What is sin(2arcsinx)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should come up with the following notation, such that:

`arcsin x = theta`

Replacing `theta` for `arcsin x` yields:

`sin(2 arcsin x) = sin(2 theta)`

Using the double angle identity, yields:

`sin(2 theta) = 2 sin theta*cos theta`

Replacing back `arcsin x` for `theta` , yields:

`sin(2 arcsin x) = 2 sin (arcsin x)*cos (arcsin x)`

You should use the following trigonometric identities that relate the trigonometric functions and its inverses, such that:

`sin (arcsin x) = x`

`cos (arcsin x) = sqrt(1 - x^2)`

Using the trigonometric identities above in double angle formula, yields:

`sin(2 arcsin x) = 2x*sqrt(1 - x^2)`

Hence, evaluating the given trigonometric expression yields `sin(2 arcsin x) = 2x*sqrt(1 - x^2)` .