Learning Journal Unit 8

1. Evaluate the cube root of z = 27 cis (240° ). Then raise them to the cube. Show the

steps of your reasoning.

Z = 27 cis (240° )
Take cube root
(Z)1/3 = (27 cis(240° ))1/3
→ cisθ = cis (θ ) + i sin (θ )
= (33)1/3 [cos (240° )+ i sin (240° )]1/3
→ (cosθ + i sinθ )n = cos nθ + i sin nθ
= (33)1/3 [cos (240° )+ i sin (240° )]
3 3
= 3 [cos 80° + i sin 80° ]
= 3 [ cos (90-10) + i sin (90-10)]
⟹cos (90-0) = sinθ
⟹sin (90-0) = cosθ
= 3[sin 10° + i cos 10° ]
21/3 = 3 sin 80°
Now rise the power to the cube
21/3 = 33(cis 80° )3 = 27[cos 240 + i sin 240]
Z = 27[cos (180+60) + i sin (180+60)]
Z = 27[-cos 60 – i sin 60]
Z = 27[-1 -i√ 3] = 27[-1-i√ 3]

2 2 2


Z = -27 [1+ i√ 3]
2




2. Evaluate