Answer
$d = \sqrt {17} $
Work Step by Step
$$\eqalign{
& {\text{Let the points: }}\underbrace {\left( {1,\frac{{5\pi }}{6}} \right)}_{\left( {{r_1},{\theta _1}} \right)}{\text{ and }}\underbrace {\left( {4,\frac{\pi }{3}} \right)}_{\left( {{r_2},{\theta _2}} \right)} \cr
& {\text{Apply }}d = \sqrt {r_1^2 + r_2^2 - 2{r_1}{r_2}\cos \left( {{\theta _1} - {\theta _2}} \right)} \cr
& d = \sqrt {{{\left( 1 \right)}^2} + {{\left( 4 \right)}^2} - 2\left( 1 \right)\left( 4 \right)\cos \left( {\frac{{5\pi }}{6} - \frac{\pi }{3}} \right)} \cr
& {\text{Simplifying}} \cr
& d = \sqrt {1 + 16 - 8\cos \left( {\frac{\pi }{2}} \right)} \cr
& d = \sqrt {17 - 8\left( 0 \right)} \cr
& d = \sqrt {17} \cr} $$
