Why is sinh called “sinus hyperbolicus” despite being just a regular e function?
5
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What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all.
The only similarity I can find is that their exponential representation looks similar.
$sin(x) = frac{1}{2i}(e^{ix}-e^{-ix})$
$sinh(x) = frac{1}{2}(e^{x}-e^{-x})$
trigonometry complex-numbers hyperbolic-functions
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
$endgroup$
add a comment |
5
$begingroup$
What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all.
The only similarity I can find is that their exponential representation looks similar.
$sin(x) = frac{1}{2i}(e^{ix}-e^{-ix})$
$sinh(x) = frac{1}{2}(e^{x}-e^{-x})$
trigonometry complex-numbers hyperbolic-functions
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
$endgroup$
2
$begingroup$
Also note the relation $$ isin(x) = sinh(ix). $$
$endgroup$
– MisterRiemann
Jan 16 at 16:39
2
$begingroup$
See Hyperbolic function.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:43
2
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See Vincenzo Riccati, Opuscula physico-mathematica, Vol.1 (1757) page 70 : "ita in hyperbola sumemus sinus et cosinus logarithmorum, quos sinus et cosinus hyperbolicos appellabimus."
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:54
1
$begingroup$
Well, I don’t know whether this helps, but the standard formulas for spherical trigonometry are mirrored by formulas in hyperbolic trigonometry by the replacement of the sine and cosine by the corresponding hyperbolic functions, when it’s a matter of side lengths. For instance, the spherical Pythagorean formula is $cos c=cos acos b$, while the corresponding hyperbolic formula is $cosh c=cosh acosh b$.
$endgroup$
– Lubin
Jan 16 at 20:22
$begingroup$
See "Geometric construction of hyperbolic trigonometric functions". In particular, my answer.
$endgroup$
– Blue
Jan 16 at 23:05
add a comment |
5
5
5
$begingroup$
What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all.
The only similarity I can find is that their exponential representation looks similar.
$sin(x) = frac{1}{2i}(e^{ix}-e^{-ix})$
$sinh(x) = frac{1}{2}(e^{x}-e^{-x})$
trigonometry complex-numbers hyperbolic-functions
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
$endgroup$
What does the sinus hyperbolicus have to do with the sinus? Their graphs do not look alike at all.
The only similarity I can find is that their exponential representation looks similar.
$sin(x) = frac{1}{2i}(e^{ix}-e^{-ix})$
$sinh(x) = frac{1}{2}(e^{x}-e^{-x})$
trigonometry complex-numbers hyperbolic-functions
trigonometry complex-numbers hyperbolic-functions
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
asked Jan 16 at 16:37
Sebastian WalkerSebastian Walker
262
262
2
$begingroup$
Also note the relation $$ isin(x) = sinh(ix). $$
$endgroup$
– MisterRiemann
Jan 16 at 16:39
2
$begingroup$
See Hyperbolic function.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:43
2
$begingroup$
See Vincenzo Riccati, Opuscula physico-mathematica, Vol.1 (1757) page 70 : "ita in hyperbola sumemus sinus et cosinus logarithmorum, quos sinus et cosinus hyperbolicos appellabimus."
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:54
1
$begingroup$
Well, I don’t know whether this helps, but the standard formulas for spherical trigonometry are mirrored by formulas in hyperbolic trigonometry by the replacement of the sine and cosine by the corresponding hyperbolic functions, when it’s a matter of side lengths. For instance, the spherical Pythagorean formula is $cos c=cos acos b$, while the corresponding hyperbolic formula is $cosh c=cosh acosh b$.
$endgroup$
– Lubin
Jan 16 at 20:22
$begingroup$
See "Geometric construction of hyperbolic trigonometric functions". In particular, my answer.
$endgroup$
– Blue
Jan 16 at 23:05
add a comment |
2
$begingroup$
Also note the relation $$ isin(x) = sinh(ix). $$
$endgroup$
– MisterRiemann
Jan 16 at 16:39
2
$begingroup$
See Hyperbolic function.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:43
2
$begingroup$
See Vincenzo Riccati, Opuscula physico-mathematica, Vol.1 (1757) page 70 : "ita in hyperbola sumemus sinus et cosinus logarithmorum, quos sinus et cosinus hyperbolicos appellabimus."
