Power calculations for an AC network

[PhysicsTest]

Homework Statement

For the circuit element , calculate (i)the instantaneous power absorbed, (ii) the real power (state whether it is delivered or absorbed), (iii) the reactive power (state whether delivered or absorbed), (iv) the power factor (state whether lagging or leading)

Relevant Equations

p = v*i

The voltage and current are given as $v(t) = 359.3\sin(\omega t + 15) volts$ and current is $i(t) = 100\cos(\omega t + 5)$. My question here is do i need to convert the voltage into cos terms or current into sin terms. suppose if i convert the voltage into cos terms, the calculations will look like
$v(t) = 359.3\cos(\omega t -75);$
$i(t) = 100 \cos(\omega t + 5);$
The general formula for $v(t) = V_{max} \cos(\omega t+\delta)$ and $i(t) = I_{max} \cos(\omega t+\beta)$
$P(t) = VI_{R}[1 + \cos[2\omega t + \delta]] + VI_{X}\sin(2(\omega t + \delta))$
$V = \frac{V_{max}} {\sqrt 2}$
$I_R = VI\cos(\delta - \beta) = 359.3 * 100/2 *\cos(-75-5) = 3119$
$I_X = VI \sin(\delta - \beta) = -17692$
$P(t) = 3119(1 + \cos(2\omega t -75))-17692(\sin(2\omega t -75))$
Is it correct?

 

[AvroArrow]

Yes, your calculations and answer are correct. The only thing to note is that the power equation you used is for the case when both voltage and current are in cosine terms. In your case, you have one in cosine terms and one in sine terms. So, you need to convert one of them (in this case voltage) to cosine terms to use the power equation.