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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 - "i")/sqrt(2)`
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Solution
Let z = `(-1 - "i")/sqrt(2) = (-1)/sqrt(2) - 1/sqrt(2)"i"`
This is of the form a + bi, where
a = `(-1)/sqrt(2)`, b = `(-1)/sqrt(2)`
∴ modulus = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt(((-1)/sqrt(2))^2 + ((-1)/sqrt(2))^2`
= `sqrt(1/2 + 1/2)`
= 1
If θ is the amplitude, then
cos θ = `"a"/"r" = -1/sqrt(2)`
and sin θ = `"b"/"r" = -1/sqrt(2)`
`∴ θ = (5pi)/4 ...[(because cos (5pi)/4 = cos(pi + pi/4) = -cos pi/4 = -1/sqrt(2)),(and sin (5pi)/4 = sin (pi + pi/4) = -sin pi/4 = -1/sqrt(2))]`
∴ the polar form of z = r(cos θ + i sin θ)
= `1(cos (5pi)/4 + "i" sin (5pi)/4)`
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