The principal value of the z and - z - of the complex number z -1-cos-isin-5-cos-isin- isA- -2- 32 cos 5-2B- -2- 16 cos 4-2C- None of theseD- -2- 32 cos 5-2

We have, z=(1+cosθ+isinθ)^5/(cosθ+isinθ)^3 ={ 1+2cos^2θ/2 1+2isinθ/2cosθ/2}^5/cosθ+isinθ =32cos^5θ/2{ cosθ/2+isinθ/2}^5/cosθ+isinθ =32cos^5θ/2{ cos5θ/2+isin5 ...

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Standard XII

Mathematics

Summation by Sigma Method

The principal...

Question

The principal value of the arg(z) and |z| of the complex number $z = \frac{(1 + c o s θ + i s i n θ)^{5}}{(c o s θ + i s i n θ)}$ is

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None of these

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Solution

The correct option is A
We have, $z = \frac{(1 + c o s θ + i s i n θ)^{5}}{(c o s θ + i s i n θ)^{3}} = \frac{{1 + 2 c o s^{2} \frac{θ}{2} - 1 + 2 i s i n \frac{θ}{2} c o s \frac{θ}{2}}^{5}}{c o s θ + i s i n θ} = \frac{32 c o s^{5} \frac{θ}{2} {c o s \frac{θ}{2} + i s i n \frac{θ}{2}}^{5}}{c o s θ + i s i n θ} = 32 c o s^{5} \frac{θ}{2} {c o s \frac{5 θ}{2} + i s i n \frac{5 θ}{2}} {c o s 3 θ - i s i n 3 θ} = 32 c o s^{5} \frac{θ}{2} {c o s {\frac{5 θ}{2} - 3 θ} + i s i n {\frac{5 θ}{2} - 3 θ}} = 32 c o s^{5} \frac{θ}{2} [c o s (\frac{- θ}{2}) + i s i n (\frac{- θ}{2})] H e n c e t h e m o d u l u s a n d a r g u m e n t o f z i s 32 [c o s (\frac{- θ}{2}) a n d (\frac{- θ}{2})] r e s p e c t i v e l y .$

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The principal value of the arg(z) and |z| of the complex number z=(1+cosθ+isinθ)5(cosθ+isinθ) is

The principal value of the arg(z) and |z| of the complex number $z = \frac{(1 + c o s θ + i s i n θ)^{5}}{(c o s θ + i s i n θ)}$ is