Question

Can you find [ (2(cos80˚+isin80˚))^3 ] ? then express the answer in rectangular form.

Answer 1

`(2((cos80^0+isin80^0))^3=-4-4isqrt3`

Explanation:

`(2((cos80^0+isin80^0))^3=2^3(cos80^0+isin80^0)^3`
By De-Moivre's theorem;
`(cosA+isinA)^n=cosnA+isinnA`
`=8(cos(3xx80^0)+isin(3xx80^0))`
`=8(cos240^0+isin240^0)`
240=180+60
`cos240^0=cos(180^0+60^0)`
`=-cos60^0=-1/2`
`sin240^0=sin(180^0+60^0)`
`=-sin60^0=-sqrt3/2`
Thus,
`8(cos240^0+isin240^0)=8xx(-1/2+i(-sqrt3)/2)`
`=-4-4isqrt3`

`(2((cos80^0+isin80^0))^3=-4-4isqrt3`