Let sin(a)=x/√6 and sin(b)=4x; so 2a+b=180/2=90. 1/√6=√6/6.
cos(a)=√(1-x^2/6), cos(b)=√(1-16x^2)
sin(2a+b)=sin90=1=sin(2a)cos(b)+cos(2a)sin(b)=2sin(a)cos(a)cos(b)+(1-2(sin(a))^2)sin(b)
sin(2a+b)=2x√6/6√((1-x^2/6)(1-16x^2))+(1-2x^2/6)4x=1
(x/3)√((6-x^2)(1-16x^2))+4x-4x^3/3=1
x√((6-x^2)(1-16x^2))+12x-4x^3=3
x^2(6-x^2)(1-16x^2)=(4x^3-12x+3)^2
x^4+24x^3+138x^2-72x+9=0
From this equation, x=0.245 or 49/200.