Q.State True or False: The least numerical value, either positive or negative of angle $\theta$ is called principal value of the inverse trigonometric function.

The principal value of an inverse trigonometric function is the angle of smallest numerical value (positive or negative) that satisfies the relation — which is exactly what the standard principal-value branches return. …

The idea

A trig equation like $\sin\theta=\frac{1}{2}$ has infinitely many solutions. To make the inverse a genuine function we pick one representative, and the convention is the angle of least numerical magnitude — the one closest to $0$ — which is precisely how the principal-value branches are defined.

Why the statement is correct

Each inverse function's principal range is built around $0$ so that it returns the numerically smallest angle:

• $\sin^{-1}$: range $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, e.g. $\sin^{-1}\frac{1}{2}=\frac{\pi}{6}$ and $\sin^{-1}\left(-\frac{1}{2}\right)=-\frac{\pi}{6}$.
• $\tan^{-1}$: range $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, e.g. $\tan^{-1}(-1)=-\frac{\pi}{4}$.

In every case the chosen angle is the one of least magnitude, positive or negative according to the sign — exactly the description in the statement.

A note on $\cos^{-1}$ …

Method: Test a definition statement against the standard branches

Steps

Step 1: Recall how principal values are chosen.

Each inverse trig branch is placed as close to $0$ as possible, so it returns the numerically smallest admissible angle (with the correct sign) for a given ratio.

Step 2: Check representative cases. …

Common Mistakes

Mistake 1: Rejecting the statement because $\cos^{-1}$ never returns a negative angle.

Why it's wrong: "least numerical value, positive or negative" means the smallest-magnitude angle within each function's OWN branch; $\cos^{-1}$ uses $[0,\pi]$, so the values it …