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Solving Simultaneous Trigonometric Equations

If α is the f...

Question

If $α$ is the fifth root of unity then which of the following is false

A

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B

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C

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D

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Solution

The correct option is D

We have $α$ = cos $\frac{2 π}{5}$ + isin $\frac{2 π}{5}$ and $1 + α + α^{2} + α^{3} + α^{4} = 0$ ...........(i)

∴ $∣ ∣ 1 + c o s \frac{2 π}{5} + i s i n \frac{2 π}{5} ∣ ∣$ = $∣ ∣ 2 c o s \frac{π}{5} (c o s \frac{π}{5} + i s i n \frac{π}{5}) ∣ ∣$ = 2cos $\frac{π}{5}$

Again from (i), |1 + $α$ | = |1 + $α$ + $α^{2}$ | = 2cos $\frac{π}{5}$

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