Find the 4th roots of the equation below. Convert the answers to rectangular form in radians.= 81 (cos+ i sin 픔)la.||JI9 (cos 13424||C.25п9 (cos 257 + i sin 254)24||9 cos(c 374 + i sin 374)242419 COS243 cos24349π24bCOS243 cosOd+ i sin243 cos+ i sin13п242+ i sin32π349п248 + i sin87128T32πT38TT+ i sin32π3+ i sin128T3b.= 3||||-3 (cosd.3 cos3πTCOS + i sin24(cosCOS: 3 cos8137T2413п2425T24COS81 cos81 (cos8T3+ i sin+ i sin+ i sin2п1 (cos ² + i sin ²)24+ i sin128π313п2425T)24371)2432π32T + i sin38TT332π3+ i sinin 128T)3

z4=81(cosπ6+isinπ6)⇒z=81(cos(2kπ+π6)+isin(2kπ+π6)14…

Answered: Find the 4th roots of the equation…

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Advanced Math

Find the 4th roots of the equation below. Convert the answers to rectangular form in radians. 24 = 81 (cos+ i sin 공)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.4: Multiple-angle Formulas

Problem 16E

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Question

Find the 4th roots of the equation below. Convert the answers to rectangular form in radians.
= 81 (cos+ i sin 픔)
la.
||
JI
9 (cos 134
24
||
C.
25п
9 (cos 257 + i sin 254)
24
||
9 cos
(c 374 + i sin 374)
24
24
1
9 COS
243 cos
243
49π
24
b
COS
243 cos
Od
+ i sin
243 cos
+ i sin
13п
24
2+ i sin
32π
3
49п
24
8 + i sin
87
128T
3
2πT
3
8TT
+ i sin
32π
3
+ i sin
128T
3
b.
= 3
||
||
-
3 (cos
d.
3 cos
3
πT
COS + i sin
24
(cos
COS
: 3 cos
81
37T
24
13п
24
25T
24
COS
81 cos
81 (cos
8T
3
+ i sin
+ i sin
+ i sin
2п
1 (cos ² + i sin ²)
24
+ i sin
128π
3
13п
24
25T)
24
371)
24
32π
32T + i sin
3
8TT
3
32π
3
+ i sin
in 128T)
3

Expert Solution

Step 1

$z^{4} = 81 (\cos \frac{π}{6} + i \sin \frac{π}{6}) \Rightarrow z = {[81 (\cos (2 k π + \frac{π}{6}) + i \sin (2 k π + \frac{π}{6})]}^{\frac{1}{4}} = [{(81)}^{\frac{1}{4}} (\cos \frac{1}{4} (2 k π + \frac{π}{6}) + i \sin \frac{1}{4} (2 k π + \frac{π}{6})], k = 0, 1, 2, 3 = [3 (\cos \frac{1}{4} (2 k π + \frac{π}{6}) + i \sin \frac{1}{4} (2 k π + \frac{π}{6})], k = 0, 1, 2, 3 = 3 [\cos (\frac{π}{24}) + isin (\frac{π}{24})], f o r k = 0 = 3 [\cos (\frac{13 π}{24}) + isin (\frac{13 π}{24})], f o r k = 1 = 3 [\cos (\frac{25 π}{24}) + isin (\frac{25 π}{24})], f o r k = 2 = 3 [\cos (\frac{37 π}{24}) + isin (\frac{37 π}{24})], f o r k = 3$