Question

Question #b9a38

Answer 1

`arctan(sqrt(3)) = pi/3 iff tan(pi/3) = sqrt(3)`

Using `tan^2 theta+1=sec^2 theta` and `cos(pi/3) = 1/2` we have:

`tan^2 (pi/3) +1=sec^2 (pi/3)`

`:. tan^2 (pi/3) +1=1/cos^2 (pi/3)`

`:. tan^2 (pi/3) +1=1/(1/2)^2`

`:. tan^2 (pi/3) +1=1/(1/4)`

`:. tan^2 (pi/3) +1=4`

`:. tan^2 (pi/3) =3`

`:. tan (pi/3) = +- sqrt(3)`

But as `pi/3` is acute `=> tan (pi/3) = sqrt(3)`

`=> arctan (sqrt(3)) = pi/3 \ \ \ ` QED