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प्रश्न
Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:
(1 + i)6
उत्तर
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
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