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Question
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`(1 + sqrt(3)"i")/2`
Solution
Let z = `(1 + sqrt(3)"i")/2 = 1/2 + sqrt(3)/2"i"`
∴ a = `1/2`, b = `sqrt(3)/2`, a, b > 0
∴ |z| = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt((1/2)^2 + (sqrt(3)/2)^2`
= `sqrt(1/4 + 3/4)`
= 1
Here `(1/2, sqrt(3)/2)` lies in 1st quadrant
amp (z) = θ = `tan^-1("b"/"a")`
= `tan^-1((sqrt(3)/2)/(1/2))`
= `tan^-1(sqrt(3))`
= `pi/3`
∴ θ = 60° = `pi/3`
∴ the polar form of z = r(cos θ + i sin θ)
= 1(cos 60° + i sin 60°)
= `1(cos pi/3 + "i" sin pi/3)`
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