Question

Sininversex.+sininverse2x= pie/3

Answer 1

Let θ=arcsinxθ=arcsin⁡x and ϕ=arcsin2xϕ=arcsin⁡2x

Hence,

x=sinθx=sin⁡θ, 2x=sinϕ2x=sin⁡ϕ

θ+ϕ=π3θ+ϕ=π3

ϕ=π3−θϕ=π3−θ

⟹sinϕ=sin(π3−θ)⟹sin⁡ϕ=sin⁡(π3−θ)

⟹sinϕ=sinπ3cosθ−cosπ3sinθ⟹sin⁡ϕ=sin⁡π3cos⁡θ−cos⁡π3sin⁡θ

⟹2x=3–√21−x2−−−−−√−x2⟹2x=321−x2−x2

⟹5x2=3–√×1−x2−−−−−√2⟹5x2=3×1−x22

Multiplying both sides by 22 and squaring,

25x2=3(1−x2)25x2=3(1−x2)

⟹25x2=3−3x2⟹25x2=3−3x2

⟹x2=328⟹x2=328

⟹x=±328