of cos θ and cos 3θ, start with cos³θ = [(e^iθ + e^−iθ) / 2]³. Expanding and grouping terms leads to:

例如，要用 cos θ 和 cos 3θ 表示 cos³θ，可以从 cos³θ = [(e^iθ + e^−iθ) / 2]³ 开始。展开并合并同类项后可得到：

cos³θ = (1/4)(cos 3θ + 3 cos θ)

Similarly, identities for sin³θ, cos⁴θ and products like sin θ cos θ can be obtained quickly. This technique is frequently examined in AQA Further Mathematics.

类似地，sin³θ、cos⁴θ 以及 sin θ cos θ 等乘积的恒等式也可以迅速得到。这种技巧在 AQA 进阶数学中经常考查。

Make sure you are comfortable using the binomial expansion and simplifying powers of e^iθ. The key is to group terms so that e^inθ and e^−inθ combine back into 2 cos nθ or 2i sin nθ.

请确保你能熟练使用二项式展开并化简 e^iθ 的幂。关键是要合并项，使 e^inθ 和 e^−inθ 组合回 2 cos nθ 或 2i sin nθ。

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10. Applications in AQA Exam Questions | AQA 考题中的应用

Euler’s formula appears in many areas of the AQA A-level Further Mathematics specification. Typical tasks include converting z = x + iy into re^iθ, solving equations of the form zⁿ = w, and using de Moivre’s theorem to find multiple-angle formulae.

欧拉公式出现在 AQA A-level 进阶数学考纲的许多领域中。典型任务包括将 z = x + iy 转换为 re^iθ，求解 zⁿ = w 形式的方程，以及利用棣莫弗定理求多倍角公式。

You may also be asked to shade regions in an Argand diagram described using exponential or modulus-argument inequalities. A solid understanding of e^iθ helps you interpret loci such as |z| = r and arg z = θ.

你也可能被要求在阿尔冈图中绘制用指数或模-辐角不等式描述的区域。对 e^iθ 的深入理解有助于你解释诸如 |z| = r 和 arg z = θ 等轨迹。

In longer proof questions, Euler’s formula can be used to establish standard results such as the sum of roots of unity being zero or particular trigonometric series. Always show clear steps and state the formula you are using.

在较长的证明题中，欧拉公式可用于建立标准结论，如单位根之和为零或某些三角级数。请始终展示清晰的步骤，并说明你使用的公式。

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11. Common Mistakes and Tips | 常见错误与技巧

One common mistake is forgetting that θ must be in radians when using Euler’s formula. If your calculator is in degree mode, your answers will be incorrect. Always check the angle unit.

一个常见错误是在使用欧拉公式时忘记 θ 必须用弧度表示。如果计算器处于角度模式，答案就会出错。务必检查角度单位。

Another frequent error is choosing the wrong quadrant for the argument. Always sketch the complex number on an Argand diagram to confirm the sign of θ, especially when converting from Cartesian form.

另一个常见错误是辐角选错象限。一定要在阿尔冈图上画出复数，以确认 θ 的符号，尤其是在从笛卡尔形式转换时。

Students also sometimes apply de Moivre’s theorem to non-integer powers without adding 2kπ to the argument, leading to missing roots. Remember to include all possible values of θ + 2kπ before dividing by n.

学生们有时在非整数次幂上应用棣莫弗定理时没有在辐角上加上 2kπ，从而漏掉根。请记住在除以 n 之前要包括 θ + 2kπ 的所有可能值。

• Always write z = re^iθ with r ≥ 0.
• Use principal arguments unless the question says otherwise.
• Check results by converting back to Cartesian form.
• Memorise e^iπ = −1, e^2πi = 1 and e^iπ/2 = i.

始终用 r ≥ 0 写出 z = re^iθ。除非题目另有说明，否则使用主辐角。通过转换回笛卡尔形式来检查结果。记住 e^iπ = −1、e^2πi = 1 和 e^iπ/2Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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