Difference between revisions of "Arcsin"

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The function $\mathrm{arcsin} \colon [-1,1] \rightarrow \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ is the [[inverse function]] of the [[sine]] function. <br />
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__NOTOC__
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The function $\mathrm{arcsin} \colon \mathbb{C} \setminus \left\{ (-\infty,-1) \bigcup (1,\infty) \right\} \rightarrow \mathbb{C}$ is defined by
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$$\rm{arcsin}(z)=-i \log \left( iz + \sqrt{1-z^2} \right),$$
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where $i$ denotes the [[imaginary number]] and $\log$ denotes the [[logarithm]]. <br />
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
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File:Arcsin.png|Graph of $\mathrm{arcsin}$ on $[-1,1]$.
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File:Arcsinplot.png|Graph of $\mathrm{arcsin}$ on $[-1,1]$.
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File:Complex arcsin.jpg|[[Domain coloring]] of [[analytic continuation]].
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File:Complexarcsinplot.png|[[Domain coloring]] of $\mathrm{arcsin}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
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<div class="toccolours mw-collapsible mw-collapsed">
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[[Arcsin as inverse sine]]<br />
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<strong>Proposition:</strong>  
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[[Derivative of arcsin]]<br />
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$$\dfrac{d}{dz} \mathrm{arcsin(z)} = \dfrac{1}{\sqrt{1-z^2}}$$
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[[Antiderivative of arcsin]] <br />
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<div class="mw-collapsible-content">
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[[Relationship between arcsin and arccsc]] <br />
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<strong>Proof:</strong> If $\theta=\mathrm{arcsin}(z)$ then $\sin(\theta)=z$. Now use [[implicit differentiation]] with respect to $z$ to get
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[[2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z]]<br />
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$$\cos(\theta)\theta'=1.$$
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The following image shows that $\cos(\mathrm{arcsin}(z))=\sqrt{1-z^2}$:
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=Videos=
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[[File:Cos(arcsin(z)).png|200px|center]]
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[https://www.youtube.com/watch?v=JGU74wbZMLg Inverse Trig Functions: Arcsin (1 October 2009)]<br />
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Hence substituting back in $\theta=\mathrm{arccos}(z)$ yields the formula
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[https://www.youtube.com/watch?v=JZ9Ku1TTeA4 What is arcsin(x)? (18 August 2011)]<br />
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$$\dfrac{d}{dz} \mathrm{arcsin(z)} = \dfrac{1}{\cos(\mathrm{arcsin(z)})} = \dfrac{1}{\sqrt{1-z^2}}. █$$
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[https://www.youtube.com/watch?v=KmHD7CsOw5Y Integrate x*arcsin(x) (25 February 2013)]<br />
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</div>
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[https://www.youtube.com/watch?v=4CY7RIUhs2s What is the inverse of arcsin(ln(x))? (28 April 2014)]<br />
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed">
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=See Also=
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<strong>Proposition:</strong>
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[[Sine]] <br />
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$$\int \mathrm{arcsin}(z) dz = \sqrt{1-z^2}+z\mathrm{arcsin}(z)+C$$
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[[Sinh]] <br />
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<div class="mw-collapsible-content">
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[[Arcsinh]]
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed">
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=References=
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<strong>Proposition:</strong>  
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*[http://mathworld.wolfram.com/InverseSine.html  Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseSine.html]<br />
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$$\mathrm{arcsin}(z) = \mathrm{arccsc}\left( \dfrac{1}{z} \right)$$
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[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0015%7CLOG_0028 On the function arc sin(x+iy)-Cayley]<br />
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
  
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<div class="toccolours mw-collapsible mw-collapsed">
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{{:Inverse trigonometric functions footer}}
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<strong>Proposition:</strong>
 
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$$\mathrm{arcsin}(z)=\sum_{k=0}^{\infty} \dfrac{\left(\frac{1}{2} \right)_n}{(2n+1)n!}x^{2n+1}$$
 
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<div class="mw-collapsible-content">
 
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<strong>Proof:</strong> █
 
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</div>
 
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</div>
 
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</div>
 
  
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=References=
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[[Category:SpecialFunction]]
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*[http://mathworld.wolfram.com/InverseSine.html  Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseSine.html]
 

Latest revision as of 23:45, 22 December 2016

The function $\mathrm{arcsin} \colon \mathbb{C} \setminus \left\{ (-\infty,-1) \bigcup (1,\infty) \right\} \rightarrow \mathbb{C}$ is defined by $$\rm{arcsin}(z)=-i \log \left( iz + \sqrt{1-z^2} \right),$$ where $i$ denotes the imaginary number and $\log$ denotes the logarithm.

  • Graph of $\mathrm{arcsin}$ on $[-1,1]$.

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

Properties

Arcsin as inverse sine
Derivative of arcsin
Antiderivative of arcsin
Relationship between arcsin and arccsc
2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z

Videos

Inverse Trig Functions: Arcsin (1 October 2009)
What is arcsin(x)? (18 August 2011)
Integrate x*arcsin(x) (25 February 2013)
What is the inverse of arcsin(ln(x))? (28 April 2014)

See Also

Sine
Sinh
Arcsinh

References

On the function arc sin(x+iy)-Cayley

Inverse trigonometric functions

Arcsin

Arccos

Arctan

Arccsc

Arcsec

Arccot
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