Difference between revisions of "Arccot"
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=References= | =References= | ||
[http://www.intmath.com/blog/which-is-the-correct-graph-of-arccot-x/6009 Which is the correct graph of arccot x?] | [http://www.intmath.com/blog/which-is-the-correct-graph-of-arccot-x/6009 Which is the correct graph of arccot x?] | ||
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| + | =See Also= | ||
| + | [[Cotangent]] <br /> | ||
| + | [[Coth]] <br /> | ||
| + | [[Arccoth]] | ||
<center>{{:Inverse trigonometric functions footer}}</center> | <center>{{:Inverse trigonometric functions footer}}</center> | ||
Revision as of 18:38, 11 November 2015
There are two functions commonly called $\mathrm{arccot}$, which refers to inverse functions of the $\mathrm{cot}$ function. First is the function $\mathrm{arccot_1}\colon \mathbb{R} \rightarrow (0,\pi)$ which results from restricting cotangent to $(0,\pi)$ and second is the function $\mathrm{arccot_2} \colon \mathbb{R} \rightarrow \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \setminus \{0\}$ which results from restricting cotangent to $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$.
- Arccots.png
Graph of $\mathrm{arccot}_1$ and $\mathrm{arccot}_2$ on $\mathbb{R}$.
- Complex ArcCot.jpg
Domain coloring of analytic continuation $\mathrm{arccot}$.
Properties
Proposition: $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{z^2+1}$$
Proof: If $y=\mathrm{arccot}(z)$ then $\cot(y)=z$. Now use implicit differentiation with respect to $z$ to get $$-\csc^2(y)y'=1.$$ Substituting back in $y=\mathrm{arccos}(z)$ yields the formula $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{\csc^2(\mathrm{arccot}(z))} = -\dfrac{1}{z^2+1}.█$$
References
Which is the correct graph of arccot x?
