Maharashtra State BoardHSC Arts 11th
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Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem: (1 + i)6 - Mathematics and Statistics

Sum

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

(1 + i)6

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Solution

Let z = 1 + i

∴ a = 1, b = 1 , i.e. a > 0, b > 0

∴ |z| = `sqrt("a"^2 + "b"^2) = sqrt(1^2 + 1^2) = sqrt(2)`

Here (1, 1) lies in 1st quadrant.

∴ amp (z) = `tan^-1("b"/"a")`

= `tan^1(1/1)`

= `pi/4`

z6 = (1 + i)6

= `[sqrt(2)(cos  pi/4 + "i" sin  pi/4)]^6`

= `8[cos  (6pi)/4 + "i"sin  (6pi)/4]`  ...[∵ (cos θ + i sin θ)n = (cos n θ + i sin n θ)]

= `8[cos  (3pi)/2 + "i"sin  (3pi)/2]`

= `8[0 + "i" (-1)]`

= – 8i

Concept: De Moivres Theorem
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