AP EAMCET202119 Aug 2021Morning ShiftMathematicsComplex NumberActual
( sin θ - i cos θ ) 3 is equal to
Options
- Ai 3 ( cos ⁡ 3 θ + i sin ⁡ 3 θ )
- Bcos ⁡ 3 θ + i sin ⁡ 3 θ
- Csin ⁡ 3 θ − i cos ⁡ 3 θ
- D( − i ) 3 ( cos ⁡ 3 θ + i sin ⁡ 3 θ )
Correct answer
D. ( − i ) 3 ( cos ⁡ 3 θ + i sin ⁡ 3 θ )
Step-by-step solution
Given ( sin θ - i cos θ ) 3 Multiplying and dividing by i , we get 1 i · i · sin θ - i cos θ 3 = - i i sin θ + cos θ 3 (Because 1 i = - i ) = - i 3 cos θ + i sin θ 3 According to De Moivre's Theorem, For n be any rational number, cos θ + i sin θ n = cos n θ + i sin n θ . = - i 3 cos 3 θ + i sin 3 θ