Are sin, cos and tan defined for other kinds of geometry other than circular and hyperbolic? Perhaps for multiple dimensions? Sorry for the open-ended question, I'm just wondering.
Asked
Active
Viewed 164 times
2
Tumbleweed53
- 307
-
2Generalized Hyperbolic Functions?, also maybe relevant: Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sinh, cosh, and tanh? – Vepir Jan 02 '21 at 15:35
-
2If you think about $\cos$ and $\sin$ as underpinning Fourier Series, then you get other orthonormal sets of plynomials/functions - Legendre and all that. – ancient mathematician Jan 02 '21 at 15:39
-
1And you shouldn't ignore the various $q$-analogs of the trigonometric functions. – ancient mathematician Jan 02 '21 at 15:40
1 Answers
0
One can define, for any real $c$, $\sin_c$ and $\cos_c$ as the solutions to the system $$\sin_c'=\cos_c\quad \cos_c' = -c\sin_c,$$subject to $\cos_c(0)=1$ and $\sin_c(0)=0$. Then we have that $$\cos_c^2(x)+c\sin_c^2(x) =1.$$In particular, $\sin_1=\sin$, $\cos_1=\cos$, as well as $\sin_{-1}=\sinh$ and $\cos_{-1}=\cosh$. We also have $\sin_0(x)=x$ for all $x$ and $\cos_0 = 1$. There are formulas such as $$\cos_c(x+y)=\cos_c(x)\cos_c(y)-c\sin_c(x)\sin_c(y)$$and$$\sin_c(x+y)=\sin_c(x)\cos_c(y)+\cos_c(x)\sin_c(y).$$These functions are used in the study of geometries of constant curvature $c$. If I remember right, Christian Bär's Elementary Differential Geometry covers some of this.
Ivo Terek
- 80,301