Q.Find the principal value of the following: $\tan^{-1}(-1)$

The key idea is that the principal value of $\tan^{-1}(x)$ lies in the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$, and we need the angle whose tangent is $-1$.

Step 1: Recall that $\tan\left(-\frac{\pi}{4}\right) = -1$. …

Why the Principal Value Matters

When you see $\tan^{-1}(-1)$, you might think "tangent is $-1$ at $-\frac{\pi}{4}$, $\frac{3\pi}{4}$, $\frac{7\pi}{4}$, and infinitely many other angles." That's correct — the equation $\tan \theta = -1$ has infinitely many solutions because tangent is periodic with period $\pi$.

But the inverse trigonometric functions are defined as functions, meaning they must give exactly one output for each input. To achieve this, we restrict the domain of the original trigonometric function so that it becomes one-to-one. For $\tan^{-1}$, the standard restriction (called the principal value branch) is:

The interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ is chosen because:

• Tangent is strictly increasing and continuous there (so it's one-to-one).
• It covers all real outputs of tangent (since $\tan\theta \to -\infty$ as $\theta \to -\frac{\pi}{2}^+$ and $\tan\theta \to \infty$ as $\theta \to \frac{\pi}{2}^-$).
• It's symmetric about 0, which makes the function odd: $\tan^{-1}(-x) = -\tan^{-1}x$.

Step-by-Step Solution

1. Set up the equation.

Let $\theta = \tan^{-1}(-1)$. By definition, this means:

and $\theta$ must lie in the principal value interval:

2. Find all angles where $\tan\theta = -1$.

Tangent equals $-1$ when sine and cosine have equal magnitude but opposite signs. The reference angle is $\frac{\pi}{4}$ (since $\tan\frac{\pi}{4}=1$). The general solution for $\tan\theta = -1$ is: …

Method: Evaluating an inverse tangent at a standard value

For any $\tan^{-1}$ of a familiar number, the goal is the one angle in the principal branch with that tangent.

Steps

Step 1: Recall the branch

$\tan^{-1}$ outputs angles in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, where tangent is strictly increasing, so each value of tangent corresponds to exactly one angle.

Step 2: Match a standard tangent

Identify the acute angle whose tangent equals the magnitude of the argument (the key value here is $\tan\frac{\pi}{4} = 1$).

Step 3: Apply oddness for the sign …

Common Mistakes

Mistake 1: Picking $\frac{3\pi}{4}$ since $\tan\frac{3\pi}{4} = -1$

Why it's wrong: $\frac{3\pi}{4}$ is outside $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, so it is not a principal value. Correct approach: choose $-\frac{\pi}{4}$, the angle with tangent $-1$ inside the branch. …