The complex number with argument frac{5 pi}{6 | vedclass

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Question

(D) Let the complex number be $z = r(\cos 	heta + i \sin 	heta)$.Given that the distance from the origin is $r = 2$ and the argument is $	heta = \f

Vedclass · Accepted Answer

$-\sqrt{3}+i$

Vedclass · Answer

(D) Let the complex number be $z = r(\cos 	heta + i \sin 	heta)$.Given that the distance from the origin is $r = 2$ and the argument is $	heta = \frac{5 \pi}{6}$.Substituting these values,we get $z = 2 \left( \cos \frac{5 \pi}{6} + i \sin \frac{5 \pi}{6} ight)$.Since $\cos \frac{5 \pi}{6} = \cos(\pi - \frac{\pi}{6}) = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}$ and $\sin \frac{5 \pi}{6} = \sin(\pi - \frac{\pi}{6}) = \sin \frac{\pi}{6} = \frac{1}{2}$.Therefore,$z = 2 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} ight) = -\sqrt{3} + i$.

The complex number with argument $\frac{5 \pi}{6}$ at a distance of $2$ units from the origin is

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