Question

How do you find the exact value of `arc tan (1/2)+arc tan (1/3)`?

Answer 1

In this way:

`arctan(1/2)+arctan(1/3)=alpha`.

If we search the tangent of both members:

`tan(arctan(1/2)+arctan(1/3))=tanalpha`

And now, using the sum angle formula of the tangent:

`(tanarctan(1/2)+tanarctan(1/3))/(1-tanarctan(1/2)tanarctan(1/3))=tanalpharArr`

`(1/2+1/3)/(1-1/2*1/3)=tanalpharArrtanalpha=(5/6)/(5/6)=1rArr`

`alpha=45°+k180°`, or, in radians: `alpha=pi/4+kpi`.