AHellelepoeowoe GELP

Using the first possible value of  B, that is,

B = arcsin(5/6)

Using the second possible value of  B, that is,

B = π - arcsin(5/6)

$A\,=\,\pi-B-C \qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\~\\ A\,=\,\pi-\arcsin(\frac56)-\frac{\pi}{6}\\~\\ A\,=\,\frac{5\pi}{6}-\arcsin(\frac56)\\~\\ \sin(A)\,=\,\sin(\frac{5\pi}{6}-\arcsin(\frac56))\\~\\ \sin(A)\,=\,(\frac12)(\frac{\sqrt{11}}{6})-(-\frac{\sqrt3}{2})(\frac56)\\~\\ \sin(A)\,=\,\frac{\sqrt{11}\,+\,5\sqrt3}{12}$

By the Law of Sines:

$\frac{\sin A}{BC}\,=\,\frac{\sin \frac{\pi}{6}}{6}\\~\\ \frac{\sin A}{BC}\,=\,\frac{1}{12}\\~\\ \frac{BC}{\sin A}\,=\,\frac{12}{1}\\~\\ BC=12\sin A\\~\\ BC\,=\,12(\frac{5\sqrt3+\sqrt{11}}{12})\\~\\ BC\,=\,5\sqrt3+\sqrt{11}$

$A\,=\,\pi-B-C\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\~\\ A\,=\,\pi-(\pi-\arcsin(\frac56))-\frac{\pi}{6}\\~\\ A\,=\,\arcsin(\frac56)-\frac{\pi}{6}\\~\\ \sin(A)\,=\,\sin(\arcsin(\frac56)-\frac{\pi}{6}) \\~\\ \sin(A)\,=\, (\frac56)(\frac{\sqrt3}{2})-(\frac{\sqrt{11}}{6})(\frac12) \\~\\ \sin(A)\,=\,\frac{5\sqrt3-\sqrt{11}}{12}$

By the Law of Sines:

$\frac{\sin A}{BC}\,=\,\frac{\sin \frac{\pi}{6}}{6}\\~\\ \frac{\sin A}{BC}\,=\,\frac{1}{12}\\~\\ \frac{BC}{\sin A}\,=\,\frac{12}{1}\\~\\ BC=12\sin A\\~\\ BC\,=\,12(\frac{5\sqrt3-\sqrt{11}}{12})\\~\\ BC\,=\,5\sqrt3-\sqrt{11}$