📘 complex numbers

1. **State the problem:** Express $$3 \frac{e^{j400} + e^{-2j100}}{e^{j00}}$$ in terms of cosine only.

1. The problem is to simplify the expression $|i \times \sqrt{3}|$. 2. Recall that $i$ is the imaginary unit with magnitude $|i|=1$.

1. **Stating the problem:** We are given the complex number $Z = 1 - i$ and we want to analyze it. 2. **Formula and rules:** A complex number is generally written as $Z = a + bi$,

1. مسئله: ساده کردن عبارت $$z = \frac{(1 + i\sqrt{3})^4 (1 - i)^5}{2\sqrt{3} (1 - i\sqrt{3})^3 (1 + i)^4}$$

1. مسئله: ساده کنید $$z = \frac{(1 + i\sqrt{3})^4 (1 - i)^5}{2\sqrt{2} (1 - i\sqrt{3})^3 (1 + i)^4}$$

1. **Énoncé du problème** : Résoudre dans $\mathbb{C}$ l'équation quadratique $(E_1) : z^2 - (3 - 2i) z + 5 - zi = 0$. 2. **Formule utilisée** : Pour une équation quadratique $az^2

1. **Énoncé du problème :** Déterminer le module et un argument des nombres complexes suivants :

1. The problem is to evaluate the magnitude of the complex number $\frac{5i}{3 - i}$.\n\n2. Recall that the magnitude of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2

1. **State the problem:** Given a complex number $z = 4 + 5i$, find the absolute value $|z|$ and the product $z \cdot z^*$, where $z^*$ is the conjugate of $z$. 2. **Recall formula

1. Problem statement: Given the complex number $z_1 = 2 + i$, find complex numbers $z = x + iy$ satisfying the given conditions. 2. Recall that for complex numbers $z = x + iy$ and

1. **State the problem:** We are given the complex number $$-2 - i\sqrt{3}$$ and it is expressed as $$x + iy$$ where $$x$$ and $$y$$ are real numbers. We need to find the value of

1. **State the problem:** Evaluate $\left( \sin \frac{\pi}{9} + i \sin \frac{7\pi}{18} \right)^{-6}$ using De Moivre's theorem. 2. **Recall De Moivre's theorem:** For a complex num

1. **State the problem:** Convert the complex number $0 + j80$ to its polar form. 2. **Recall the polar form of a complex number:** A complex number $z = x + jy$ can be expressed i

1. **State the problem:** Convert each complex number $z_k$ into the form $\cos\theta + i\sin\theta$ by finding the angle $\theta$.

1. The problem is to express each complex number $z_k = x + yi$ in the form $\cos \theta + i \sin \theta$, where $\theta$ is the argument (angle) of the complex number. 2. Recall t

1. We are asked to find the square root of the complex number $\sqrt{2} - i$ and express it in the form $a + bi$ where $a$ and $b$ are real numbers. 2. Recall that for a complex nu

1. **State the problem:** Find the modulus and argument of the complex number $$\frac{1 + \sin\theta + i \cos\theta}{1 + \sin\theta - i \cos\theta}$$ and show that for $$\theta = \

1. **State the problem:** Find the modulus and argument of the complex number $$z = \frac{1 + \sin \theta + i \cos \theta}{1 + \sin \theta - i \cos \theta}$$ and then show that $$\

1. **State the problem:** Given $z = 4 \sqrt{3} e^{\frac{\pi i}{3}} - 4 e^{\frac{5\pi i}{6}}$, express $z$ in the form $re^{i\theta}$. Then show that $$\frac{z}{8} + i\left(\frac{z

1. **Problem Statement:** Define the modulus of a complex number and explain its geometric interpretation with an example. 2. **Definition:** The modulus of a complex number $z = a

1. The problem is to find the power of a complex number and express it in rectangular form. 2. The power of a complex number $z = r(\cos \theta + i \sin \theta)$ raised to the $n$t