Question-21296

Question Number 21296 by NECx last updated on 19/Sep/17

Commented by NECx last updated on 19/Sep/17

please help me solve this . It was

solved before but i cant assess the

answer again .

please help

Commented by NECx last updated on 20/Sep/17

i think then it was solved by

smalso 334 \dots . something like that

Commented by NECx last updated on 20/Sep/17

please help

Answered by Joel577 last updated on 20/Sep/17

{2 \sin}^{- 1} (x \sqrt{6}) + \sin^{- 1} (4 x) = \frac{π}{2}

\sin^{- 1} x \sqrt{6} = α \to \sin α = x \sqrt{6}

\sin^{- 1} 4 x = β \to \sin β = 4 x

\sin ({2 \sin}^{- 1} x \sqrt{6} + \sin^{- 1} 4 x) = \sin \frac{π}{2}

\sin (2 α + β) = 1

\sin 2 α \cos β + \cos 2 α \sin β = 1

2 \sin α \cos α \cos β + (1 - {2 \sin}^{2} α) \sin β = 1

2 \sin α \cos α \cos β = 2 . x \sqrt{6} . \sqrt{1 - 6 x^{2}} . \sqrt{1 - 16 x^{2}}

= 2 x \sqrt{6 (1 - 6 x^{2}) (1 - 16 x^{2})}

(1 - {2 \sin}^{2} α) \sin β = (1 - 12 x^{2}) 4 x

= 4 x - 48 x^{3}

2 x \sqrt{6 (1 - 6 x^{2}) (1 - 16 x^{2})} + 4 x - 48 x^{3} = 1

4 x^{2} \sqrt{576 x^{4} - 132 x^{2} + 6} = 48 x^{3} - 4 x + 1

4 x^{2} (576 x^{4} - 132 x^{2} + 6) = 2304 x^{6} - 384 x^{4} + 96 x^{3} + 16 x^{2} - 8 x + 1

2304 x^{6} - 528 x^{4} + 24 x^{2} = 2304 x^{6} - 384 x^{4} + 96 x^{3} + 16 x^{2} - 8 x + 1

144 x^{4} + 96 x^{3} - 8 x^{2} - 8 x + 1 = 0

If we plot it into Cartecian plane, it shows

x = - \frac{1}{2} or x = \frac{1}{6}

Commented by Joel577 last updated on 20/Sep/17

If someone have shorter ways to solve this,

please explain it to us . Thanks

Commented by NECx last updated on 20/Sep/17

thanks boss

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