Solve-arcsin-2x-arcsin-x-3-arcsin-x-

Question Number 193676 by lapache last updated on 18/Jun/23

S o l v e

a r c s i n (2 x) + a r c s i n (x \sqrt{3}) = a r c s i n (x)

Answered by Frix last updated on 18/Jun/23

\sin^{- 1} t \in R \Leftrightarrow - 1 ⩽ t ⩽ 1

f (x) = \sin^{- 1} 2 x + \sin^{- 1} \sqrt{3} x - \sin^{- 1} x

- \frac{1}{2} ⩽ x ⩽ \frac{1}{2}

f^{'} (x) = \frac{2}{\sqrt{1 - 4 x^{2}}} + \frac{\sqrt{3}}{\sqrt{1 - 3 x^{2}}} - \frac{1}{\sqrt{1 - x^{2}}} > 0 \forall x \in (- \frac{1}{2}; \frac{1}{2})

\Rightarrow only solution is x = 0

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