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Question
Express the following numbers in the form x + iy:
`"e"^((-4pi)/3"i")`
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Solution
Let z = `"e"^((-4pi)/3"i")` = reiθ
∴ r = 1 and θ = `(-4pi)/3`
∴ polar form of z = r(cos θ + i sin θ)
= `1[cos ((-4pi)/3) + "i" sin((-4pi)/3)]`,
Where `cos((-4pi)/3) = cos (4pi)/3 = cos(pi + pi/3)`
= `-cos pi/3 = -1/2`
and `sin((-4pi)/3) -sin (4pi)/3 = -sin(pi + pi/3)`
= `-(-sin pi/3) = sqrt(3)/2`
∴ z = `-1/2 + sqrt(3)/2"i"`, which is of the form x + iy.
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