check the picture below.
[tex]\bf \textit{Law of Cosines}\\\\ c^2 = a^2+b^2-(2ab)cos(C)\implies c = \sqrt{a^2+b^2-(2ab)cos(C)} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{cases} a=500\\ b=600\\ C=45^o \end{cases}\implies c=\sqrt{500^2+600^2-[2(500)(600)]cos(45^o)} \\\\\\ c=\sqrt{610000-600000\cdot \frac{\sqrt{2}}{2}}\implies c\approx \sqrt{185735.93}\implies c\approx 430.97[/tex]
