Question

An alternating current is given by i = (3 sin t + 4 cos t)a. The rms current is given by

Answer 1

You can differentiate the expression to get the answer

Answer 2

Answer:

RMS stands for root mean square

So first we square

${i}^{2} = {a}^{2} ( 9 \sin {}^{2} (t) + 16 \cos {}^{2} (t) + 12(2 \sin(t) \cos(t) ) \\ {i}^{2} = {a}^{2} ( 9 \sin {}^{2} (t) + 16 \cos {}^{2} (t) + 12 \sin(2t) )$

Now average

You can use formula

$avg = \frac{ \int_{0} ^{T} f(t) dt }{T}$

from ( 0 , π )

Or be smart

Sin2x has zero average value . Sin^2x and cos^2x 's average value is 1/2 .

So

${i}^{2} = {a}^{2} (16 + 9) \frac{1}{2} = \frac{25 {a}^{2} }{2}$

Now take root and done

$i_{rms} = \sqrt{ \frac{25 {a}^{2} }{2} } = \frac{5a}{ \sqrt{2} }$