Question #ad560

2 Answers
Ratnaker Mehta
Feb 16, 2018

# {6(cos75+isin75)}^3=-108sqrt2(1+i)#.

Explanation:

De' Moivre's Theorem :

#{r(costheta+isintheta)}^n=r^n{cos(ntheta)+isin(ntheta)}#.

#:. {6(cos75+isin75)}^3=6^3{cos(3xx75)+isin(3xx75)}#,

#=216{cos(225)+isin(225)}#,

#=216{cos(180+45)+isin(150+45)}#,

#=216(-cos45-isin45)#,

#=216(-1/sqrt2-i/sqrt2)#,

#rArr {6(cos75+isin75)}^3=-108sqrt2(1+i)#.

Shiva Prakash M V
Feb 16, 2018

#-4.242-4.242i#

Explanation:

#6(cos75^@+isin75^@)^3=6(cos(3xx75^@+isin(3xx75^@)#
#=6(cos225^@+isin225^@)#
#cos225=-1/sqrt2, sin225^@=-1/sqrt2#
Substituting
#6(cos75^@+isin75^@)^3=6((-1)/sqrt2)+i((-1)/sqrt2)#

#=-6/sqrt2-i6/sqrt2#
#-4.242-4.242i#

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