Certainly! Let’s work through the simplification of the expression using De Moivre's Theorem:
Given the expression:
[tex]\[ \frac{\cos 7 \theta + i \sin 7 \theta}{\cos 2 \theta - i \sin 2 \theta} \][/tex]
Step 1: Express the terms in exponential form using Euler's formula:
Euler's formula states:
[tex]\[ e^{i\theta} = \cos \theta + i \sin \theta \][/tex]
Thus:
[tex]\[ \cos 7\theta + i \sin 7\theta = e^{i \cdot 7\theta} \][/tex]
[tex]\[ \cos 2\theta - i \sin 2\theta = \cos 2\theta + i \sin(-2\theta) = e^{-i \cdot 2\theta} \][/tex]
Step 2: Substitute these exponential forms into the expression:
So, the expression becomes:
[tex]\[ \frac{e^{i \cdot 7\theta}}{e^{-i \cdot 2\theta}} \][/tex]
Step 3: Use the properties of exponents to simplify the fraction:
Recall the property of exponents that states [tex]\(\frac{e^a}{e^b} = e^{a - b}\)[/tex]:
Therefore, we have:
[tex]\[ \frac{e^{i \cdot 7\theta}}{e^{-i \cdot 2\theta}} = e^{i \cdot 7\theta - (-i \cdot 2\theta)} \][/tex]
[tex]\[ = e^{i \cdot 7\theta + i \cdot 2\theta} \][/tex]
[tex]\[ = e^{i \cdot (7\theta + 2\theta)} \][/tex]
[tex]\[ = e^{i \cdot 9\theta} \][/tex]
Hence, the simplified form of the given expression is:
[tex]\[ e^{i \cdot 9\theta} \][/tex]
So, using De Moivre's Theorem, we have simplified:
[tex]\[ \frac{\cos 7 \theta + i \sin 7 \theta}{\cos 2 \theta - i \sin 2 \theta} = e^{i \cdot 9\theta} \][/tex]
