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Question
Find the modulus and amplitude of the following complex numbers.
−4 − 4i
Solution
Let z = −4 − 4i.
Here, a = −4, b = −4 i.e., a < 0, b < 0
∴ |z| = `sqrt(xa^2 + y^2)`
= `sqrt((-4)^2 + (-4)^2`
= `sqrt(16 + 16)`
= `sqrt(32)`
= `4sqrt(2)`
Here, (-4, -4) lies in 3rd quadrant.
∴ amp (z) = `tan^-1(y/x) + pi`
= `tan^-1((-4)/(-4)) + pi`
`= tan^-1(1) = tan^-1 (tan pi/4)`
`= pi/4`
Hence, modulus = `4sqrt2` and
amplitude = `tan^-1 (1) or (pi/4)`
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