Determining initial conditions for a simple circuit
Firstly, note that \$\small 14.1\$ is probably \$\small 10\sqrt 2\$.

At \$\small \omega=1000\$ rad/sec, \$\small X_C=-j10\$. Hence, the voltage across the capacitor, relative to a sinusoidal voltage source of amplitude \$\small 10\sqrt2\$, but with zero phase angle, \$\small \phi = 0^o\$, would be:

\$\small V_C=10\sqrt2\:\dfrac{-j10}{10-j10}= 10\sqrt 2\:\dfrac{-j}{1-j}=10\sqrt 2\:\dfrac{1-j}{2} \equiv 10\:\large\angle\small{-45^o}\$.

However, the source actually has a phase angle, \$\small \phi=135^o\$, hence the actual \$\small V_C\$ is:

\$\small V_C=10\:\large\angle\small (135-45)=10\:\large \angle\small {90^o}\$.

Thus, when \$\small t=0\$, the capacitor voltage is: \$\small V_C=10\:sin(90^o)=10V\$, which is the required initial condition.