anonymous
  • anonymous
Find the fifth roots of 243(cos 300° + i sin 300°).
Mathematics
  • Stacey Warren - Expert brainly.com
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schrodinger
  • schrodinger
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anonymous
  • anonymous
@jim_thompson5910
jim_thompson5910
  • jim_thompson5910
Given z = r*[ cos( theta ) + i*sin( theta ) ] To find the five 5th roots of z Use the formula the formula: r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] where k = 0,1,2,3,4 -------------------------------------------------------------------------------------------------------- First 5th root: k = 0 r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] (243)^(1/5)*[ cos( (300+360*k)/5 ) + i*sin( (300+360*k)/5 ) ] (243)^(1/5)*[ cos( (300+360*0)/5 ) + i*sin( (300+360*0)/5 ) ] 3*[ cos( (300+360*0)/5 ) + i*sin( (300+360*0)/5 ) ] 3*[ cos( (300+0)/5 ) + i*sin( (300+0)/5 ) ] 3*[ cos( 300/5 ) + i*sin( 300/5 ) ] 3*[ cos( 60 ) + i*sin( 60 ) ] ------------------------------------------------------------------- Second 5th root: k = 1 r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] (243)^(1/5)*[ cos( (300+360*k)/5 ) + i*sin( (300+360*k)/5 ) ] (243)^(1/5)*[ cos( (300+360*1)/5 ) + i*sin( (300+360*1)/5 ) ] 3*[ cos( (300+360*1)/5 ) + i*sin( (300+360*1)/5 ) ] 3*[ cos( (300+360)/5 ) + i*sin( (300+360)/5 ) ] 3*[ cos( 660/5 ) + i*sin( 660/5 ) ] 3*[ cos( 132 ) + i*sin( 132 ) ] ------------------------------------------------------------------- Third 5th root: k = 2 r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] (243)^(1/5)*[ cos( (300+360*k)/5 ) + i*sin( (300+360*k)/5 ) ] (243)^(1/5)*[ cos( (300+360*2)/5 ) + i*sin( (300+360*2)/5 ) ] 3*[ cos( (300+360*2)/5 ) + i*sin( (300+360*2)/5 ) ] 3*[ cos( (300+720)/5 ) + i*sin( (300+720)/5 ) ] 3*[ cos( 1020/5 ) + i*sin( 1020/5 ) ] 3*[ cos( 204 ) + i*sin( 204 ) ] ------------------------------------------------------------------- Fourth 5th root: k = 3 r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] (243)^(1/5)*[ cos( (300+360*k)/5 ) + i*sin( (300+360*k)/5 ) ] (243)^(1/5)*[ cos( (300+360*3)/5 ) + i*sin( (300+360*3)/5 ) ] 3*[ cos( (300+360*3)/5 ) + i*sin( (300+360*3)/5 ) ] 3*[ cos( (300+1080)/5 ) + i*sin( (300+1080)/5 ) ] 3*[ cos( 1380/5 ) + i*sin( 1380/5 ) ] 3*[ cos( 276 ) + i*sin( 276 ) ] ------------------------------------------------------------------- Fifth 5th root: k = 4 r^(1/n)*[ cos( (theta+360*k)/n ) + i*sin( (theta+360*k)/n ) ] (243)^(1/5)*[ cos( (300+360*k)/5 ) + i*sin( (300+360*k)/5 ) ] (243)^(1/5)*[ cos( (300+360*4)/5 ) + i*sin( (300+360*4)/5 ) ] 3*[ cos( (300+360*4)/5 ) + i*sin( (300+360*4)/5 ) ] 3*[ cos( (300+1440)/5 ) + i*sin( (300+1440)/5 ) ] 3*[ cos( 1740/5 ) + i*sin( 1740/5 ) ] 3*[ cos( 348 ) + i*sin( 348 ) ] ------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------- Given z = 243 * [ cos( 300 ) + i*sin( 300 ) ], the five 5th roots of z are: 3*[ cos( 60 ) + i*sin( 60 ) ] 3*[ cos( 132 ) + i*sin( 132 ) ] 3*[ cos( 204 ) + i*sin( 204 ) ] 3*[ cos( 276 ) + i*sin( 276 ) ] 3*[ cos( 348 ) + i*sin( 348 ) ] Note: all angles are in degrees I used the formulas found on this page: http://www.mathxtc.com/Downloads/NumberAlg/files/NthRootsComplex.pdf

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