How do you evaluate $tan^-1(sqrt3/3)$ without a calculator?

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Answer 1 · 2016-09-29T21:23:21

$tan^(-1) (sqrt(3)/3) = pi/6$

Consider a triangle with sides in ratio $1:sqrt(3):2$

This is a right angled triangle since:

$1^2+sqrt(3)^2 = 1+3 = 4 = 2^2$

In fact, it is one half of an equilateral triangle, hence has angles $pi/6$ , $pi/3$ , $pi/2$ ...

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Now $tan(theta) = "opposite"/"adjacent"$

Hence we find:

$tan (pi/6) = 1/sqrt(3) = sqrt(3)/3$

Since $pi/6$ radians is in Q1, this is the value of $tan^(-1) (sqrt(3)/3)$

How do you evaluate tan^-1(sqrt3/3)tan^-1(sqrt3/3) without a calculator?