Arctan root(3) is equal to pi/3. Why??

Question

It's math homework, the question is what is arctan root(3) equal to, and I found pi/3 using my calculator, but how do I figure that out on paper?

Answer 1

Draw an equilateral triangle ABC with all its angles 60 deg. (= pi / 3 rad). The sides are all equal. Call then 2 units in length.

Drop a perpendicular from A meeting BC at D. That bisects BC, making BD = 1. It also bisects the angle BAC, making angle DAC equal 30 deg.

In triangle ADC:

AC = 2

CD = 1

Angle ACD = 60 deg.

Angle CAD = 30 deg.

AD^2 = AC^2 - CD^2 = 4 - 1 = 3.

AD = sqrt(3).

You can now read off all the ratios for 30 and 60 deg, and in particular, tan(pi / 3) = sqrt(3).