1.

Change the following complex numbers into polar form. 1+7i / (2-i)² .​

Answer»

SOLUTIONS:-

  • Z => 1 + 7i / (2 - i)²

=> 1 + 7i / (2 - i)²

=> 1 + 7i / 4 + i² - 4i

=> 1 + 7i / 4 - 1 - 4i

=> 1 + 7i / 3 - 4i × 3 + 4i / 3 + 4i

=> 3 + 4i + 21i + 28i² / 3² + 4²

=> 3 + 4i + 21i - 28 / 3² + 4²

=> -25 + 25i / 25

=> -1 + i

  • Let rCos∅ = -1 and rSin∅ = 1

On SQUARING and adding, we obtain

=> r²(Cos²∅ + Sin²∅) = 2

=> r² = 2 [Cos²∅ + Sin²∅ = 1]

=> r = √2 [Convertionally, r > 0]

Therefore,

√2Cos∅ = -1 and √2Sin∅ = 1

=> Cos∅ = -1/√2 and Sin∅ = 1/√2

• ∅ = π - π/4 = 3π/4 [As ∅ lies in (ii) quadrant]

Z => rCos∅ + i rSin∅

=> √2Cos3π/4 + i √2Sin3π/4

=> √2(Cos3π/4 + i Sin3π/4)

This is the required polar form.


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