Two alternating currents i1 and i2 flow into a circuit node

Two alternating currents i_1 and i_2 flow into a circuit node. The output current i is given by i_1 + i_2. If i_1 = 5 sin (50 t + pi/3) i_2 = 6 cos 50t calculate the amplitude of i and the first time that it occurs.

Solution

first we need to calculate the amplitude of i

5 sin ( 50t +60)
= 5 ( sin 50t cos60 + sin60cos50t )
= 2.5 sin 50t + 4.33 cos 50t

2.5 sin 50t + 4.33 cos 50t + 6 cos 50t
= 2.5 sin 50t +10.33 cos 50t

R² = 2.5²+10.33²
R = 113
R= 10.6


so i(resultant Amplitude) = 10.6