Capacitance value - AC RC Circuit - Impedance Method

Here is the circuit:

simulate this circuit – Schematic created using CircuitLab

$V_i = \frac{8}{\sqrt 2} \angle0° V\\ V_c = \frac{6}{\sqrt 2} \angle45° V\\ R = 1000\space\Omega$

I'd like to know how to find the capacitance with these given values. What I've tried:

$V_c = V_i \times \frac{X_c}{R\space\space+\space\space X_c} \\ X_c = \frac{V_c\space R}{V_i\space\space - \space\space V_c}$

Here is the problem, just solving for Xc, I got a complex number with a angle different than -90°. So I forced Xc to be -90°, then when I solve for the capacitance c, I get a complex number:

$X_c = 1058.7141\angle 93.4716182° \\ \frac{1}{j\frac{500}{2\pi}c} = 1058.7141\angle 93.4716182° \\ c = -0.00001184768784 + j\space 7.18682779 \times 10^{-7}$

What am I possibly missing?

It's a malformed question because with these components, the phase angle of $V_C$ must be $-90^\circ$. The only way it could be $45^\circ$ would be if there were also an inductance in the circuit.
• @JoãoPauloOliveiraFernandes: The equations are correct up to point at which you get a value for the complex impedence, but it's not really $X_C$ but rather $X_N$ where $N$ is some strange compound part with the impedance you calculated. A complex capacitance has no real physical sense. Dec 13, 2014 at 18:24