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Let z=cos theta I sin theta Then the v...

Let `z=cos theta I sin theta ` Then the value of `underset(m=1)overset(16) Im (z^(2m-1))` at
`theta = 2^@ ` is

A

`1/(sin 2^@)`

B

`1/(3 sin 2^@)`

C

`1/(2 sin 2^@)`

D

`1/(4 sin 2^@)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given that, `z = cos theta + I sin theta = e^(I theta)`
`therefore underset(u=1)overset(15)Sigma Iu ( zeta ^(2u-1))=underset(u=1)overset(15)SigmaIu(epsi^(theta))^(2 u-1)=underset(u=1)overset(15)SigmaIue^((2 u-1)theta)`
`= sin theta + sin 3 theta + sin 5 theta + ....+ sin 29 theta`
`=(sin((theta + 29theta)/2)sin ((15xx 2 theta )/2))/(sin""((2 theta)/2))`
`(sin ( 15 theta) sin (15 theta))/(sin theta)=1/(4 sin2^@)`
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