Calculating the current of an AC circuit
What ever you have done is correct except at the last step. 

The current flows from +ve terminal to -ve terminal and the current \$i(t)\$ marked in the schematic is in the same direction. So the voltage you have to consider is +U. That will change the phase angle by \$\pi\$. ie.,
$$I = \left( U\over Z_{total}\right) = \left( 10\over (1000+5000\sqrt2)+j(1200+5000\sqrt2)\right)$$
$$ = 0.000865e^{-j(0.007976+\pi)}$$ 
$$= 0.000865\sin(1000t-0.007976-\pi)$$

The ac current through the circuit is going to be a sinusoid with same frequency as the voltage source, phase shifted (lag) by \$\mathbf{0.007976+\pi}\$ and peak value of \$0.865mA\$. 

Since there are reactive elements in the circuit, the current will have a phase difference from the voltage. So this makes perfect sense.