.
#2costheta+2sintheta=sqrt6#
#costheta+sintheta=sqrt6/2#, #color(red)(Equation-1)#
#(costheta+sintheta)^2=6/4=3/2#
#cos^2theta+sin^2theta+2sinthetacostheta=3/2#
#1+2sinthetacostheta=3/2#
#2sinthetacostheta=3/2-1=1/2#
#sinthetacostheta=1/4#
From #color(red)(Equation-1)#:
#costheta=sqrt6/2-sintheta#
Let's plug this in:
#sintheta(sqrt6/2-sintheta)=1/4#
#sqrt6/2sintheta-sin^2theta=1/4#
#sin^2theta-sqrt6/2sintheta+1/4=0#
Let's multiply the equation by #4#:
#4sin^2theta-2sqrt6sintheta+1=0#
Let #sintheta=x#
#4x^2-2sqrt6x+1=0#
Using the quadratic formula:
#x=(-b+-sqrt(b^2-4ac))/(2a)#:
#x=(2sqrt6+-sqrt(24-4(4)(1)))/(2(4))=(2sqrt6+-sqrt8)/8=(2sqrt6+-2sqrt2)/8#
#x=(sqrt6+-sqrt2)/4#
#x=0.97, 0.26#
#sintheta=0.97, :. theta=arcsin(0.97), :. theta=75.93^@ and 104.7^@#
#sintheta=0.26, :. theta=arcsin(0.26), :. theta=15.07^@ and 164.93^@#