Difference between revisions of "Arccot"
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[[File:Complex ArcCot.jpg|500px]] | [[File:Complex ArcCot.jpg|500px]] | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Proposition:</strong> | ||
| + | $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{z^2+1}$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> 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 | ||
| + | $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{\csc^2(\mathrm{arccot}(z))} = -\dfrac{1}{z^2+1}.█$$ | ||
| + | </div> | ||
| + | </div> | ||
=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?] | ||
Revision as of 02:04, 28 October 2014
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 $\mathrm{arccot_2} \colon \mathbb{R} \rightarrow \left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right) \setminus \{0\}$ which results from restricting cotangent to $\left( -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right)$.
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 $$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{\csc^2(\mathrm{arccot}(z))} = -\dfrac{1}{z^2+1}.█$$
