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Question: If \(\cos\theta - i\sin\theta\), then sum \(\frac{(\cos\theta + i\sin\theta)^{4}}{(\sin\theta + i\co...

If cos⁡θ−isin⁡θ\cos\theta - i\sin\theta, then sum (cos⁡θ+isin⁡θ)4(sin⁡θ+icos⁡θ)5\frac{(\cos\theta + i\sin\theta)^{4}}{(\sin\theta + i\cos\theta)^{5}}to 1000 terms is equal to.

A

1

B

– 1

C

I

D

0

Answer

0

Explanation

Solution

β=−q1/3ω\beta = - q^{1/3}\omegaup to 1000 terms

=i(1−i1000)1−i== \frac { i \left( 1 - i ^ { 1000 } \right) } { 1 - i } = x+yω+zω2xω+yω2+z=1ω.zω+yω2+zxω+yω2+z=1ω=ω2\frac{x + y\omega + z\omega^{2}}{x\omega + y\omega^{2} + z} = \frac{1}{\omega}.\frac{z\omega + y\omega^{2} + z}{x\omega + y\omega^{2} + z} = \frac{1}{\omega} = \omega^{2}.