TS EAMCET · Maths · Complex Number
If \(\theta=\frac{\pi}{6}\), then the \(10^{\text {th }}\) term of the series \(1+(\cos \theta+i \sin \theta)^1+(\cos \theta+i \sin \theta)^2+\ldots\) is
- A \(-1\)
- B \(-i\)
- C \(\frac{1}{2}+\frac{\sqrt{3} i}{2}\)
- D \(1\)
Answer & Solution
Correct Answer
(B) \(-i\)
Step-by-step Solution
Detailed explanation
\(T_{10}=(\cos \theta+i \sin \theta)^9\) \(\begin{aligned} & =\cos 9 \theta+i \sin 9 \theta=\cos \frac{9 \pi}{6}+\sin \frac{9 \pi}{6} \\ & =\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}=0+i(-1)=-i\end{aligned}\)
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