Question 4
Convert the given complex number in polar form: β 1 + i
Given π§ = β1+ π
Let polar form be
γπ§ = π (cosγβ‘ΞΈ+π sinβ‘ΞΈ)
From (1) & (2)
β 1+ π = r ( cosβ‘ΞΈ + π sinβ‘ΞΈ)
β 1+ π = rγ cosγβ‘ΞΈ + π r sinβ‘ΞΈ
Adding ( 3 ) and ( 4 )
1 + 1 = π2 cos2 ΞΈ+ π2 sin2ΞΈ
2 = π2 ( cos2 ΞΈ+ sin2 ΞΈ)
2 = π2 Γ 1
2 = π2
β2 = π
π = β2
Finding argument
β 1+ π = rγ cosγβ‘ΞΈ + π r sinβ‘ΞΈ
Hence, sin ΞΈ = 1/β2 & cos ΞΈ = (β 1)/β2
Hence, sin ΞΈ = 1/β2 & cos ΞΈ = (β 1)/β2
Here,
sin ΞΈ is positive and cos ΞΈ is negative,
Hence, ΞΈ lies in IInd quadrant
Argument = 180Β° β 45Β°
= 135Β°
= 135Β° Γ π/180o
= ( 3 π)/4
So argument of z = ( 3 π)/4
Hence π = β2 and ΞΈ = 3π/4
Polar form of z = r (cos ΞΈ + sin ΞΈ)
= β2 (cos (( 3 π)/4)+ π sin(( 3 π)/4))
Made by
Davneet Singh
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo
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