The average power dissipated ⟨P⟩ in an AC circuit with voltage V and current I is given by:
⟨P⟩=VrmsIrmscosϕ,
where Vrms is the root mean square (RMS) value of the voltage, Irms is the RMS value of the current, and ϕ is the phase difference between the voltage and current.
The given voltage is V=20sin200πt, so the peak voltage V0 is 20 V. The RMS value of the voltage Vrms is:
Vrms=2V0=220=102V.
Similarly, the given current is I=10sin(200πt+3π), so the peak current I0 is 10 A. The RMS value of the current Irms is:
Irms=2I0=210=52A.
The current leads the voltage by 3π, meaning the phase difference ϕ is:
ϕ=3π=60∘.
Since ϕ=60∘:
cosϕ=cos60∘=21.
Now, substitute the values of Vrms, Irms, and cosϕ into the formula for average power: