Question

How do you find the cube root of `81 (cos (pi/12) + isin (pi/12))`?

Answer 1

#3^(4/3)(cos 5^o +i sin 5^o), 3^(4/3)(cos 125 o +i sin 125^o) and 3^(4/3)(cos 245^o +i sin 245^o)#.

Explanation:

`(81(cos (pi/12)+i sin( pi/12))^(1/3)`

`=81^(1/3)(cispi/12)^(1/3)`
`3^(4/3)(cis((1/3(2kpi+pi/12)), k = 0, 1, 2`

`=3^(4/3)(cis 5^o, cis 125^o, cis 245^o)`, using `pi = 180^o`

`=3^(4/3)(cos 5^o + i sin 5^o)`,

`3^(4/3)(cos 125^o +i sin 125^o)` and

`3^(4/3)(cos 245^o +i sin 245^o)`.