Find the exact value of sin(tan^-1 1/2)?

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What are the steps for this?

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Answer 1 · 2018-03-22T23:28:00

#sin(arctan(1/2)) = 1/sqrt5#

Let #x = arctan(1/2)#.

Then #x in (0,pi/2)# and:

#tanx = 1/2#

#sinx/cosx = 1/2#

#sin^2x/cos^2x = 1/4#

#sin^2x/(1-sin^2x) =1/4#

#4sin^2x = 1-sin^2x#

#5sin^2x = 1#

#sin^2x = 1/5#

#sinx = 1/sqrt5#

and we take the root with the positive sign because the angle is in the first quadrant.