A series $RLC$ circuit is driven by a generator at frequency $1000 Hz$. The inductance is $90.0 mH$; capacitance is $0.500 \mu F$; and the phase constant has a magnitude of $60.0°$ (Take ${\mathrm{\pi }}^2=10$)

$$\begin{gathered} \omega L=90\times {10}^{-3}\times 2\mathrm{\pi }\times 1000=180\mathrm{\pi } \\ \frac{1}{\mathrm{\omega C}}=\frac{1}{0.5\times {10}^{-6}}\times 2\mathrm{\pi }\times 1000=\frac{1000}{\mathrm{\pi }} \end{gathered}$$

$\Rightarrow$circuit is inductive $V_L>V_R$

$\Rightarrow$voltage leads the current

$$\begin{gathered} \tan \varphi =\frac{\omega L-{\displaystyle \frac{1}{\omega C}}}{R}=\frac{800}{\mathrm{\pi }R} \\ R=\frac{800}{\mathrm{\pi }\sqrt{3}} \end{gathered}$$

at resonance, $\omega =\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{90\times {10}^{-3}\times 0.5\times {10}^{-6}}}rad/s$

$=\frac{1}{\sqrt{45\times {10}^{-9}}}=\frac{{10}^5}{15\times \sqrt{2}}=\frac{\sqrt{2}}{3}\times {10}^4$ rad/s