How to solve : \(\displaystyle{\arcsin{{x}}}-{\arcsin{{\left(\frac{{x}}{{2}}\right)}}}={\arcsin{{\left({\frac{{{x}\sqrt{{3}}}}{{{2}}}}\right)}}}\) Why can't I

Hanzikoval1pa

Hanzikoval1pa

Answered question

2022-03-31

How to solve : arcsinxarcsin(x2)=arcsin(x32)
Why can't I figure this one out?? Is it possible to cancel out the arcsins? I know from graphing on my calculator that the answers are -1,0, and 1, but want help getting there by hand.

Answer & Explanation

Lana Hamilton

Lana Hamilton

Beginner2022-04-01Added 12 answers

this makes sense ony if |x|<1
arcsinxarcsinx2=arcsinx32
sin(arcsinxarcsinx2)=sinarcsinx32
sin(arcsinx)cos(arcsinx2)cos(arcsinx)sin(arcsinx2)=x32
x1(x2)2x21x2=x32
x4x2x1x2=x3
x2(52x22(4x2)(1x2))=3x2
Hence
x=0  or  52x22(4x2)(1x2)=3
However,
52x22(4x2)(1x2)=3
(4-x2)(1-x2)=1-x2
(4-x2)=(1-x2)    or    1-x2=0
But (4x2)=(1x2) is impossible
Conclusion x{1,0,1} is the set of solutions

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