The Tangent Function and the Inverse Tangent Function

Tangent, $y=\tan x=\dfrac{\sin x}{\cos x}$, is used to find heights and distances (a building, mountain, or flagpole) from a known angle and a known side, or — inverted — an unknown angle from two known sides. Since $\cos x=0$ at $x=\ldots,-\tfrac{3\pi}2,-\tfrac{\pi}2,\tfrac{\pi}2,\tfrac{3\pi}2,\ldots$, the domain of $y=\tan x$ is $\left\{x\in\mathbb{R}:x\ne(2n+1)\tfrac{\pi}2,\,n\in\mathbb{Z}\right\}=\bigcup_{k=-\infty}^\infty\left(\tfrac{(2k+1)\pi}2,\tfrac{(2k+3)\pi}2\right)$, and its range is …