SOLUTION: prove that 2 arctan(2/3)=arcsin(12/13)

Question 476290: prove that 2 arctan(2/3)=arcsin(12/13)
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
prove that 2 arctan(2/3)=arcsin(12/13)
Let arctan(2/3) = x
Then tan(x) = 2/3
Then sin(x) = 2/sqrt(2^2 + 3^2) = 2/sqrt(13)
And cos(x) = 3/sqrt(13)
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Take the sin of both sides of the original equation to get:
sin[2arctan(2/3)] = 2sin[arctan(2/3)*cos[arctan(2/3)]
= 2[2/sqrt(13)*3/sqrt(13)] = 12/13
And sin[arcsin(12/13)] = 12/13
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So 2*arctan(2/3) = arcsin(12/13)
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Cheers,
Stan H.

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Cheers,
Stan H.
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