A+2 2 ~A+ 3 ~A is equal to which of the following? 1. 4 2 ~A ^2 (...

Let $\mathrm{A}=30^{{}^{\circ}}$$\begin{aligned} \Rightarrow & \sin \mathrm{A}+2 \sin 2 \mathrm{ A}+\sin 3 \mathrm{ A}=\sin 30^{{}^{\circ}}+2 \sin 60^{{}^{\circ}}+\sin 90^{{}^{\circ}} \\ \quad & \frac{1}{2}+\frac{2 \sqrt{3}}{2}+1=\frac{2 \sqrt{3}+3}{2} \\ & \left(\because 2 \cos ^2 \mathrm{ A}=1+\cos 2 \mathrm{ A}\right) \end{aligned}$Now, $4 \sin 2 \mathrm{ A} \cos ^2\left(\frac{\mathrm{ A}}{2}\right)=2 \sin 2 \mathrm{ A}[1+\cos \mathrm{A}]=2 \sin 60^{{}^{\circ}}\left[1+\cos 30^{{}^{\circ}}\right]=\frac{2 \sqrt{3}+3}{2}$ Also, $\sin 2 \mathrm{ A}=2 \sin \mathrm{ A} \cos \mathrm{ A} \& \sin ^2 \mathrm{ A}+\cos ^2 \mathrm{ A}=1\begin{aligned} & 2 \sin 2 \mathrm{ A}\left[\sin \frac{\mathrm{ A}}{2}+\cos \frac{\mathrm{A}}{2}\right]^2 \\ & =2 \sin 2 \mathrm{ A}\left[\sin ^2 \frac{\mathrm{ A}}{2}+\cos ^2 \frac{\mathrm{ A}}{2}+2 \sin \frac{\mathrm{ A}}{2} \cos \frac{\mathrm{ A}}{2}\right] \\ & =2 \sin 2 \mathrm{ A}[1+\sin \mathrm{A}]=2 \sin 60^{{}^{\circ}}\left[1+\sin 30^{{}^{\circ}}\right] \\ & =\frac{3 \sqrt{3}}{2} \end{aligned}$$\begin{aligned} & & 8 \sin A \cos A \cos ^2\left(\frac{\mathrm{ A}}{2}\right) \\ & =4 \sin \mathrm{ A} \cos \mathrm{ A}[1+\cos \mathrm{A}] \\ & =4 \sin 30^{{}^{\circ}} \cos 30^{{}^{\circ}}\left[1+\cos 30^{{}^{\circ}}\right] \\ & =\frac{2 \sqrt{3}+3}{2} \end{aligned}