Question

Find the fifth roots of 243(cos 240° + i sin 240°).
i have this:
r^5=243
Theta=240 degrees
root:3
240/5+(0+360/3)=168
240/5+(1+360/3)=169
240/5+(2+360/3=170
making the roots: 3, 168, 169,170

Answer 1

@jim_thompson5910 @satellite73

Answer 2

divide the angle by 5

Answer 3

one answer is \(3\left(\cos(48)+i\sin(48)\right)\)

Answer 4

r^5=243
Theta=240 degrees
root:3
240/5+(0+360/5)=168
240/5+(1+360/5)=169
240/5+(2+360/5=170?????????
making the roots: 3, 168, 169,170

Answer 5

i have no idea what this means
240/5+(0+360/3)=168

Answer 6

i think you may be confused
the roots are not real numbers, they are complex numbers
your answers will all look like \(a+bi\) or \(r(\cos(\theta)+i\sin(\theta))\)

Answer 7

and in each case when it looks like
\[r(\cos(\theta)+i\sin(\theta))\] you will have \(r=3\)
that doesn't change, only \(\theta\) changes

Answer 8

there will be five answers because you are asked for the fifth roots
the first angle you find by dividing \(240\) by \(5\) to get \(48\)

Answer 9

wait now im super confused

Answer 10

that is why i had the first answer above as
\[3(\cos(48)+i\sin(48))\]

Answer 11

lets go slow then
this is the original question

Find the fifth roots of
\(243(\cos (240°) + i \sin (240°))\)

right?

Answer 12

yea

Answer 13

ok is it clear to you that this is a complex number? a number that we could also write as \(a+bi\) ?

Answer 14

yes

Answer 15

in fact lets do it
since \(\cos(240)=-\frac{1}{2}\) and \(\sin(240)=-\frac{\sqrt3}{2}\) this number is
\[-\frac{232}{2}-\frac{232\sqrt3}{2}i\] not that this helps us find the answer, just so we understand that we are looking for five numbers \(a+bi\) with \((a+bi)^5=-\frac{232}{2}-\frac{232\sqrt3}{2}i\)

Answer 16

at least we see now that \(3\) makes no sense as an answer, because \(3^5=243\)
not \(-\frac{243}{2}-\frac{243\sqrt3}{2}i\)

Answer 17

but to find the 5 answers in trig from is real easy
just divide the angle by 5
the angle is \(240\) and \(240\div 5=48\) so one answer is
\[3(\cos(48)+i\sin(48))\]

Answer 18

and you are not even expected to write this number as \(a+bi\) because you do not know what \(\cos(48)\) is
just leave it in trig form as it is

Answer 19

that is one of the five answers
to get the next one, add \(360\) to \(240\) get \(600\) then divide that by \(5\) and get \(120\)

Answer 20

then the next complex number will be
\[3(\cos(120)+i\sin(120))\]

Answer 21

so i just basically keep doing that 5 times?

Answer 22

yes
keep adding 360
on the fifth time you will be where you started, at an angle co-terminal with 48

Answer 23

for the next one do i add 360 to 600?

Answer 24

@satellite73