singinscottnud

2022-12-30

Find the principal value of ${\mathrm{tan}}^{-1}\left(-1\right)$

Jonah Curtis

Expert

Let $y={\mathrm{tan}}^{-1}\left(-1\right)\phantom{\rule{0ex}{0ex}}⇒\mathrm{tan}y=-1$
We are aware that the major value branch of ${\mathrm{tan}}^{-1}x$ has a range of $\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$.
$⇒\mathrm{tan}y=\mathrm{tan}\left(-\frac{\pi }{4}\right)$
$\therefore y=-\frac{\pi }{4}$
Thus, the principal value of ${\mathrm{tan}}^{-1}\left(-1\right)$ is $-\frac{\pi }{4}$.

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