Principal Values of Inverse Trigonometric Functions

This short section gathers the domain and principal-value range of all six inverse trigonometric functions in one place, now that each has been defined individually in the six preceding sub-sections: $\sin^{-1}:[-1,1]\to\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$; $\cos^{-1}:[-1,1]\to[0,\pi]$; $\tan^{-1}:\mathbb{R}\to\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$; $\text{cosec}^{-1}:\mathbb{R}-(-1,1)\to\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)-\{0\}$; $\sec^{-1}:\mathbb{R}-(-1,1)\to[0,\pi]-\left\{\dfrac{\pi}{2}\right\}$; $\cot^{-1}:\mathbb{R}\to(0,\pi)$. This table (reproduced as the Table note attached to this section) is the reference every property and worked example in the next section relies on to check …