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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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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=Properties=
 
=Properties=
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[[Arcsin as inverse sine]]<br />
 
[[Derivative of arcsin]]<br />
 
[[Derivative of arcsin]]<br />
 
[[Antiderivative of arcsin]] <br />
 
[[Antiderivative of arcsin]] <br />
 
[[Relationship between arcsin and arccsc]] <br />
 
[[Relationship between arcsin and arccsc]] <br />
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[[2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z]]<br />
  
 
=Videos=
 
=Videos=
[https://www.youtube.com/watch?v=JGU74wbZMLg Inverse Trig Functions: Arcsin]<br />
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[https://www.youtube.com/watch?v=JGU74wbZMLg Inverse Trig Functions: Arcsin (1 October 2009)]<br />
[https://www.youtube.com/watch?v=KmHD7CsOw5Y Integrate x*arcsin(x)]<br />
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[https://www.youtube.com/watch?v=JZ9Ku1TTeA4 What is arcsin(x)? (18 August 2011)]<br />
[https://www.youtube.com/watch?v=JZ9Ku1TTeA4 What is arcsin(x)?]<br />
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[https://www.youtube.com/watch?v=KmHD7CsOw5Y Integrate x*arcsin(x) (25 February 2013)]<br />
[https://www.youtube.com/watch?v=4CY7RIUhs2s What is the inverse of arcsin(ln(x))?]<br />
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[https://www.youtube.com/watch?v=4CY7RIUhs2s What is the inverse of arcsin(ln(x))? (28 April 2014)]<br />
  
 
=See Also=
 
=See Also=
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[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]
[[Category:Definition]]
 

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