$\begin{align}  & \sin \theta =2\sin \tfrac{\theta }{2}\cos \tfrac{\theta }{2}=\frac{2\sin \tfrac{\theta }{2}\cos \tfrac{\theta }{2}}{1} \\ & \text{Note: }{{\sin }^{2}}\tfrac{\theta }{2}+{{\cos }^{2}}\tfrac{\theta }{2}=1 \\ & \sin \theta =\frac{2\sin \tfrac{\theta }{2}\cos \tfrac{\theta }{2}}{{{\sin }^{2}}\tfrac{\theta }{2}+{{\cos }^{2}}\tfrac{\theta }{2}} \\ & \text{Divide both the numerator and denominator } \\ & \text{by }{{\cos }^{2}}\tfrac{\theta }{2} \\ & \sin \theta =\frac{\tfrac{2\sin \tfrac{\theta }{2}\cos \tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}}{\tfrac{{{\sin }^{2}}\tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}+\tfrac{{{\cos }^{2}}\tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}}=\frac{2\tan \tfrac{\theta }{2}}{{{\tan }^{2}}\tfrac{\theta }{2}+1} \\ & \sin \theta =\frac{2\tan \tfrac{\theta }{2}}{{{\tan }^{2}}\tfrac{\theta }{2}+1} \\ & \cos \theta ={{\cos }^{2}}\tfrac{\theta }{2}-{{\sin }^{2}}\tfrac{\theta }{2}=\frac{{{\cos }^{2}}\tfrac{\theta }{2}-{{\sin }^{2}}\tfrac{\theta }{2}}{1} \\ & \cos \theta =\frac{{{\cos }^{2}}\tfrac{\theta }{2}-{{\sin }^{2}}\tfrac{\theta }{2}}{{{\sin }^{2}}\tfrac{\theta }{2}+{{\cos }^{2}}\tfrac{\theta }{2}} \\ & \text{Divide both the numerator and denominator } \\ & \text{by }{{\cos }^{2}}\tfrac{\theta }{2} \\ & \cos \theta =\frac{\tfrac{{{\cos }^{2}}\tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}-\tfrac{{{\sin }^{2}}\tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}}{\tfrac{{{\sin }^{2}}\tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}+\tfrac{{{\cos }^{2}}\tfrac{\theta }{2}}{{{\cos }^{2}}\tfrac{\theta }{2}}}=\frac{1-{{\tan }^{2}}\tfrac{\theta }{2}}{{{\tan }^{2}}\tfrac{\theta }{2}+1} \\ & \cos \theta =\frac{1-{{\tan }^{2}}\tfrac{\theta }{2}}{{{\tan }^{2}}\tfrac{\theta }{2}+1} \\ & \text{Now let }\tan \tfrac{\theta }{2}=t \\ & \sin \theta =\frac{2t}{1+{{t}^{2}}},\text{ }\cos \theta =\frac{1-{{t}^{2}}}{{{t}^{2}}+1} \\ & 55\cos \theta -48\sin \theta =51 \\ & 55\left( \frac{1-{{t}^{2}}}{1+{{t}^{2}}} \right)-48\left( \frac{2t}{1+{{t}^{2}}} \right)=51 \\ & 55(1-{{t}^{2}})-48(2t)=51(1+{{t}^{2}}) \\ & 106{{t}^{2}}+96t-4=0 \\ & t=\frac{-96\pm \sqrt{{{96}^{2}}-4(106)(-4)}}{2(106)} \\ & t=-0.04378\text{ or }-0.8619 \\ &  \\ & \text{when }t=-0.04378,\text{ }\tan \tfrac{\theta }{2}=-0.04378 \\ & \frac{\theta }{2}={{\tan }^{-1}}-0.04378={{177.5}^{\circ }} \\ & \theta ={{355}^{\circ }} \\ & \text{when }t=-0.8619 \\ & \frac{\theta }{2}={{\tan }^{-1}}(-0.8619)={{139.24}^{\circ }} \\ & \theta ={{278.48}^{\circ }} \\ & \theta ={{355}^{\circ }}\text{, }{{278.48}^{\circ }} \\\end{align}$