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JEE Main202331 Jan 2023Evening ShiftMathematicsComplex NumberActual

The complex number z = i - 1 cos π 3 + i sin π 3 is equal to:

Options

  1. A2 i cos 5 π 12 - i sin 5 π 12
  2. Bcos π 12 - i sin π 12
  3. C2 cos π 12 + i sin π 12
  4. D2 cos 5 π 12 + i sin 5 π 12

Correct answer

D. 2 cos 5 π 12 + i sin 5 π 12

Step-by-step solution

Given, z = i - 1 cos π 3 + i sin π 3 ⇒ z = 2 - 1 2 + i 2 cos π 3 + i sin π 3 Now argument of complex number - 1 2 + i 2 is 3 π 4 So, by euler's formula e i θ = cos θ + i sin θ we get, ⇒ z = 2 e i 3 π / 4 e i π / 3 ⇒ z = 2 e i 5 π / 12 ⇒ z = 2 cos 5 π 12 + i sin 5 π 12

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