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Question
Find the modulus and amplitude of the following complex numbers.
7 − 5i
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Solution
Let z = 7 − 5i
Here a = 7, b = − 5 i.e., a < 0, b > 0
∴ |z| = `sqrt(a^2 + b^2)`
= `sqrt(7^2 + (-5)^2`
= `sqrt(49 + 25)`
= `sqrt(74)`
Here, (7, -5) lies in 4th quadrant.
amp (z) = θ = `tan^-1("b"/"a")+2pi`
θ = `tan^-1((-5)/7)+2pi`
θ = `-tan^-1(5/7) + 2pi` ...[∵ tan(– θ) = – tanθ]
Hence, modulus = `sqrt(74)` and amplitude = `-tan^-1(5/7) + 2pi`
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