Math Problem Statement
explain
Explain the simplification of a complex fraction involving real and imaginary parts (complex conjugates).
Solution
This handwritten solution involves the simplification of a complex fraction using the concepts of complex conjugates, separating the real and imaginary parts of a complex expression.
Here’s a detailed explanation of each step:
1. Expression Overview
The problem involves simplifying:
= \frac{1 - |k|(\cos\theta + j\sin\theta)}{1 + |k|(\cos\theta + j\sin\theta)}$$ Where: - $$|k|$$ is the magnitude (a scalar value), - $$\cos\theta + j\sin\theta$$ is the exponential form of a complex number. The goal is to simplify this expression into real and imaginary components by **separating** the numerator and denominator. --- ### **2. Using Complex Conjugates** To simplify the fraction, multiply both numerator and denominator by the **complex conjugate** of the denominator. The conjugate of $$1 + |k|(\cos\theta + j\sin\theta)$$ is: $$1 + |k|\cos\theta - j|k|\sin\theta$$ This eliminates the imaginary component in the denominator when simplified. --- ### **3. Simplification** #### **Numerator:** Multiply: \[ \Big[1 - |k|(\cos\theta + j\sin\theta)\Big] \times \Big[1 + |k|\cos\theta - j|k|\sin\theta\Big] \] Expanding step by step: $$= (1)(1) + (1)(|k|\cos\theta) - j(1)(|k|\sin\theta)$$ $$- |k|(\cos\theta)(1) - |k|^2(\cos^2\theta) + j|k|^2(\cos\theta)(\sin\theta)$$ $$- j|k|(\sin\theta)(1) - j|k|^2(\cos\theta)(\sin\theta) + |k|^2(\sin^2\theta)$$ Combine terms: $$\text{Real part: } 1 - |k|^2\cos^2\theta - |k|^2\sin^2\theta = 1 - |k|^2$$ $$\text{Imaginary part: } -j[|k|(\sin\theta) + |k|(\sin\theta)] = -2j|k|(\sin\theta)$$ Thus, the numerator becomes: $$1 - |k|^2 - 2j|k|(\sin\theta)$$ --- #### **Denominator:** Multiply: \[ \Big[1 + |k|(\cos\theta + j\sin\theta)\Big] \times \Big[1 + |k|\cos\theta - j|k|\sin\theta\Big] \] Using the difference of squares: $$(a + b)(a - b) = a^2 - b^2$$ Here: $$a = 1 + |k|\cos\theta, \quad b = j|k|\sin\theta$$ So: $$(1 + |k|\cos\theta)^2 - (j|k|\sin\theta)^2$$ Simplify each term: 1. $$(1 + |k|\cos\theta)^2 = 1 + 2|k|\cos\theta + |k|^2\cos^2\theta$$, 2. $$-(j|k|\sin\theta)^2 = +|k|^2\sin^2\theta$$. Combine: $$1 + 2|k|\cos\theta + |k|^2(\cos^2\theta + \sin^2\theta)$$ Since $$\cos^2\theta + \sin^2\theta = 1$$, we get: $$1 + 2|k|\cos\theta + |k|^2$$ --- ### **4. Final Simplified Form** Now the fraction becomes: $$\frac{1 - |k|^2 - 2j|k|\sin\theta}{1 + 2|k|\cos\theta + |k|^2}$$ This separates the real and imaginary parts: - **Real part:** $$\frac{1 - |k|^2}{1 + 2|k|\cos\theta + |k|^2}$$, - **Imaginary part:** $$-\frac{2|k|\sin\theta}{1 + 2|k|\cos\theta + |k|^2}$$. --- ### Key Notes: 1. The denominator simplifies using the **difference of squares** property. 2. The numerator and denominator were expanded, grouped, and simplified carefully. 3. The final result clearly separates real and imaginary components. --- Do you want further clarification or any other explanation? Let me know! --- **Related Questions:** 1. What is the role of the complex conjugate in simplifying fractions involving complex numbers? 2. How does the identity $$\cos^2\theta + \sin^2\theta = 1$$ simplify such expressions? 3. Can you explain how to separate real and imaginary parts in a general complex expression? 4. How do you interpret the physical significance of real and imaginary components in electrical engineering (e.g., admittance)? 5. What mathematical tools can be used to handle similar expressions involving $$e^{j\theta}$$? --- **Tip:** When working with complex fractions, always simplify the denominator first by using its conjugate—it ensures clarity in real and imaginary separation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Complex Conjugates
Simplification of Fractions
Real and Imaginary Parts
Formulas
(a + b)(a - b) = a^2 - b^2
cos^2(θ) + sin^2(θ) = 1
Theorems
Difference of Squares
Trigonometric Identity
Suitable Grade Level
Undergraduate Mathematics or Engineering