Example 13 - Find modulus, argument of (1 + i)/(1 - i) - Examples

Example 13  - Chapter 5 Class 11 Complex Numbers - Part 2
Example 13  - Chapter 5 Class 11 Complex Numbers - Part 3
Example 13  - Chapter 5 Class 11 Complex Numbers - Part 4
Example 13  - Chapter 5 Class 11 Complex Numbers - Part 5
Example 13  - Chapter 5 Class 11 Complex Numbers - Part 6
Example 13  - Chapter 5 Class 11 Complex Numbers - Part 7

 

 

 

  1. Chapter 5 Class 11 Complex Numbers (Term 1)
  2. Serial order wise

Transcript

Example, 13 Find the modulus and argument of the complex numbers: (i) (1 + 𝑖)/(1 βˆ’ 𝑖) , First we solve (1 + 𝑖)/(1 βˆ’ 𝑖) Let 𝑧 = (1 + 𝑖)/(1 βˆ’ 𝑖) Rationalizing the same = (1 + 𝑖)/(1 βˆ’ 𝑖) Γ— (1 + 𝑖)/(1 + 𝑖) = (( 1 + 𝑖 ) ( 1 + 𝑖 ))/("(" 1 βˆ’ 𝑖 ) (1 + 𝑖 )) Using (a – b) (a + b) = a2 βˆ’ b2 = ( 1+ 𝑖 )2/( ( 1 )2 βˆ’ ( 𝑖 )2) Using ( a + b )2 = a2 + b2 + 2ab = ((1)2 + (𝑖)2 + 2𝑖)/( (1)2 βˆ’ (𝑖)2) Putting i2 = βˆ’ 1 = ((1)2 + (βˆ’1) + 2𝑖)/( 1βˆ’ (βˆ’ 1) ) = (1 βˆ’1 + 2𝑖)/( 1 + 1) = ( 2𝑖)/( 2) = 𝑖 = 0 + 𝑖 Hence, 𝑧 = 0 + 𝑖 Method 1 To calculate modulus of z z = 0 + i Complex number z is of the form x + 𝑖y Hence x = 0 and y = 1 Modulus of z = √(π‘₯^2+𝑦2) = √(( 0 )2+(1)2) = √(0+1) = √1 = 1 Modulus of z = 1 Method 2 To calculate modulus of z We have , z = 0 + 𝑖 Let z = r ( cos ΞΈ + 𝑖 sin ΞΈ ) Here r is modulus, and ΞΈ is argument From (1) and (2) 0 + 𝑖 = r ( cos ΞΈ + 𝑖 sin ΞΈ ) 0 + 𝑖 = r cos ΞΈ + 𝑖r sin ΞΈ Comparing real part 0 = r cos ΞΈ Squaring both sides (0)2 = ( π‘Ÿ cos⁑θ )2 0 = r2 cos2 ΞΈ Adding (3) and (4) 0 + 1 = r2 cos2 ΞΈ + r2 sin2 ΞΈ 1=π‘Ÿ2 (cos2 ΞΈ+sin2 ΞΈ) 1 = r2 (1) 1 = r2 1 = r β‡’ Modulus of z = 1 Finding argument 0 + 𝑖 = r cos ΞΈ + 𝑖r sin ΞΈ Comparing real part 0 = r cos ΞΈ Put r = 1 0 = 1 Γ— cos ΞΈ 0 = cos ΞΈ cos ΞΈ = 0 Hence, cos ΞΈ = 0 & sin ΞΈ = 1 Since, sin ΞΈ is positive and cos ΞΈ is zero Hence, ΞΈ lies in Ist quadrant So, Argument = 90Β° = 90 Γ— πœ‹/180 = πœ‹/2 Hence, argument of z = πœ‹/2

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.