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:54
1
$begingroup$
Well, I don’t know whether this helps, but the standard formulas for spherical trigonometry are mirrored by formulas in hyperbolic trigonometry by the replacement of the sine and cosine by the corresponding hyperbolic functions, when it’s a matter of side lengths. For instance, the spherical Pythagorean formula is $cos c=cos acos b$, while the corresponding hyperbolic formula is $cosh c=cosh acosh b$.
$endgroup$
– Lubin
Jan 16 at 20:22
$begingroup$
See "Geometric construction of hyperbolic trigonometric functions". In particular, my answer.
$endgroup$
– Blue
Jan 16 at 23:05
2
2
$begingroup$
Also note the relation $$ isin(x) = sinh(ix). $$
$endgroup$
– MisterRiemann
Jan 16 at 16:39
$begingroup$
Also note the relation $$ isin(x) = sinh(ix). $$
$endgroup$
– MisterRiemann
Jan 16 at 16:39
2
2
$begingroup$
See Hyperbolic function.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:43
$begingroup$
See Hyperbolic function.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:43
2
2
$begingroup$
See Vincenzo Riccati, Opuscula physico-mathematica, Vol.1 (1757) page 70 : "ita in hyperbola sumemus sinus et cosinus logarithmorum, quos sinus et cosinus hyperbolicos appellabimus."
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:54
$begingroup$
See Vincenzo Riccati, Opuscula physico-mathematica, Vol.1 (1757) page 70 : "ita in hyperbola sumemus sinus et cosinus logarithmorum, quos sinus et cosinus hyperbolicos appellabimus."
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:54
1
1
$begingroup$
Well, I don’t know whether this helps, but the standard formulas for spherical trigonometry are mirrored by formulas in hyperbolic trigonometry by the replacement of the sine and cosine by the corresponding hyperbolic functions, when it’s a matter of side lengths. For instance, the spherical Pythagorean formula is $cos c=cos acos b$, while the corresponding hyperbolic formula is $cosh c=cosh acosh b$.
$endgroup$
– Lubin
Jan 16 at 20:22
$begingroup$
Well, I don’t know whether this helps, but the standard formulas for spherical trigonometry are mirrored by formulas in hyperbolic trigonometry by the replacement of the sine and cosine by the corresponding hyperbolic functions, when it’s a matter of side lengths. For instance, the spherical Pythagorean formula is $cos c=cos acos b$, while the corresponding hyperbolic formula is $cosh c=cosh acosh b$.
$endgroup$
– Lubin
Jan 16 at 20:22
$begingroup$
See "Geometric construction of hyperbolic trigonometric functions". In particular, my answer.
$endgroup$
– Blue
Jan 16 at 23:05
$begingroup$
See "Geometric construction of hyperbolic trigonometric functions". In particular, my answer.
$endgroup$
– Blue
Jan 16 at 23:05
add a comment |
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2
$begingroup$
Also note the relation $$ isin(x) = sinh(ix). $$
$endgroup$
– MisterRiemann
Jan 16 at 16:39
2
$begingroup$
See Hyperbolic function.
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:43
2
$begingroup$
See Vincenzo Riccati, Opuscula physico-mathematica, Vol.1 (1757) page 70 : "ita in hyperbola sumemus sinus et cosinus logarithmorum, quos sinus et cosinus hyperbolicos appellabimus."
$endgroup$
– Mauro ALLEGRANZA
Jan 16 at 16:54
1
$begingroup$
Well, I don’t know whether this helps, but the standard formulas for spherical trigonometry are mirrored by formulas in hyperbolic trigonometry by the replacement of the sine and cosine by the corresponding hyperbolic functions, when it’s a matter of side lengths. For instance, the spherical Pythagorean formula is $cos c=cos acos b$, while the corresponding hyperbolic formula is $cosh c=cosh acosh b$.
$endgroup$
– Lubin
Jan 16 at 20:22
$begingroup$
See "Geometric construction of hyperbolic trigonometric functions". In particular, my answer.
$endgroup$
– Blue
Jan 16 at 23:05