What is $\operatorname{arccot}(-1)$? According to me $\operatorname{arccot}(-1)$ should be equal to $3\pi/4$ because 
$\operatorname{arccot}(-x) = \pi - \operatorname{arccot}(x)$. 
But in my book it is given to be $-\pi$/4...
also on wolfram alpha $\operatorname{arccot}(-1)$ is given as $-\pi/4$...
Please help!
 A: The $\cot$ function is periodic with period $\pi$  To make an inverse function, you need to choose a range, as $\cot (-\frac \pi 4) =\cot( \frac {3\pi} 4)=\cot (\frac {7\pi} 4)=-1$  The usual choice is that the result is between $0$ and $\pi$, so $\operatorname{arccot} (-1)=\frac {3\pi}4$, but that is a matter of convention.  You can't maintain all the relations you would like when choosing the principal values.
A: Since the cotangent function is periodic with period $\pi,$ then it doesn't have an inverse function in the usual sense. Instead, we restrict the function to a representative portion of its domain so that this restricted version will have an inverse. The typical restriction is to the interval $(0,\pi),$ on which the cotangent function is both continuous and one-to-one, and so the arccotangent function has $(0,\pi)$ as the typical range. Given that choice of restrictions, your answer is the correct one. I'm not sure why the book's answer and Wolfram Alpha's differ, but I suspect that it is because they take a restriction to $(-\pi/2,\pi/2],$ instead (which gives the function a discontinuity).
