📘 complex numbers

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

1. **Problem Statement:** Find the values of $n$ such that $\left(1 + \sqrt{3}i\right)^n$ is a real number. 2. **Formula and Important Rules:**

1. **Problem Statement:** Find the modulus and argument of the complex number

1. **Problem Statement:** Calculate the following complex expressions step-by-step:

1. **Problem:** Calculate the following complex expressions: i. $(1 - 2i)^4$

1. مسئله: یافتن ریشه‌های معادله $z^6 = -2 + 6 \sqrt{3} i$ در دستگاه اعداد مختلط است. 2. ابتدا عدد مختلط سمت راست را به فرم مثلثاتی تبدیل می‌کنیم. برای این کار، قدر مطلق و آرگومان آ

1. **State the problem:** Given $x = \cos \theta - i \sin \theta$, find the value of $x - \frac{1}{x}$. 2. **Recall the expression for $x$:** We have $x = \cos \theta - i \sin \the

1. The problem states: "The nth roots of unity form a regular polygon on joining on an Argand diagram." We need to determine if this statement is true or false. 2. Recall that the

1. The problem is to find the value of $z_1 + z_2 - z_3$ given $z_1 = 2 + j3$. 2. However, the values of $z_2$ and $z_3$ are not provided in the problem statement.

1. Problem 18. State the problem: In the Argand diagram the point P represents the complex number $z$ and we are given $|z-1-i|=\sqrt{2}$. 2. Interpretation and locus: The equation

1. Problem: Simplify the following complex quotients and express each result in the form $a+ib$. 2. (a) Compute $\frac{20}{3+i}$.