Question

Express in simplest form: sin-1[3x – 4x3]

Answer 1

$\textbf{Given:}$

$sin^{-1}(3x-4x^3)$

$\text{Take x=sin\theta }$

$\implies\,\theta=sin^{-1}x$

$\text{Now,}$

$sin^{-1}(3x-4x^3)$

$=sin^{-1}(3sin\theta-4sin^3\theta)$

$\text{Using,}$

$\boxed{\bf\,sin3A=3sinA-4\,sin^3A}$

$=sin^{-1}(sin3\theta)$

$\text{We know that}$

$\boxed{\bf\,sin^{-1}(sinx)=x}$

$=3\theta$

$=3\,sin^{-1}x$

$\therefore\textbf{The simplest form of \bf\,sin^{-1}(3x-4x^3) is \bf\,3\,sin^{-1}x }$

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