Question

3 (b) simplify \\(\dfrac{\cos(\omega t) - \mathrm{j}\sin(\omega t)}{\cos(2\omega t) - \mathrm{j}\sin(2\omega t)}\\) and leave the result in polar form.

Explanation:

Step1: Apply Euler's Formula

Recall Euler's formula: $e^{-j\theta} = \cos(\theta) - j\sin(\theta)$. Rewrite numerator and denominator:
Numerator: $\cos(\omega t) - j\sin(\omega t) = e^{-j\omega t}$
Denominator: $\cos(2\omega t) - j\sin(2\omega t) = e^{-j2\omega t}$

Step2: Simplify the exponential fraction

Use exponent rule $\frac{e^a}{e^b}=e^{a-b}$:
$\frac{e^{-j\omega t}}{e^{-j2\omega t}} = e^{-j\omega t - (-j2\omega t)} = e^{j\omega t}$

Step3: Convert to polar form

Polar form of $e^{j\theta}$ is $1\angle\omega t$ (magnitude 1, angle $\omega t$).

Answer:

$1\angle\omega t$