Polar Coordinates And Complex Numbers Worksheets With Answers - Math 611b Page 45
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50Ex 3 p.601 AMC
3
8i
Find
or the cube roots of 8i. Check your work by graphing on the TI-83.
Solution
π
π
=
+
Convert 8i to polar form. r = 8, 2 = B /2
8
i
8[cos
i
sin ]
2
2
⎡
⎤
⎛
θ
π
⎞
⎛
θ
π
⎞
1
1
+
+
2
k
2
k
=
+
p
p
⎢
⎥
⎜
⎟
⎜
⎟
z
r
cos
i
sin
Using DeMoivre’s Theorem
⎝
⎠
⎝
⎠
⎣
p
p
⎦
r = 8, 2 = B /2, p = 3, k = 0,1,2
π
π
⎡
⎤
⎛
⎞
⎛
⎞
π
π
+
+
⎢
2
k
2
k
⎥
⎜
⎟
⎜
⎟
1
1
2
2
=
+
⎢
⎥
⎜
⎟
⎜
⎟
3
3
8
8 cos
i
sin
3
3
⎢
⎜
⎟
⎜
⎟
⎥
⎢
⎥
⎝
⎠
⎝
⎠
⎣
⎦
π
π
⎡
⎤
1
1
⎛ ⎞
⎛ ⎞
=
+
=
+
3
3
⎜ ⎟
⎜ ⎟
8
8 cos
⎢
i
sin
⎥
3
i
when k = 0
⎝ ⎠
⎝ ⎠
⎣
⎦
6
6
π
π
⎡
⎤
⎛
⎞
⎛
⎞
1
1
5
5
=
+
= −
+
3
3
⎜
⎟
⎜
⎟
8
8 cos
⎢
i
sin
⎥
3
i
when k = 1
⎝
⎠
⎝
⎠
⎣
⎦
6
6
π
π
⎡
⎤
⎛
⎞
⎛
⎞
1
1
3
3
=
+
= −
⎜
⎟
⎜
⎟
3
3
8
8 cos
⎢
i
sin
⎥
2
i
when k = 2
⎝
⎠
⎝
⎠
⎣
⎦
2
2
Using the TI-83 (use the steps on p.602).
45
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