The common principal solution of the equation | vedclass

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(D) For $\sin 	heta = -\frac{1}{2}$,the principal values are $	heta = \pi + \frac{\pi}{6} = \frac{7 \pi}{6}$ and $	heta = 2 \pi - \frac{\pi}{6} = \

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None of these

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(D) For $\sin 	heta = -\frac{1}{2}$,the principal values are $	heta = \pi + \frac{\pi}{6} = \frac{7 \pi}{6}$ and $	heta = 2 \pi - \frac{\pi}{6} = \frac{11 \pi}{6}$.For $	an 	heta = \frac{1}{\sqrt{3}}$,the principal values are $	heta = \frac{\pi}{6}$ and $	heta = \pi + \frac{\pi}{6} = \frac{7 \pi}{6}$.The common value in both sets is $	heta = \frac{7 \pi}{6}$.However,the principal solution for $	an 	heta$ is defined in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$,where $	an 	heta = \frac{1}{\sqrt{3}}$ gives $	heta = \frac{\pi}{6}$.Since $\sin \frac{\pi}{6} = \frac{1}{2} 
eq -\frac{1}{2}$,there is no common principal solution.

The common principal solution of the equations $\sin \theta = -\frac{1}{2}$ and $\tan \theta = \frac{1}{\sqrt{3}}$ is

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