( +i )^n is equal to - Vidyadip

MCQ

$(\sin \theta+i \cos \theta)^n$ is equal to

A
$\cos n \theta+i \sin n \theta$
B
$\sin n \theta+i \cos n \theta$
✓
$\cos n\left(\frac{\pi}{2}-\theta\right)+i \sin n\left(\frac{\pi}{2}-\theta\right)$
D
$\cos n\left(\frac{\pi}{2}-\theta\right)-i \sin n\left(\frac{\pi}{2}-\theta\right)$

✓

Answer

Correct option: C.

$\cos n\left(\frac{\pi}{2}-\theta\right)+i \sin n\left(\frac{\pi}{2}-\theta\right)$

(C)
$(\sin \theta+ i \cos \theta)^{ n }=\left[\cos \left(\frac{\pi}{2}-\theta\right)+ i \sin \left(\frac{\pi}{2}-\theta\right)\right]^{ n }$
$=\cos n\left(\frac{\pi}{2}-\theta\right)+i \sin n\left(\frac{\pi}{2}-\theta\right)$

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