Trigonometric Equations

Given : sin

%5Ctheta

, cos

%5Ctheta

, tan

%5Ctheta

are in G.P.

%5CRightarrow%20%5C%2C%20%5C%2C%20%5C%2C%20cos%5E2%20%5Ctheta%20%5C%2C%20%3D%20%5C%2C%20sin%5Ctheta%20%5C%2C%20tan%5Ctheta%20%5C%2C%20%5CRightarrow%20%5C%2C%20cos%5E3%20%5Ctheta%20%5C%2C%20%3D%20%5C%2C%20sin%5E2%5Ctheta

%5CRightarrow%20%5C%2C%20%5C%2C%20cos%5E3%20%5Ctheta%20%5C%2C%20%3D1-cos%5E2%20%5Ctheta

%5CRightarrow%20%5C%2C%20%5C%2C%20%5C%2C%20(cos%5E3%5Ctheta%20%5C%2C%20%5C%2C%20%2B%20%5C%2C%20cos%5E2%20%5Ctheta)%20%5C%2C%20%3D%201%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%3Cbr%20%2F%3E%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20...(1)

Cubic both sides, we have

cos%5E9%20%5Ctheta%20%2B%20cos%5E6%20%5Ctheta%20%2B%203%20cos%5E5%20%5Ctheta%20.%20(cos%5E3%20%5Ctheta%2B%20cos%5E2%20%5Ctheta)%20%3D1

%5CRightarrow%20%5C%2C%20%5C%2C%20%5C%2C%20cos%5E9%20%5Ctheta%20%2B%20cos%5E6%20%5Ctheta%20%2B%203cos%5E5%20%5Ctheta%20%3D1

%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C

[Using equation (1)]

%5CRightarrow%20%5C%2C%20%5C%2C%20%5C%2C%20cos%5E9%20%5Ctheta%20%2B%20cos%5E6%20%5Ctheta%20%2B%203cos%5E5%20%5Ctheta%20-%201%3D0

Let

%5Ctheta%20%3D%20sin%5E%7B-1%7D%5Cleft%5Bsin%5Cfrac%7B5%5Cpi%7D%7B3%7D%5Cright%5D

%5CRightarrow%20sin%5C%2C%5Ctheta%20%3D%20sin%5Cfrac%7B5%5Cpi%20%7D%7B3%7D%20%3D%20sin%5Cleft%5B2%5Cpi-%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cright%5D

%5CRightarrow%20sin%5C%2C%5Ctheta%20%3D%20-sin%5Cfrac%7B%5Cpi%20%7D%7B3%7D%20%3D%20sin%20%5Cleft(%5Cfrac%7B-%5Cpi%20%7D%7B3%7D%5Cright)

%5Cleft(%5Cbecause%20%5C%2Csin%5C%2C%20%5Cleft(-%20%5Ctheta%20%5Cright)%3D%20%5C%2C-%20%5C%2Csin%20%5C%2C%5Ctheta%5Cright)

Therefore, principal value of

sin%5E%7B-1%7D

%5Cleft%5Bsin%5Cfrac%7B5%5Cpi%20%7D%7B3%7D%5Cright%5D

%5Cfrac%7B-%5Cpi%20%7D%7B3%7D

, as principal value of

sin%5E%7B-%7D

%5E%7B1%7D%20%5C%2Cx

lies between

%5Cfrac%7B-%5Cpi%20%7D%7B2%7D

and

%5Cfrac%7B%5Cpi%20%7D%7B2%7D.

Chapter Questions
6 MCQs