📘 complex numbers

1. مسئله: باید مقدار $$\cosh(2 + \frac{\pi i}{4})$$ را به صورت $$a + ib$$ که $$a$$ و $$b$$ اعداد حقیقی هستند، بیان کنیم. 2. فرمول مورد استفاده: تابع هذلولی کسینوس برای عدد مختلط $$

1. The problem involves the complex number in polar form: $z = r(\cos\theta + i\sin\theta)$. 2. This is the polar representation of a complex number, where $r$ is the magnitude (di

1. **Problem Statement:** Use De Moivre's Theorem to find $(\cos \theta + i \sin \theta)^n$ for given $\theta$ and integer $n$. 2. **Formula:** De Moivre's Theorem states that for

1. **Problem Statement:** Given a complex number $z_0$ on the positive real axis between 0 and 1, and points $A$, $B$, $C$, and $D$ representing complex numbers in the plane, deter

1. **State the problem:** We need to find the magnitude $|z|$ of the complex number $$z = \frac{(1 + i)^5}{(\sqrt{3} - i)^2}.$$\n\n2. **Recall the formula for magnitude:** For any

1. **Problem:** Given complex numbers $z_1 = 1 + i$, $z_2 = 3 - 2i$, and $z_3 = -2 + 3i$, calculate the following: 2. **Formula and rules:**

1. **Problem:** Given complex numbers $z_1 = 1 + i$, $z_2 = 3 - 2i$, and $z_3 = -2 + 3i$, calculate $z_1 \times z_2 + 2z_3^*$. 2. **Formula and rules:**

1. مسئله: اعداد مختلط داده شده را به شکل دکارتی بنویسید. 2. فرمول‌ها و قوانین مهم:

1. **Problem 1:** Compute $$\frac{z_1^3 + z_2^3 + z_3^3}{z_1 z_2 z_3}$$ where $$z_1 = 1 + i\sqrt{3}, z_2 = \sqrt{3} - i, z_3 = -1 + i$$. 2. **Step 1:** Calculate each cube.

1. The question "why it -j?" likely refers to the use of the imaginary unit in electrical engineering or complex numbers. 2. In mathematics and engineering, the imaginary unit is d

1. **State the problem:** We are given a complex number $z$ that satisfies the equation $$z + i|z| = 12 + 9i.$$ We need to find the imaginary part of $z$. 2. **Express $z$ in terms

1. **Problem:** Write $\frac{3 + 2i}{4 - 3i}$ in the form $a + bi$. 2. **Formula and rules:** To write a complex fraction in the form $a + bi$, multiply numerator and denominator b

1. Problem: Pretvoriti kompleksni broj $z = -\cos\left(\frac{7\pi}{10}\right) - i\sin\left(\frac{7\pi}{10}\right)$ u trigonometrijski zapis. 2. Formula: Kompleksni broj u trigonome

1. **Problem:** Write each expression in the standard form $a + bi$. **a)** $(-3 - 2i)^2 + \frac{1 + i}{8 - 5i}$

1. **State the problem:** Find the cube roots of the complex number $i$. 2. **Formula and explanation:** To find the cube roots of a complex number, express it in polar form $z = r

1. **Problem Statement:** Find the multiplicative identity in complex numbers. 2. **Concept:** The multiplicative identity is the complex number which, when multiplied by any compl

1. **State the problem:** Given $\frac{1}{z_1} = 2 + 3i$ and $z_2 = 4 + 2i$, verify that $\frac{z_1}{z_2} = \frac{z_1}{z_2}$. This means we need to find $z_1$ from the first equati

1. **State the problem:** We are given two complex numbers in polar form: $z = 2 \operatorname{cis}(-\frac{\pi}{4})$ and $w = \sqrt{3} \operatorname{cis}(\frac{\pi}{6})$. We need t

1. **State the problem:** We are given two complex numbers in polar form:

1. **Stel het probleem vast:** We hebben een complex getal met modulus 5 en argument $\frac{\pi}{2}$. We moeten de correcte notatie van dit complex getal vinden in de vorm $a + bj$

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