Difference between revisions of "Arcsin"

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The $\mathrm{arcsin}$ function is the inverse function of the [[sine]] function. <br />
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__NOTOC__
[[File:Arcsin.png|500px]]
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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 />
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<div align="center">
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<gallery>
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File:Arcsinplot.png|Graph of $\mathrm{arcsin}$ on $[-1,1]$.
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File:Complexarcsinplot.png|[[Domain coloring]] of $\mathrm{arcsin}$.
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</gallery>
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</div>
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=Properties=
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[[Arcsin as inverse sine]]<br />
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[[Derivative of arcsin]]<br />
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[[Antiderivative of arcsin]] <br />
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[[Relationship between arcsin and arccsc]] <br />
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[[2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z]]<br />
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=Videos=
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[https://www.youtube.com/watch?v=JGU74wbZMLg Inverse Trig Functions: Arcsin (1 October 2009)]<br />
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[https://www.youtube.com/watch?v=JZ9Ku1TTeA4 What is arcsin(x)? (18 August 2011)]<br />
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[https://www.youtube.com/watch?v=KmHD7CsOw5Y Integrate x*arcsin(x) (25 February 2013)]<br />
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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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=See Also=
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[[Sine]] <br />
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[[Sinh]] <br />
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[[Arcsinh]]
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=References=
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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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[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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{{:Inverse trigonometric functions footer}}
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[[Category:SpecialFunction]]

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.

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