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Question
Select the correct answer from the given alternatives:
The modulus and argument of `(1 + "i"sqrt(3))^8` are respectively
Options
2 and `(2pi)/3`
256 and `(8pi)/3`
256 and `(2pi)/3`
64 and `(4pi)/3`
Solution
256 and `(2pi)/3`
Explanation:
Let z = `(1 + "i"sqrt(3))^8` = [r (cos θ + i sin θ)]8
r cos θ = 1, r sin θ = `sqrt(3)`,
r = `sqrt(1 + 3)`
= 2
∴ cos θ = `1/2`, sin θ = `sqrt(3)/2`
∴ arg z = `pi/3`
z = `[2(cos pi/3 + "i"sin pi/3)]^8`
= `2^8 (cos (8pi)/3 + "i" sin (8pi)/3)`
= `256[cos(2pi + (2pi)/3) + "i"sin(2pi + (2pi)/3)]`
= `256[cos (2pi)/3 + "i"sin (2pi)/3]`
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