Starting from \(\displaystyle{R}=\sqrt{{{66}}},{a}={\arcsin{{\frac{{{7}}}{{\sqrt{{{65}}}}}}}}\) we have

\(\displaystyle\sqrt{{{65}}}{\sin{{\left({x}+{a}\right)}}}={6}\)

\(\displaystyle\Rightarrow{x}={\arcsin{{\frac{{{6}}}{{\sqrt{{{65}}}}}}}}-{a}={a}{r}{c}{i}{s}{n}{\frac{{{6}}}{{\sqrt{{{65}}}}}}-{\arcsin{{\frac{{{7}}}{{\sqrt{{{65}}}}}}}}\)

Using

\(\displaystyle{\arcsin{{u}}}-{\arcsin{{v}}}={\arcsin{{\left({u}\sqrt{{{1}-{v}^{{2}}}}-{v}\sqrt{{{1}-{u}^{{2}}}}\right)}}}\)

\(\displaystyle{x}={\arcsin{{\left({\frac{{{6}}}{{\sqrt{{{65}}}}}}\cdot{\frac{{{4}}}{{\sqrt{{{65}}}}}}-{\frac{{{7}}}{{\sqrt{{{65}}}}}}\cdot{\frac{{\sqrt{{{65}-{6}^{{2}}}}}}{{\sqrt{{{65}}}}}}\right)}}}\)

\(\displaystyle{x}={\arcsin{{\left({\frac{{{24}-{7}\sqrt{{{29}}}}}{{{65}}}}\right)}}}\)

\(\displaystyle\sqrt{{{65}}}{\sin{{\left({x}+{a}\right)}}}={6}\)

\(\displaystyle\Rightarrow{x}={\arcsin{{\frac{{{6}}}{{\sqrt{{{65}}}}}}}}-{a}={a}{r}{c}{i}{s}{n}{\frac{{{6}}}{{\sqrt{{{65}}}}}}-{\arcsin{{\frac{{{7}}}{{\sqrt{{{65}}}}}}}}\)

Using

\(\displaystyle{\arcsin{{u}}}-{\arcsin{{v}}}={\arcsin{{\left({u}\sqrt{{{1}-{v}^{{2}}}}-{v}\sqrt{{{1}-{u}^{{2}}}}\right)}}}\)

\(\displaystyle{x}={\arcsin{{\left({\frac{{{6}}}{{\sqrt{{{65}}}}}}\cdot{\frac{{{4}}}{{\sqrt{{{65}}}}}}-{\frac{{{7}}}{{\sqrt{{{65}}}}}}\cdot{\frac{{\sqrt{{{65}-{6}^{{2}}}}}}{{\sqrt{{{65}}}}}}\right)}}}\)

\(\displaystyle{x}={\arcsin{{\left({\frac{{{24}-{7}\sqrt{{{29}}}}}{{{65}}}}\right)}}}\)