Difference between revisions of "Arccot"

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There are two functions commonly called $\mathrm{arccot}$, which refers to inverse functions of the [[cotangent | $\mathrm{cot}$]] 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 $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$.  
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__NOTOC__
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The [[function]] $\mathrm{arccot} \colon \mathbb{R} \rightarrow \left( - \dfrac{\pi}{2}, \dfrac{\pi}{2} \right] \setminus \{0\}$ is the [[inverse function]] of the [[cotangent]] function.  
  
 
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File:Arccots.png|Graph of $\mathrm{arccot}_1$ and $\mathrm{arccot}_2$ on $\mathbb{R}$.
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File:Arccotplot.png|Graph of $\mathrm{arccot}$ on $\mathbb{R}$.
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File:Complex ArcCot.jpg|[[Domain coloring]] of [[analytic continuation]] $\mathrm{arccot}$.
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File:Complexarccotplot.png|[[Domain coloring]] of $\mathrm{arccot}$.
 
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=Properties=
 
=Properties=
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[[Derivative of arccot]]
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<strong>Proposition:</strong>
 
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$$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{z^2+1}$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> If $y=\mathrm{arccot}(z)$ then $\cot(y)=z$. Now use [[implicit differentiation]] with respect to $z$ to get
 
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$$-\csc^2(y)y'=1.$$
 
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Substituting back in $y=\mathrm{arccos}(z)$ yields the formula
 
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$$\dfrac{d}{dz} \mathrm{arccot}(z) = -\dfrac{1}{\csc^2(\mathrm{arccot}(z))} = -\dfrac{1}{z^2+1}.█$$ 
 
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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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<center>{{:Inverse trigonometric functions footer}}</center>
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=See Also=
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[[Cotangent]] <br />
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[[Coth]] <br />
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[[Arccoth]]
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{{:Inverse trigonometric functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 03:44, 6 July 2016

The function $\mathrm{arccot} \colon \mathbb{R} \rightarrow \left( - \dfrac{\pi}{2}, \dfrac{\pi}{2} \right] \setminus \{0\}$ is the inverse function of the cotangent function.

  • Graph of $\mathrm{arccot}$ on $\mathbb{R}$.

  • Domain coloring of $\mathrm{arccot}$.

Properties

Derivative of arccot

References

Which is the correct graph of arccot x?

See Also

Cotangent
Coth
Arccoth

Inverse trigonometric functions

Arcsin

Arccos

Arctan

Arccsc

Arcsec

Arccot
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