Working with Hyperbolic Sine and Cosine
Definition of sinh and cosh
Hyperbolic Sine
sinh x = (e^x - e^-x) / 2
Hyperbolic Cosine
cosh x = (e^x + e^-x) / 2
Properties
Adding sinh to cosh
cosh x + sinh x
= (e^x + e^-x) / 2 + (e^x - e^-x) / 2
= (2*e^x + e^-x - e^-x) / 2
= (2*e^x) / 2
= e^x
Subtracting sinh from cosh
cosh x - sinh x
= (e^x + e^-x) / 2 - (e^x - e^-x) / 2
= (e^x + e^-x - e^x + e^-x) / 2
= (2*e^-x) / 2
= e^-x
Multiplication of sinh and cosh
(I)
cosh x * cosh y
= (e^x + e^-x) / 2 * (e^y + e^-y) / 2
= (e^x * e^y + e^x * e^-y + e^-x * e^y + e^-x * e^-y) / 4
sinh x * sinh y
= (e^x - e^-x) / 2 * (e^y - e^-y) / 2
= (e^x * e^y - e^x * e^-y - e^-x * e^y + e^-x * e^-y) / 4
cosh x * cosh y + sinh x * sinh y
= (e^x * e^y + e^x * e^-y + e^-x * e^y + e^-x * e^-y) / 4
+ (e^x * e^y - e^x * e^-y - e^-x * e^y + e^-x * e^-y) / 4
= (2 * e^x * e^y + 2 * e^-x * e^-y) / 4
= (e^x * e^y + e^-x * e^-y) / 2
= (e^(x+y) + e^-(x+y)) / 2
= cosh(x + y)
(II)
sinh x * cosh y
= (e^x - e^-x) / 2 * (e^y + e^-y) / 2
= (e^x * e^y + e^x * e^-y - e^-x * e^y - e^-x * e^-y) / 4
sinh y * cosh x
= (e^y - e^-y) / 2 * (e^x + e^-x) / 2
= (e^x * e^y - e^x * e^-y - e^-x * e^-y + e^-x * e^y) / 4
sinh x * cosh y + sinh y * cosh x
= (e^x * e^y + e^x * e^-y - e^-x * e^y - e^-x * e^-y) / 4
+ (e^x * e^y - e^x * e^-y - e^-x * e^-y + e^-x * e^y) / 4
= (2 * e^x * e^y - 2 * e^-x * e^-y) / 4
= (e^(x+y) - e^-(x+y)) / 2
= sinh(x + y)
(III)
cosh^2 x
= ((e^x + e^-x) / 2)^2
= (e^(2*x) + 2 * e^x * e^-x + e^-(2*x)) / 4
= (e^(2*x) + 2 + e^-(2*x)) / 4
sinh^2 x
= ((e^x - e^-x) / 2)^2
= (e^(2*x) - 2 * e^x * e^-x + e^-(2*x)) / 4
= (e^(2*x) - 2 + e^-(2*x)) / 4
cosh^2 x + sinh^2 x
= (e^(2*x) + 2 + e^-(2*x)) / 4 + (e^(2*x) - 2 + e^-(2*x)) / 4
= (2*e^(2*x) + 2*e^-(2*x)) / 4
= (e^(2*x) + e^-(2*x)) / 2
= cosh(2*x)
Summary
cosh x + sinh x = e^x
cosh x - sinh x = e^-x
cosh x * cosh y + sinh x * sinh y = cosh(x + y)
sinh x * cosh y + sinh y * cosh x = sinh(x + y)
cosh^2 x + sinh^2 x = cosh(2*x)
Source:
Oldham, Keith, Myland, Jan, and Spainer, Jerome. An Atlas of Functions 2nd Edition. Springer: New York. ISBN 978-0-387-48806-6 2009
Stay safe and sane this Black Friday,
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Definition of sinh and cosh
Hyperbolic Sine
sinh x = (e^x - e^-x) / 2
Hyperbolic Cosine
cosh x = (e^x + e^-x) / 2
Properties
Adding sinh to cosh
cosh x + sinh x
= (e^x + e^-x) / 2 + (e^x - e^-x) / 2
= (2*e^x + e^-x - e^-x) / 2
= (2*e^x) / 2
= e^x
Subtracting sinh from cosh
cosh x - sinh x
= (e^x + e^-x) / 2 - (e^x - e^-x) / 2
= (e^x + e^-x - e^x + e^-x) / 2
= (2*e^-x) / 2
= e^-x
Multiplication of sinh and cosh
(I)
cosh x * cosh y
= (e^x + e^-x) / 2 * (e^y + e^-y) / 2
= (e^x * e^y + e^x * e^-y + e^-x * e^y + e^-x * e^-y) / 4
sinh x * sinh y
= (e^x - e^-x) / 2 * (e^y - e^-y) / 2
= (e^x * e^y - e^x * e^-y - e^-x * e^y + e^-x * e^-y) / 4
cosh x * cosh y + sinh x * sinh y
= (e^x * e^y + e^x * e^-y + e^-x * e^y + e^-x * e^-y) / 4
+ (e^x * e^y - e^x * e^-y - e^-x * e^y + e^-x * e^-y) / 4
= (2 * e^x * e^y + 2 * e^-x * e^-y) / 4
= (e^x * e^y + e^-x * e^-y) / 2
= (e^(x+y) + e^-(x+y)) / 2
= cosh(x + y)
(II)
sinh x * cosh y
= (e^x - e^-x) / 2 * (e^y + e^-y) / 2
= (e^x * e^y + e^x * e^-y - e^-x * e^y - e^-x * e^-y) / 4
sinh y * cosh x
= (e^y - e^-y) / 2 * (e^x + e^-x) / 2
= (e^x * e^y - e^x * e^-y - e^-x * e^-y + e^-x * e^y) / 4
sinh x * cosh y + sinh y * cosh x
= (e^x * e^y + e^x * e^-y - e^-x * e^y - e^-x * e^-y) / 4
+ (e^x * e^y - e^x * e^-y - e^-x * e^-y + e^-x * e^y) / 4
= (2 * e^x * e^y - 2 * e^-x * e^-y) / 4
= (e^(x+y) - e^-(x+y)) / 2
= sinh(x + y)
(III)
cosh^2 x
= ((e^x + e^-x) / 2)^2
= (e^(2*x) + 2 * e^x * e^-x + e^-(2*x)) / 4
= (e^(2*x) + 2 + e^-(2*x)) / 4
sinh^2 x
= ((e^x - e^-x) / 2)^2
= (e^(2*x) - 2 * e^x * e^-x + e^-(2*x)) / 4
= (e^(2*x) - 2 + e^-(2*x)) / 4
cosh^2 x + sinh^2 x
= (e^(2*x) + 2 + e^-(2*x)) / 4 + (e^(2*x) - 2 + e^-(2*x)) / 4
= (2*e^(2*x) + 2*e^-(2*x)) / 4
= (e^(2*x) + e^-(2*x)) / 2
= cosh(2*x)
Summary
cosh x + sinh x = e^x
cosh x - sinh x = e^-x
cosh x * cosh y + sinh x * sinh y = cosh(x + y)
sinh x * cosh y + sinh y * cosh x = sinh(x + y)
cosh^2 x + sinh^2 x = cosh(2*x)
Source:
Oldham, Keith, Myland, Jan, and Spainer, Jerome. An Atlas of Functions 2nd Edition. Springer: New York. ISBN 978-0-387-48806-6 2009
Stay safe and sane this Black Friday,
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
