To compute $\tan^{-1}(-1)$, also written as $\arctan(-1)$, we seek the angle $\theta$ such that:

$\tan(\theta) = -1$

Key Information:

1. The range of the $\arctan$ function is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, which means the output is limited to this interval.

2. Within this range, the angle where $\tan(\theta) = -1$ is:

$\theta = -\frac{\pi}{4}$

Final Answer:

$\tan^{-1}(-1) = -\frac{\pi}{4}$

Would you like to explore this further or see a graphical explanation?

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Here are some related questions to consider:

1. What is the value of $\tan^{-1}(1)$?
2. How does the range of $\tan^{-1}(x)$ affect its outputs for negative inputs?
3. What is the tangent of angles beyond the range of $\tan^{-1}(x)$?
4. How can $\arctan(x)$ be visualized on the unit circle?
5. How does the derivative of $\arctan(x)$ relate to its computation?

Tip: Always remember that $\arctan(x)$'s output is constrained within its principal range, even though $\tan(x)$ repeats for all real numbers.