I've a parallel RL circuit:
simulate this circuit – Schematic created using CircuitLab
And I know that (the RMS input-current) \$\overline{\text{I}}_{\space\text{in}}=2\$ A, (the RMS input-voltage) \$\overline{\text{V}}_{\space\text{in}}=200\$ V, the input frequency is \$50\$ Hz and \$\overline{\text{P}}_{\space\text{in}}=69.44\$ W now I need to find \$\text{R}\$ and \$\text{L}\$:
$$ \begin{cases} \overline{\text{I}}_{\space\text{in}}=\frac{\overline{\text{V}}_{\space\text{in}}}{\left|\underline{\text{Z}}_{\space\text{in}}\right|}=\frac{\overline{\text{V}}_{\space\text{in}}}{\sqrt{\frac{R\omega L}{R^2+(\omega L)^2}}}\\ \\ P=\overline{\text{V}}_{\space\text{in}}\cdot\overline{\text{I}}_{\space\text{in}}\cdot\cos\left(\varphi_{\space\text{in}}\right)=\overline{\text{V}}_{\space\text{in}}\cdot\overline{\text{I}}_{\space\text{in}}\cdot\cos\left(\frac{\pi}{2}-\arctan\left(\frac{\omega L}{R}\right)\right) \end{cases}\tag1 $$
Using the given values:
$$ \begin{cases} 2=\frac{200}{\sqrt{\frac{R\cdot2\pi\cdot50L}{R^2+(2\pi\cdot50L)^2}}}\\ \\ 69.44=200\cdot2\cdot\cos\left(\frac{\pi}{2}-\arctan\left(\frac{2\pi\cdot50L}{R}\right)\right) \end{cases}\tag2 $$
But when I tried to solve the system I get imaginary numbers, what is my mistake?
EDIT:
I can write:
$$\cos\left(\frac{\pi}{2}-\arctan\left(\frac{2\pi\cdot50L}{R}\right)\right)=\frac{2\pi\cdot50L}{R}\cdot\frac{1}{\sqrt{1+\left(\frac{2\pi\cdot50L}{R}\right)^2}}\tag3$$
So, I get:
$$ \begin{cases} 2=\frac{200}{\sqrt{\frac{R\cdot2\pi\cdot50L}{R^2+(2\pi\cdot50L)^2}}}\\ \\ 69.44=200\cdot2\cdot\frac{2\pi\cdot50L}{R}\cdot\frac{1}{\sqrt{1+\left(\frac{2\pi\cdot50L}{R}\right)^2}} \end{cases}\tag4 $$
Now, for example we get:
$$2=\frac{200}{\sqrt{\frac{R\cdot2\pi\cdot50L}{R^2+(2\pi\cdot50L)^2}}}\space\Longleftrightarrow\space L=\frac{R\pm Ri\sqrt{3999999999999}}{200000000\pi}\tag5$$