Questions
_MATH1013_1ABCD_2025 Subsection 2.2 (closed on 20 Sep)
Short answer
Find the standard form of the complex number with modulus [math: 2] and argument [math: π5]\dfrac {\pi }{5}. (Correct the answer to 2 decimal places.)
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Step-by-Step Analysis
We’re asked to convert a complex number from polar (modulus-argument) form to standard rectangular form.
The given modulus is r = 2 and the argument is θ = π/5 (......Login to view full explanationLog in for full answers
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( 6 − 6 𝑖 − 9 3 − 27 𝑖 ) 256 = _ _ _ _ _ _ _ _ _ _ Hints: Convert the complex numbers to polar form. If 𝑧 = 𝑟 ( cos 𝜃 + 𝑖 sin 𝜃 ) then 𝑧 𝑛 = 𝑟 𝑛 ( cos 𝑛 𝜃 + 𝑖 sin 𝑛 𝜃 ) . If 𝑧 1 = 𝑟 1 ( cos 𝜃 1 + 𝑖 sin 𝜃 1 ) and 𝑧 2 = 𝑟 2 ( cos 𝜃 2 + 𝑖 sin 𝜃 2 ) then: 𝑧 1 𝑧 2 = 𝑟 1 𝑟 2 [ cos ( 𝜃 1 + 𝜃 2 ) + 𝑖 sin ( 𝜃 1 + 𝜃 2 ) ] and 𝑧 1 𝑧 2 = 𝑟 1 𝑟 2 [ cos ( 𝜃 1 − 𝜃 2 ) + 𝑖 sin ( 𝜃 1 − 𝜃 2 ) ] .
The polar form of 𝑧 = − 6 7 + 3 2 7 𝑖 is: Hint: 𝑧 = 𝑎 + 𝑏 𝑖 = 𝑟 ( cos 𝜃 + 𝑖 sin 𝜃 ) where 𝑟 = 𝑎 2 + 𝑏 2 and 𝜃 = tan − 1 𝑏 𝑎 . Also, don't forget to plot the complex number on the Argand diagram.
Given four complex numbers 𝑧 1 = 2 + 3 𝑖 , 𝑧 2 = − 7 − 5 𝑖 , 𝑧 3 = − 9 + 7 𝑖 , 𝑧 4 = 2 + 5 𝑖 . Calculate | 𝑧 3 ¯ 𝑧 1 + 𝑧 2 𝑧 4 | . Formulae: If 𝑧 = 𝑎 + 𝑏 𝑖 then 𝑧 ¯ = 𝑎 − 𝑏 𝑖 and | 𝑧 | = 𝑎 2 + 𝑏 2 . 𝑎 + 𝑏 𝑖 𝑐 + 𝑑 𝑖 = ( 𝑎 + 𝑏 𝑖 ) ( 𝑐 − 𝑑 𝑖 ) ( 𝑐 + 𝑑 𝑖 ) ( 𝑐 − 𝑑 𝑖 ) = ⋯
The figure shows the Argand diagram together with the complex number \(z\). If \(d=13\) and the imaginary part of \(z\) is \(-5\), find \(z\).
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