Answer
$R = 707~\Omega$
Work Step by Step
Ir is given that the impedance is $~~Z = 1000~\Omega$
It is given that $\phi = 45^{\circ}$
We can find an expression for $X_L - X_C$:
$tan~\phi = \frac{X_L-X_C}{R}$
$X_L-X_C = R~tan~\phi$
We can find the resistance $R$:
$Z = \sqrt{R^2+(X_L-X_C)^2} = 1000~\Omega$
$\sqrt{R^2+(R~tan~\phi)^2} = 1000~\Omega$
$\sqrt{R^2~(1+~tan^2~\phi)} = 1000~\Omega$
$R^2~(1+~tan^2~\phi) = (1000~\Omega)^2$
$R^2 = \frac{(1000~\Omega)^2}{1+~tan^2~\phi}$
$R = \frac{1000~\Omega}{\sqrt{1+~tan^2~\phi}}$
$R = \frac{1000~\Omega}{\sqrt{1+~tan^2~45^{\circ}}}$
$R = \frac{1000~\Omega}{\sqrt{2}}$
$R = 707~\Omega$
