Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTRIGONOMETRIC FUNCTIONS4 Marks
Question
Find the general solution for each of the following equations: $\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$
✓
Answer
Given: $\sin\text{x}+\sin3\text{x}+\sin5\text{x}=0$ $\Rightarrow2\sin\Big(\frac{5\text{x}+\text{x}}{2}\Big)\cos\Big(\frac{5\text{x}-\text{x}}{2}\Big)+\sin3\text{x}=0$ $\Rightarrow2\sin3\text{x}\cos2\text{x}+\sin3\text{x}=0$ $\Rightarrow\sin3\text{x}(2\cos2\text{x}+1)=0$ $\Rightarrow\sin3\text{x}=0$ or $2\cos2\text{x}+1=0$ $\Rightarrow3\text{x}=\text{n}\pi$ or $\cos2\text{x}=\frac{-1}{2}=\cos\frac{2\pi}{3},\text{n}\in\text{Z}$ $\Rightarrow\text{x}=\frac{\text{n}\pi}{3}$ where $2\text{x}=2\text{n}\pi\pm\frac{2\pi}{3},\text{n}\in\text{Z}$ $\Rightarrow\text{x}=\frac{\text{n}\pi}{3}$ where $\text{x}=\text{n}\pi\pm\frac{\pi}{3},\text{n}\in\text{Z}$
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