The continued product of the four values of -cos-3- i sin-3-3 - 4A- -1B- 1C- -2D- 2

Given: [cos(π3) nbsp;+ isin(π3)]^34 Let z=(cosπ + i sinπ)^34 ⇒ z= nbsp;(cosπ + i sinπ)^1/4 ⇒ z^4=cosπ + i sinπ ⇒ z^4 = 1 If z_1,z_2,z_3,z_4 are the roots, ...

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Standard XII

Mathematics

Geometrical Representation of Argument and Modulus

The continued...

Question

The continued product of the four values of ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$ is

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Solution

The correct option is B $1$
We have, ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$
$= (cos π + i sin π)^{1 / 4}$
Let $z = (cos π + i sin π)^{1 / 4}$
$∴ z^{4} = cos π + i sin π = - 1$
If $z_{1}, z_{2}, z_{3}, z_{4}$ are the roots then
$z_{1} z_{2} z_{3} z_{4} = - 1 (- 1)^{4 - 1} = 1$

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Geometrical Representation of Argument and Modulus

Standard XII Mathematics

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The continued product of the four values of [cos(π3)+isin(π3)]3/4 is

The correct option is B 1We have, [cos(π3)+isin(π3)]3/4 =(cosπ+isinπ)1/4 Let z=(cosπ+isinπ)1/4 ∴z4=cosπ+isinπ=−1 If z1,z2,z3,z4 are the roots then z1z2z3z4=−1(−1)4−1=1

The continued product of the four values of ${[cos (\frac{π}{3}) + i sin (\frac{π}{3})]}^{3 / 4}$ is