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Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 9

Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 10
Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 11 Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 12 Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 13 Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 14 Misc 5 - Chapter 5 Class 11 Complex Numbers - Part 15

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Misc 5 Convert the following in the polar form: (ii) ( 1 + 3𝑖)/(1 βˆ’ 2𝑖) Let z = ( 1 + 3𝑖)/(1 βˆ’ 2𝑖) Rationalizing = (1 + 3𝑖)/(1 βˆ’ 2𝑖) Γ— (1 + 2𝑖)/(1 + 2𝑖) = ((1 + 3𝑖) (1 + 2𝑖 ))/((1 βˆ’ 2𝑖) (1 + 2𝑖)) = (1 (1 + 2𝑖) + 3𝑖 (1 + 2𝑖 ))/((1 βˆ’ 2𝑖) (1 + 2𝑖)) = (1 +2𝑖 +3𝑖 +6𝑖2)/((1 βˆ’ 2𝑖) (1 + 2𝑖)) = (1 + 5𝑖 +6𝑖2)/((1 βˆ’2𝑖) (1 +2𝑖)) Using (a – b) (a + b) = a2 – b2 = (1 + 5𝑖 +6𝑖2)/((1)2βˆ’ (2𝑖)2) = (1 + 5𝑖 +6𝑖2)/(1 βˆ’4𝑖2) Putting 𝑖2 = –1 = (1 + 5𝑖 +6 (βˆ’1 ))/(1 βˆ’4 ( βˆ’1 )) = (1 + 5𝑖 βˆ’6)/(1 +4) = (1 βˆ’6 + 5𝑖)/5 = (βˆ’ 5 + 5𝑖)/5 = (βˆ’ 5 ( βˆ’1 + 𝑖 ))/5 = - 1 + 𝑖 Hence, z = – 1 + 𝑖 Let polar form be z = π‘Ÿ (cos⁑θ+𝑖sin⁑θ ) From (1) and (2) –1 + 𝑖 = r (cos ΞΈ + 𝑖 sin ΞΈ) –1 + 𝑖 = r cos ΞΈ + 𝑖 r sin ΞΈ Comparing real part βˆ’ 1 = r cos ΞΈ Squaring both sides (βˆ’ 1 )2 =( π‘Ÿ cos⁑θ )^2 1 = π‘Ÿ2 cos2ΞΈ Adding (3) and (4) 1 + 1 = π‘Ÿ2 cos2 ΞΈ + π‘Ÿ2 sin2 ΞΈ 1 + 1 = r2 cos2 ΞΈ + r2 sin2 ΞΈ 2 = r2 ( cos2 ΞΈ + sin2 ΞΈ ) 2 = r2 Γ— 1 2 = r2 √2 = r r = √2 Now finding argument –1 + 𝑖 = r cos ΞΈ + 𝑖 r sin ΞΈ Comparing real part βˆ’ 1 = r cos ΞΈ Putting r =√2 βˆ’ 1 = √2 cos ΞΈ (βˆ’ 1 )/√2 = cos ΞΈ cos ΞΈ = (βˆ’ 1 )/√2 Hence, cos ΞΈ = (βˆ’ 1 )/√2 & sin ΞΈ = ( 1)/√2 Hence, cos ΞΈ = (βˆ’ 1 )/√2 & sin ΞΈ = ( 1)/√2 Since, sin ΞΈ is positive and cos ΞΈ is negative, Hence, ΞΈ lies in IInd quadrant Argument = 180Β° – 45Β° = 135Β° = 135Β° Γ— πœ‹/(180Β°) = 3πœ‹/4 Hence, argument of 𝑧 = 3πœ‹/4 Hence π‘Ÿ = √2 and ΞΈ = 3πœ‹/4 Polar form of 𝑧=π‘Ÿ (cos⁑θ+sin⁑θ ) = √2 (cos(3πœ‹/4)+sin(3πœ‹/4))

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.