Sin(2sin^-1x)=?

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Sin(2sin^-1x)=?

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Answer 1 · 2018-06-12T08:27:25

$sin (2 arcsin x) = 2 x \sqrt{1 - x^{2}}$ $sin(2arcsinx) = 2xsqrt(1-x^2)$

Explanation:

Knowing that:

$sin (2 α) = 2 sin α cos α$ $sin(2alpha) = 2sin alpha cos alpha$

we have:

$sin (2 arcsin x) = 2 sin (arcsin x) cos (arcsin x)$ $sin(2arcsinx) = 2sin(arcsinx)cos (arcsinx)$

Now, by definition:

$sin (arcsin x) = x$ $sin(arcsinx) = x$

Let $y = arcsin x$ $y = arcsinx$ , so that $x = sin y$ $x = siny$ with $y \in (- \frac{π}{2}, \frac{π}{2})$ $y in (-pi/2,pi/2)$ and $cos y > 0$ $cos y >0$ . Then:

$cos y = \sqrt{1 - {sin}^{2} y} = \sqrt{1 - x^{2}}$ $cosy = sqrt(1-sin^2y) = sqrt(1-x^2)$

Finally:

$sin (2 arcsin x) = 2 x \sqrt{1 - x^{2}}$ $sin(2arcsinx) = 2xsqrt(1-x^2)$

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Sin(2sin^-1x)=?

1 Answer

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