The Inverse Hyperbolic Function
Output of MD2002 Program ABH
Question and Answer
Questions
Read Symbol defintion
Q01 |
- Inverse hyperbolic functions
Q02 |
- Composition of function
Q03 |
- Prove that Arcsinh(sinh(X))=X
Q04 |
- The graphics of the inverse hyperbolic functions
Q05 |
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Q06 |
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Q07 |
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Q08 |
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Q09 |
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Q10 |
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Answers
Q01. Inverse hyperbolic functions
A1. Answer
(a) arcsinh(X) = ln(X+SQR(X^2+1))
(b) arccosh(X) = ln(X+SQR(X^2-1))
(c) arctanh(X) = ln((1+X)/(1-X))/2
(d) arccsvh(X) = ln(1/X + SQR(X^2+1))
(e) arcsech(X) = ln(1/X + SQR(X^2-1)
(f) arccoth(X) = ln(1-X)/(1+X))/2
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Q02. Composition of function
A2. sinh(arcsinh(x))=x
sinh(arcsinh(x))=(exp(arcsinh(x)-exp(-arcsinh(x)))/2
=exp(ln(X+SQR(X^2+1)))-exp(-ln(X+SQR(X^2+1)))/2
=((X+SQR(X^2+1))-(1/(X+SQR(X^2+1)))/2
=((X+SQR(X^2+1))+(X-SQR(X^2+1)))/2
=X
Note 1 :
exp(-ln(X+SQR(X^2+1)))=1/exp(ln(X+SQR(X^2+1)))
=1/(X+SQR(X^2+1))
=(X-SQR(X^2-1))/(X+SQR(X^2+1)*(X-SQR(X^2+1)
=(X-SQR(X^2-1))/(-1)
Note 2 : exp(ln(X+SQR(X^2+1)))=X+SQR(X^2+1)
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Q03. Prove that Arcsinh(sinh(X))=X
arcsinh(sinh(X))=ln(sinh(X)+SQR(sinh(X)^2+1))
=ln((exp(X)-exp(-X))/2+(exp(X)+exp(-X))/2)
=ln(exp(X))
=X
Note :
SQR(sinh(X)^2+1)=SQR((exp(X)-exp(-X))/2)^2+1)
=SQR((exp(X)+exp(-X))/2)^2)
=(exp(X)+exp(-X))/2
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Q04. The graphics of the inverse hyperbolic functions
A4. Click
Program ABH
y=arcsinh(x) .... Program 06 07
y=arcsinh(x) .... Program 06 08
y=arcsinh(x) .... Program 06 09
y=arcsinh(x) .... Program 06 10
y=arcsinh(x) .... Program 06 11
y=arcsinh(x) .... Program 06 12
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Q05. Reference
MD2002 ZM50
MD2002 TH.txt
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Q06. Answer
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Q07. Answer
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Q08. Answer
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Q09. Answer
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Q10. Answer
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