evaluate the expression below (for angles measured in radians). arctan(tan(-2π/3))

Explanation:

Step1: Recall the property of tangent function

The tangent - function (y = \tan x) has a period of (\pi), i.e., (\tan(x + k\pi)=\tan x,k\in\mathbb{Z}). So, (\tan\left(-\frac{2\pi}{3}\right)=\tan\left(-\frac{2\pi}{3}+\pi\right)=\tan\frac{\pi}{3}).

Step2: Recall the range of the arctangent function

The range of the inverse - tangent function (y = \arctan x) is (\left(-\frac{\pi}{2},\frac{\pi}{2}\right)). And (\arctan(\tan x)=x) when (x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)). Since (\tan\left(-\frac{2\pi}{3}\right)=\tan\frac{\pi}{3}), and (\arctan(\tan x)) gives the angle in the range (\left(-\frac{\pi}{2},\frac{\pi}{2}\right)) with the same tangent value. So, (\arctan\left(\tan\left(-\frac{2\pi}{3}\right)\right)=-\frac{\pi}{3}).

Answer:

(-\frac{\pi}{3})