Find the general solution of the equation `(sqrt(3)-1)costheta+(sqrt(3)+1)sintheta=2`.

Options

A. `2npi+-(pi)/(4)-(5pi)/(12)`
B. `2npi+-(pi)/(4)+(5pi)/(12)`
C. `2npi+-pi-(3pi)/(12)`
D. None of these

Answer

A

Explanation

Given , `(sqrt(3)-1)costheta+(sqrt(3)+1)sintheta=2` …(i)
Let `(sqrt(3)-1)=rcosalphaand(sqrt(3)+1)=rsinalpha`
`thereforer^(2)(cos^(2)alpha+sin^(2)alpha)=(sqrt(3)-1)^(2)+(sqrt(3)+1)^(2)`
`rArrr^(2)=3+1-2sqrt(3)+3+1+2sqrt(3)`
`rArrr^(2)=8rArrr=2sqrt(2)` Now , `(rsinalpha)/(rcosalpha)=(sqrt(3)+1)/(sqrt(3)-1)=-(sqrt(3)+1)/((1-sqrt(3)))=-tan((pi)/(6)+(pi)/(4))`
`rArrtanalpha=-tan((5pi)/(12))rArralpha=-(5pi)/(12)` From Eq . (i) becomes,
`(cosalphacostheta+sinalphasintheta)=2`
`rArr2sqrt(2)cos(alpha-theta)=2`
`rArrcos(theta-alpha)=(1)/(sqrt(2))`
`rArrtheta-alpha=2npi+-(pi)/(4)`
`thereforetheta=2npi+-(pi)/(4)-(5pi)/(12)`