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Trigonometric Equations — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Equations5 Marks
Question
Solve the following equations:
$\sin\text{x}+\sin2\text{x}+\sin3=0$
✓

Answer

We have,
$\sin\text{x}+\sin2\text{x}+\sin3=0$
$\Rightarrow\sin2\text{x}+2\sin2\text{x}.\cos\text{x}=0$
$\Rightarrow\sin2\text{x}+(1+2\cos\text{x})=0$
$\Rightarrow\text{Either}$
$\sin2\text{x}=0$ or $1+2\cos\text{x}=0$
$\Rightarrow2\text{x}=\text{n}\pi,\text{n}\in\text{z}$ or $\cos\text{x}=-\frac{1}{2}=\cos\Big(\pi-\frac{\pi}{3}\Big)$
$\Rightarrow\text{x}=\frac{\text{n}\pi}{2},\text{n}\in\text{z}$ or $\text{x}=2\text{m}\pi\pm\frac{2\pi}{3},\text{m}\in\text{z}$
Thus,
$\text{x}=\frac{\text{n}\pi}{2},\text{n}\in\text{z}$ or $\text{x}=2\text{m}\pi\pm\frac{2\pi}{3},\text{m}\in\text{z} $

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