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Question
Find the modulus and amplitude of the following complex numbers.
−8 + 15i
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Solution
Let z = −8 + 15i
Here, a = −8 , b = 15, i.e., a < 0, b > 0
∴ |z| = `sqrt("a"^2 + "b"^2)`
= `sqrt((-8)^2 + 15^2`
= `sqrt(64 + 225)`
= `sqrt(289)`
= 17
Here, (-8, 15) lies in 2nd quadrant.
∴ amp (z) = ` tan^-1("b"/"a") + pi`
= `tan^-1(15/(-8)) + pi`
= `-tan^-1(15/8) + pi` ...[∵ tan–1(– θ) = – tan–1θ]
Hence, modulus = 17 and amplitude = `-tan^-1(15/8) + pi`.
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