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

What are the steps for this?

1 Answer
Mar 22, 2018

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

Explanation:

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.