Solve the following equation \(\displaystyle{2}{\arccos{{\left({x}\right)}}}-{\arcsin{{\left({2}{x}\sqrt{{{1}-{x}^{{2}}}}\right)}}}={0}\)

Oliver Tyler

Oliver Tyler

Answered question

2022-03-23

Solve the following equation
2arccos(x)arcsin(2x1x2)=0

Answer & Explanation

Jazlyn Mitchell

Jazlyn Mitchell

Beginner2022-03-24Added 14 answers

Note that x[1,1]. Let x=cos(t) with t[0,π] then sin(t)0 and
2arccos(cos(t))=arcsin(2cos(t)1cos2(t))=arcsin(2cos(t)sin(t))

=arcsin(sin(2t))
Note for t[0,π]arccos(cos(t))=t and
arcsin(sin(2t))=(2t, if t[0,π/4]),(π-2t, if t[π/4,3π/4]),(-2π+2t, if t

[3π/4,π]):

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