I would like to calculate the reactive power of a given two-pole which has the following voltage and current characteristic.

\$u(t) = 2+5sin(\omega t)+2sin(2\omega t+\frac{\pi}{6});i(t) = 10+3cos(\omega t-\frac{\pi}{6})+8sin(3\omega t)\$

So I know the reactive power is calculated by one the following formulas:

enter image description here

I started to work with phasors , I converted all the sin() functions into cos() like

\$2sin(2\omega t+\frac{\pi}{6}) = 2cos(\frac{\pi}{2}-(2\omega t+\frac{\pi}{6})) = 2cos(\frac{\pi}{3}-2\omega t) \$ then complex phasor \$e^{j(\frac{\pi}{3}-2\omega t)}\$ so i this this procedure for every trigonometric function in \$ u(t) \$ and \$i(t)\$

then i used the following formula:

enter image description here

and then simplified the whole expression, i hoped that the exponents will turn out to be the same but they differ.

This question is a small part of an signals and systems exam it's going for 2 points out of 30 so i am guessing (since my calculation taking so long) there is a faster and simplier way to get the right answer ? The following picture is just showing that all the exponents are different:

enter image description here


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