Concept:The general equation for a sinusoidal AC is \(i(t) = I_{\text{new}} \sin(\omega_{\text{new}} t)\), where \(I_{\text{new}}\) is the amplitude and \(\omega_{\text{new}}\) is the angular frequency.
Formula / Reaction:$$I_{\text{new}} = \frac{i_0}{2}, \quad \omega_{\text{new}} = 2\omega \implies i = \frac{1}{2}i_0 \sin(2\omega t)$$
Solution:- Original waveform: \(i = i_0 \sin(\omega t)\).
- Half amplitude \(\implies I_{\text{new}} = \frac{1}{2} i_0\).
- Double frequency \(\implies \omega_{\text{new}} = 2\omega\).
- Substituting these parameters gives \(i = \frac{1}{2} i_0 \sin(2\omega t)\).
Why other options are incorrect:- Opt_A: Represents doubled amplitude (\(2i_0\)) with unchanged frequency.
- Opt_B: Contains an inverse trigonometric function, which is mathematically invalid for an AC wave.
- Opt_D: \(i = \frac{1}{2} i_0 \sin(\omega t)\) has half amplitude but retains the original frequency \(\omega\).
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