Complex numbers：練習題

Given that \(z = \cos\theta + i\sin\theta\), use the identity \(z^n + z^{-n} = 2\cos(n\theta)\) to express \(\cos^3\theta\) in terms of multiple angles.

Use De Moivre's Theorem to find the exact value of the expression \(\left(\cos\frac{\pi}{10} + i\sin\frac{\pi}{10}\right)^5\).

The locus of points in the Argand plane satisfying the equation \(|z - 3i| = |z - 4|\) is described as:

Given two complex numbers \(z_1 = 4e^{i\frac{\pi}{3}}\) and \(z_2 = 2e^{-i\frac{\pi}{6}}\), find the product \(z_1 z_2\) in the form \(re^{i\theta}\), where \(r > 0\) and \(-\pi < \theta \le \pi\).

Sketch the locus of a point represented by the complex number \( z \) such that \( |z - 2i| = 3 \).

Use de Moivre’s theorem to find the exact value of \( (1 + i)^8 \).

By considering the roots of the equation \( z^n - 1 = 0 \), prove that \( \sum_{k=0}^{n-1} \cos\left(\frac{2k\pi}{n}\right) = 0 \) for any integer \( n > 1 \).

Given that the complex number \( z \) is such that \( |z| = 2 \) and \( \arg(z) = \frac{\pi}{3} \), find the exact value of \( z^6 \) using de Moivre’s theorem.

Express your final answer in Cartesian form \( a + bi \).

Consider the complex number \( z = \cos \theta + i \sin \theta \).

(a) Use de Moivre’s theorem to show that \( z^n + \frac{1}{z^n} = 2 \cos(n\theta) \).
(b) By expanding \( \left( z + \frac{1}{z} \right)^4 \), find an expression for \( \cos^4 \theta \) in terms of \( \cos(4\theta) \) and \( \cos(2\theta) \).
(c) Hence, evaluate \( \int_{0}^{\frac{\pi}{2}} \cos^4 \theta \, d\theta \).