Question

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

What are the steps for this?

Answer 1

`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.