Assume:

$${v_s}(t) = {V_s} sin(\omega t)$$

and

$${v_C}(t) = {V_C} sin( \omega t - \theta ) .$$

Then

$$C \frac {d {v_C}(t)}{dt }$$ i(t) =

$$C {V_C} \omega cos (\omega t - \theta ) .$$ =

KVL:

v_s(t) = v_C(t) + v_R(t)

= v_C(t) + Ri(t) .

$${V_s} sin( \omega t) {V_C} \left [ sin( \omega t - \theta ) + \omega RC cos ( \omega t - \theta ) \right ]$$ =

$${V_C} \sqrt {1 + ( \omega RC)^2} \left [ \frac{1}{\sqrt {1 + ( \o... ...frac {\omega RC}{\sqrt {1 + ( \omega RC)^2} } cos (\omega t - \theta ) \right ]$$ =

$${V_C} \sqrt {1 + ( \omega RC)^2 } \left [ cos (\delta ) sin (\omega t - \theta ) + sin (\delta ) cos (\omega t - \theta ) \right ]$$ =

$${V_C} \sqrt {1 + ( \omega RC)^2 } sin ( \omega t - \theta + \delta)$$ =

where:

$$tan(\delta ) = \omega RC .$$

Therefore:

$${V_C} = \frac{{V_s}}{ \sqrt {1 + ( \omega RC)^2}}$$

and

$$\theta = \delta = tan^{-1} (\omega RC) .$$

Furthermore,

$${v_R}(t) = {V_s} \frac {\omega RC}{\sqrt {1 + (\omega RC)^2}} cos (\omega t - \delta)$$

and

$${v_C}(t) = {V_s} \frac{1}{ \sqrt {1 + (\omega RC)^2)}} sin(\omega t - \delta) .$$