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How do you simplify #sin (2 * arcsin (x))#?

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92
Nov 8, 2016

Answer:

The answer is #=2xsqrt(1-x^2) #

Explanation:

Let #y=arcsinx#, then #x=siny#

#sin(2arcsinx)=sin2y=2sinycosy#

#cos^2y+sin^2y=1#

#cos^2y=1-x^2##=>##cosy=sqrt(1-x^2)#

#:.sin(2arcsinx)=2xsqrt(1-x^2) #

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Dean R. Share
Jun 15, 2018

Answer:

If we interpret #arcsin a# as all the solutions to #sin x = a# then

#sin(2 arcsin x) = 2 (sin arcsin x)(cos arcsin x) = 2 x cos arcsin (x/1) = pm 2x sqrt{1-x^2}#

Explanation:

#arcsin (x/1)# refers to a right triangle, opposite #x#, hypotenuse #1# so adjacent #sqrt{1-x^2}#. The sign is ambiguous so we prepend #pm#.

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