📚 Applying de Moivre's Theorem to Trigonometric Identities | 棣莫弗定理在三角恒等式建立中的应用
In AQA A-Level Mathematics, de Moivre's theorem is much more than a statement about complex powers. It is a standard, elegant method for establishing trigonometric identities involving multiple angles and for expressing powers of sin θ and cos θ as sums of sines and cosines of integer multiples of θ.
在 AQA A-Level 数学中，棣莫弗定理不仅仅是一个关于复数幂的结论。它也是证明多倍角三角恒等式、将 sin θ 和 cos θ 的幂表示为整数倍角正弦与余弦之和的一种标准而优雅的方法。
1. Statement of de Moivre's Theorem | 棣莫弗定理的陈述
For any real θ and any integer n, de Moivre' s usually a positive integer, because we use binomial expansion to compare real and imaginary parts.
对任意实数 θ 和任意整数 n，棣莫弗定理指出 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。在 AQA 考题中，n 通常是正整数，因为我们需要用二项式展开来比较实部与虚部。
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
modulus 1 and argument θ, so raising it to the power n gives a complex number with modulus 1 and argument nθ.
复数 z = cos θ + i sin θ 的模为 1，辐角为 θ，因此对它取 n 次幂后所得复数的模仍为 1，辐角变为 nθ。
2. Core Strategy: Expand and Compare Real and Imaginary Parts | 核心策略：展开并比较实部与虚部
y, start from z = cos θ + i sin θ. Then write zⁿ in two different ways: first by de Moivre' parts produces cosine identities, while equating imaginary parts produces sine identities.
要建立三角恒等式，可设 z = cos θ + i sin θ。然后用两种方法写出 zⁿ：一方面使用棣莫弗定理，另一方面用二项式定理展开 (cos θ + i sin θ)ⁿ。比较实部可得到余弦恒等式，比较虚部可得到正弦恒等式。
zⁿ = (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
This comparison works because two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.
这种比较之所以成立，是因为两个复数相等当且仅当它们的实部相等且虚部相等。
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