| \[sin\text{ }A+2\text{ }sin\,2A+sin\text{ }3A\]is equal to which of the following? |
| 1. \[4\sin 2A{{\cos }^{2}}\left( \frac{A}{2} \right)\] |
| 2. \[2\sin 2A{{\left( \sin \frac{A}{2}+\cos \frac{A}{2} \right)}^{2}}\] |
| 3. \[8\sin A\cos A{{\cos }^{2}}\left( \frac{A}{2} \right)\] |
| Select the correct answer using the code given below: |
A) 1 and 2 only
B) 2 and 3 only
C) 1 and 3 only
D) 1, 2 and 3
Correct Answer: C
Solution :
Let \[A=30{}^\circ \] \[\Rightarrow \sin A+2\sin 2A+\sin 3A\] \[=\sin 30{}^\circ +2\sin 60{}^\circ +\sin 90{}^\circ \] \[=\frac{1}{2}+\frac{2\sqrt{3}}{2}+1=\frac{2\sqrt{3}+3}{2}\] \[(\because \,\,2{{\cos }^{2}}A=1+\cos 2A)\] Now, \[4\sin 2A{{\cos }^{2}}\left( \frac{A}{2} \right)=2\sin 2A\ [1+\cos A]\] \[=2\sin 60{}^\circ [1+\cos 30{}^\circ ]=\frac{2\sqrt{3}+3}{2}\] Also, \[\sin 2A=2\sin \,A\,\cos A\] & \[{{\sin }^{2}}A+{{\cos }^{2}}A=1\] \[2\sin 2A{{\left[ \sin \frac{A}{2}+\cos \frac{A}{2} \right]}^{2}}\] \[=2\sin 2A\left[ {{\sin }^{2}}\frac{A}{2}+{{\cos }^{2}}\frac{A}{2}+2\sin \frac{A}{2}\cos \frac{A}{2} \right]\] \[=2\sin 2A\left[ 1+\sin A \right]=2\sin 60{}^\circ \left[ 1+\sin 30{}^\circ \right]=\frac{3\sqrt{3}}{2}\]& \[8\sin A\,\,\cos A\,{{\cos }^{2}}\left( \frac{A}{2} \right)\] \[=4\sin A\cos A[1+\cos A]\] \[=4\sin 30{}^\circ cos30{}^\circ [1+cos30{}^\circ ]\] \[=\frac{2\sqrt{3}+3}{2}\]
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