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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(2) + 1/sqrt(2)"i"`
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Solution
Let z = `(1)/sqrt(2) + (1)/sqrt(2)"i"`
∴ a = `(1)/sqrt(2)`, b = `(1)/sqrt(2)`, a > 0, b > 0
∴ |z| = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt(((1)/sqrt(2))^2 + ((1)/sqrt(2))^2`
= `sqrt(1/2 + 1/2)`
= 1
Here `(1/sqrt(2), 1/sqrt(2))` lies in 1st quadrant
amp (z) = θ = `tan^-1("b"/"a")`
= `tan^-1((1/sqrt(2))/(1/sqrt(2)))`
= tan–1(1) = `pi/4`
∴ θ = 45° = `pi/4`
∴ the polar form of z = r(cos θ + i sin θ)
= 1(cos 45° + i sin 45°)
= `1(cos pi/4 + "i"sin pi/4)`
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