Question

Given that sin(2θ)=2/3, the value of
sin^6θ+cos^6θ can be written as a/b with a and b as coprime positive integers. Find a+b.

Answer 1

sin^6 θ+cos^6 θ
= (sin^2 Ø)^3 + (cos^2 Ø)^3 ---- the sum of cubes
= (sin^2 Ø + cos^2 Ø)( (sin^2 Ø)^2 - (sin^2 Ø)(cos^2 Ø) + (cos^2 Ø)^2 )
= (1)( (sin^2 Ø)^2 - (sin^2 Ø)(cos^2 Ø) + (cos^2 Ø)^2 )

now, if sin 2Ø = 1/3
2sinØcosØ = 2/3
sinØcosØ = 1/3
and
(sin^2 Ø)^2 + (cos^2 Ø)^2
= (sin^2 Ø + cos^2 Ø)^2 - 2(sin^2 Ø)(cos^2 Ø)
= (1 - 2(sinØcosØ)^2)
= 1 - 2(1/9) = 7/9

then (1)( (sin^2 Ø)^2 - (sin^2 Ø)(cos^2 Ø) + (cos^2 Ø)^2 )
= (1)(7/9 - 1/9)
= 6/9
= 2/3

so for you a+b stuff, a=2 and b=3
and a+b= 5