if z = a ( cos x + i sin x ) ⇒⇒⇒ ∴ z^n = a^n ( cos nx + i sin nx)
Part (1) ∴ [ 3 (cos 27° + i sin 27°) ]⁵ = (3⁵) ( cos 5*27° + i sin 5*27°) = 243 ( cos 135° + i sin 135°) ⇒⇒⇒⇒ Polar form ================================= Part (2)
The angle 135° in the second quadrant ⇒ sin is (+) and cos is (-) and its reference angle = 180° - 135° = 45° sin 45° = cos 45°= 1/√2 = (√2)/2 ∴ sin 135° = (√2)/2 and cos 135° = -(√2)/2 ∴ 243 ( cos 135° + i sin 135°) = 243 [ -(√2)/2 + i (√2)/2 ]
= - 243(√2)/2 + 243 (√2)/2 i ⇒⇒⇒⇒ standard form =========================================== The correct answers are options (1) and (3)
option (1) ⇒⇒⇒ 243 ( cos 135° + i sin 135°) ⇒⇒⇒⇒ Polar form option (3) ⇒⇒⇒ - 243(√2)/2 + 243 (√2)/2 i ⇒⇒⇒⇒ standard form
