Complex Numbers：練習問題

Given the complex numbers \( z_1 = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \) and \( z_2 = 3e^{i\frac{\pi}{4}} \), find the modulus and principal argument of the product \( z_1 z_2 \).

Let \( \omega \) be a primitive 5th root of unity. Evaluate the sum \( \sum_{r=0}^{4} \omega^{2r} \).

By using de Moivre’s theorem, express \( \cos(4\theta) \) in terms of \( \cos \theta \).

Given the complex number \( z = \sqrt{3} - i \), find the exponential form of \( z \) in the form \( re^{i\theta} \), where \( r > 0 \) and \( -\pi < \theta \le \pi \).

Find the two square roots of the complex number \( z = 3 + 4i \).

Given the complex number \( z = \sqrt{3} - i \), find the modulus \( |z| \) and the principal argument \( \text{arg}(z) \), where \( -\pi < \text{arg}(z) \le \pi \).

Find the two square roots of the complex number \( 3 + 4i \), giving your answers in the form \( x + iy \) where \( x \) and \( y \) are real constants.

The complex number \( w \) is a root of the equation \( z^n = 1 \).
(i) Use de Moivre's theorem to show that the \( n \) distinct roots of unity form the vertices of a regular \( n \)-gon on an Argand diagram.
(ii) Given that \( n=5 \) and \( \omega = e^{i\frac{2\pi}{5}} \), simplify the expression \( (1-\omega)(1-\omega^2)(1-\omega^3)(1-\omega^4) \) by considering the polynomial \( P(z) = \frac{z^5-1}{z-1} \).

A complex number \( z \) satisfies the equation \( z^6 = 1 \).
(a) Find all six roots of the equation in the form \( e^{i\theta} \), where \( -\pi < \theta \le \pi \).
(b) The roots are represented by points \( P_0, P_1, P_2, P_3, P_4, P_5 \) on an Argand diagram, forming a regular hexagon inscribed in a unit circle. Find the exact area of this hexagon.
(c) Let \( P_0 \) be the point corresponding to \( z = 1 \). By considering the expression \( \frac{z^6 - 1}{z - 1} \), or otherwise, show that the product of the lengths of the chords from \( P_0 \) to all other vertices, \( |P_0P_1| \times |P_0P_2| \times |P_0P_3| \times |P_0P_4| \times |P_0P_5| \), is equal to 6.

(a) Use de Moivre’s theorem to show that \(\cos(5\theta) = 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta\).
(b) Hence, show that the four roots of the equation \(16x^4 - 20x^2 + 5 = 0\) are of the form \(x = \cos \alpha\), and state the values of \(\alpha\) in the interval \(0 < \alpha < \pi\).
(c) By considering the product of the roots of the equation in part (b), show that \(\cos \left(\frac{\pi}{10}\right) \cos \left(\frac{3\pi}{10}\right) = \frac{\sqrt{5}}{4}\).
(d) On an Argand diagram, the roots of the equation \(z^5 = 1\) form a regular pentagon. Sketch this pentagon and label the vertex representing the root with the smallest positive argument in modulus-argument form.