Answer
$d \approx 12.0652$
Work Step by Step
$$\eqalign{
& {\text{Let the points: }}\underbrace {\left( {8,\frac{{7\pi }}{4}} \right)}_{\left( {{r_1},{\theta _1}} \right)}{\text{ and }}\underbrace {\left( {5,\pi } \right)}_{\left( {{r_2},{\theta _2}} \right)} \cr
& {\text{Apply }}d = \sqrt {r_1^2 + r_2^2 - 2{r_1}{r_2}\cos \left( {{\theta _1} - {\theta _2}} \right)} \cr
& d = \sqrt {{{\left( 8 \right)}^2} + {{\left( 5 \right)}^2} - 2\left( 8 \right)\left( 5 \right)\cos \left( {\frac{{7\pi }}{4} - \pi } \right)} \cr
& {\text{Simplifying}} \cr
& d = \sqrt {64 + 25 - 80\cos \left( {\frac{{3\pi }}{4}} \right)} \cr
& d = \sqrt {89 - 80\left( { - \frac{{\sqrt 2 }}{2}} \right)} \cr
& d = \sqrt {89 + 40\sqrt 2 } \cr
& d \approx 12.0652 \cr} $$
