Principal Value Branches

2. Principal Value Branches

Definition. Among all the intervals on which a given trigonometric function is one-one and onto its usual range, the principal value branch is the one interval singled out by convention -- chosen to contain, or lie as close as possible to, the origin, and to keep the function continuous and strictly monotonic throughout. For $x$ in the domain of the inverse function, the unique value of the inverse lying in this branch is called the principal value of that inverse trigonometric function at $x$.

The six principal value branches.

Why each branch is chosen.

• $\sin x$ restricted to $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ increases strictly from $-1$ to $1$, so it is one-one onto $[-1,1]$; this interval is centred at the origin, so it is the natural choice.
• $\cos x$ is not one-one on any interval centred at the origin, since it is an even function ($\cos(-\theta)=\cos\theta$); the closest interval to the origin on which it is one-one and decreases strictly from $1$ to $-1$ is $[0,\pi]$.
• $\tan x$ is undefined at $x=\pm\frac{\pi}{2}$, so its branch $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ must be open at both ends; on it, $\tan x$ increases strictly from $-\infty$ to $\infty$.
• $\cot x$ is undefined at $x=0$ and $x=\pi$, so its branch $(0,\pi)$ is open at both ends, on which it decreases strictly from $+\infty$ to $-\infty$.
• $\sec x$ is undefined at $x=\frac{\pi}{2}$, so the branch $[0,\pi]$ has that one point removed.
• $\text{cosec}\,x$ is undefined at $x=0$, so the branch $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ has that one point removed. …