Decide P controller gain K so that phase margin is 50 deg
From you question I understand that you have written one equation and have unknown variables in it; viz, \$\omega_c\$ and K. If you write one more equation, you may be able to solve for K. To that end,
Definition of gain cross over frequency is
\$
\begin{align}
|GF|_{s=j\omega_c} ={}&amp; 1\\
\frac{K}{|1+j\omega_c|^2} ={}&amp; 1\\
K^2={}&amp; (1+\omega_c^2)^2 \\
\end{align}\$
Solve for \$\omega_c\$ in terms of K.
In the phase equation you already have with you, substitute \$\omega_c\$ with the expression containing K. Now that equation has only one unknown, viz. K. Solve!