JEE Main202324 Jan 2023Evening ShiftMathematicsComplex NumberActual
The value of 1 + sin 2 π 9 + i cos 2 π 9 1 + sin 2 π 9 - i cos 2 π 9 3 is
Options
- A- 1 2 1 - i 3
- B1 2 1 - i 3
- C- 1 2 3 - i
- D1 2 3 + i
Correct answer
C. - 1 2 3 - i
Step-by-step solution
Given, 1 + sin 2 π 9 + i cos 2 π 9 1 + sin 2 π 9 - i cos 2 π 9 3 Now let z = sin 2 π 9 + i cos 2 π 9 , So, z ¯ = sin 2 π 9 - i cos 2 π 9 = 1 z So, 1 + sin 2 π 9 + i cos 2 π 9 1 + sin 2 π 9 - i cos 2 π 9 3 = 1 + z 1 + z ¯ 3 = 1 + z 1 + 1 z 3 = z 3 1 + z 1 + z 3 = z 3 = sin 2 π 9 + i cos 2 π 9 3 = i 3 cos 2 π 9 - i sin 2 π 9 3 = - i cos 3 × 2 π 9 - i sin 3 × 2 π 9 = - i cos 2 π 3 - i sin 2 π 3 = - i