Trigonometric Equations

%5Cleft%5B%5Cfrac%7B1%2B%5Csin%20%5Cfrac%7B%5Cpi%7D%7B8%7D%2Bi%20%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B8%7D%7D%7B1%2B%5Csin%20%5Cfrac%7B%5Cpi%7D%7B8%7D-i%20%5Ccos%20%5Cfrac%7B%5Cpi%7D%7B8%7D%7D%5Cright%5D%5E%7Bn%7D

%3D%5Cleft%5B%5Cfrac%7B1%2B%5Ccos%20%5Calpha%2Bi%20%5Csin%20%5Calpha%7D%7B1%2B%5Ccos%20%5Calpha-i%20%5Csin%20%5Calpha%7D%5Cright%5D%5E%7Bn%7D%20%5Cquad%5Cleft(%5Cright.

Put

%5Cleft.%5Calpha%3D%5Cfrac%7B%5Cpi%7D%7B2%7D-%5Cfrac%7B%5Cpi%7D%7B8%7D%5Cright)

%3D%5Cleft%5B%5Cfrac%7B2%20%5Ccos%20%5E%7B2%7D%20%5Cfrac%7B%5Calpha%7D%7B2%7D%2B2%20i%20%5Csin%20%5Cfrac%7B%5Calpha%7D%7B2%7D%20%5Ccos%20%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%7B2%20%5Ccos%20%5E%7B2%7D%20%5Cfrac%7B%5Calpha%7D%7B2%7D-2%20i%20%5Csin%20%5Cfrac%7B%5Calpha%7D%7B2%7D%20%5Ccos%20%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%5Cright%5D%5E%7Bn%7D

%3D%5Cleft%5B%5Cfrac%7B%5Ccos%20%5E%7B%5Calpha%7D%2Bi%20%5Csin%20%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%7B%5Ccos%20%5Cfrac%7B%5Calpha%7D%7B2%7D-i%20%5Csin%20%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%5Cright%5D%5E%7Bn%7D

%3D%5Cleft(e%5E%7B2%20i%20%5Cfrac%7B%5Calpha%7D%7B2%7D%7D%5Cright)%5E%7Bn%7D%3De%5E%7Bi%20n%20%5Calpha%7D

%3De%5E%7Bi%20n%5Cleft(%5Cfrac%7B3%20%5Cpi%7D%7B8%7D%5Cright)%7D%3D%5Ccos%20%5Cfrac%7B3%20n%20%5Cpi%7D%7B8%7D%2Bi%20%5Csin%20%5Cfrac%7B3%20n%20%5Cpi%7D%7B8%7D

For

n%3D4

, we get imaginary part.

The correct answer is B:0
Let

z_1%3Dcos%5C%2C%20%5Calpha%20%5C%2C%2B%20%5C%2Ci%20%5C%2Csin%5C%2C%20%5Calpha%3De%5E%7Bi%5Calpha%7D

z_2%20%3D%20cos%20%5C%2C%5Cbeta%20%5C%2C%2B%5C%2C%20i%20%5C%2Csin%20%5C%2C%5Cbeta%3De%5E%7Bi%5Cbeta%7D

and

z_3%20%3D%20cos%5C%2C%20%5Cgamma%5C%2C%20%2B%20%5C%2Ci%20%5C%2Csin%20%5C%2C%5Cgamma%3De%5E%7Bi%5Cgamma%7D

Then,

z_1%20%2B%202z_2%20%2B%203z_3%20%3D%20%5Cleft(%20cos%5C%2C%20%5Calpha%20%5C%2C%2B%20%5C%2C2%20%5C%2Ccos%20%5C%2C%5Cbeta%20%2B%203%20%5C%2Ccos%5C%2C%20%5Cgamma%5Cright)%20%5C%2C%2B%5C%2C%20i%20%5Cleft(sin%20%5C%2C%5Calpha%5C%2C%2B%20%5C%2C2%5C%2Csin%20%5C%2C%5Cbeta%20%5C%2C%2B%20%5C%2C3%20%5C%2Csin%20%5C%2C%5Cgamma%5Cright)%3D%200

a%2Bb%2Bc%3D0

, then,

a%5E3%2Bb%5E3%2Bc%5E3%3D3abc

(z_1)%5E3%2B(2z_2)%5E3%2B(3z_3)%5E3%3D3(z_1)(2z_2)(3z_3)

⇒

e%5E%7B3i%5Calpha%7D%2Be%5E%7B3i%5Cbeta%7D%2Be%5E%7B3i%5Cgamma%7D%3D18e%5E%7Bi(%5Calpha%2B%5Cbeta%2B%5Cgamma)%7D

⇒

(cos3%5Calpha%2B8cos2%5Cbeta%2B27cos3%5Cgamma)%2Bi(sin3%5Calpha%2B8sin2%5Cbeta%2B27sin%5Calpha)%3D18%5E%7Bi%5Calpha%7D

⇒

(cos3%5Calpha%2B8cos2%5Cbeta%2B27cos3%5Cgamma)%2Bi(sin3%5Calpha%2B8sin2%5Cbeta%2B27sin%5Cgamma)%3D18(cos%5Cpi%2Bisin%5Cpi)

%5Ctherefore%20sin3%5Calpha%2B8sin3%5Cbeta%2B27sin%5Cgamma%3D0%5Cspace%20(%5Ctherefore%20cos%5Cpi%3D-1%2Csin%5Cpi%3D0)

Chapter Questions
7 MCQs