Question

If the effective value of current
$i=2\sin 100\pi t+2\cos (100\pi t+{30}^{\circ })$ is $\sqrt{x}A$. Find $x$

Answer 1

$cos\theta =sin({90}^o-\theta )$
$sin\alpha +sin\beta =2sin\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}$
$i=2sin100\pi t+2cos(100\pi t+{30}^o)=2sin100\pi t+2sin({90}^o-(100\pi t+{30}^o\left)\right)=2sin100\pi t+2sin({60}^o-100\pi t)=2\ast 2\ast sin\frac{(100-100)\pi t+{60}^o}{2}\ast cos\frac{(100+100)\pi t-{60}^o}{2}=4\ast sin{30}^o\ast cos(100\pi t-{30}^o)=4\ast 1/2\ast cos(100\pi t-{30}^o)=2cos(100\pi t-{30}^o)$
therefore, $I_o=2A{I}_{rms}=2/\sqrt{2}=\sqrt{2}A$