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This is a question of complex nos. The pricipal value of the arg (z) and |z| of the complex no. z=(1+cos theta+i sin theta)^5/(cos theta + i sin theta) ^3 is?? - theta/2 , 32 cos^5 theta/2 theta/2 , 32 cos^5 theta/2 -theta/2, 16 cos^4 theta/2 none of these

This is a question of complex nos.
 
The pricipal value of the arg (z) and |z| of the complex no. z=(1+cos theta+i sin theta)^5/(cos theta + i sin theta) ^3 is??
 
  1. - theta/2 , 32 cos^5 theta/2
  2. theta/2 , 32 cos^5 theta/2
  3. -theta/2, 16 cos^4 theta/2
  4. none of these

Grade:11

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
5 years ago
using de moviesr theorm we can simplify the above complex number as
z = \frac{(1+cos\theta + isin\theta)^5}{e^{i3\theta}} = \frac{(2cos^2\frac{\theta}{2} + 2isin\frac{\theta}{2}cos\frac{\theta}{2})^5}{e^{i3\theta}}
z = 32cos^5\frac{\theta}{2} \frac{e^{i5\theta/2}}{e^{i3\theta}} = 32cos^5\frac{\theta}{2} e^{-i\theta/2}

hence the correct answer is option 1

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