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The parameter for the normal trigonometric functions represents the length of the opposite and adjacent sides of a triangle in a unit circle. The parameter is the angle of the triangle that is located at the radius. The vertex that touches the circle has the coordinates of $(\cos{\theta},\sin{\theta})$.

From what I understand, the hyperbolic trig functions represent a triangle that touches a unit hyperbola ($x^2-y^2=1$). The coordinates of the vertex that touches the hyperbola is $(\cosh{t},\sinh{t})$, but what does the parameter represent here?

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This parameter $t$, sometimes called "hyperbolic angle" can be interpreted as an area (https://en.wikipedia.org/wiki/Hyperbolic_angle). See also (https://www.revolvy.com/topic/Hyperbolic%20function&item_type=topic) See also the Russian Encyclopedia of Mathematics, which has often interesting points of view (https://www.encyclopediaofmath.org/index.php/Hyperbolic_functions).

Important remark: Interpreting the parameter as an area could already be done for circular functions ($\sin, \cos$ ...). In fact, we are accustomed to consider $\theta$ in $\cos(\theta)$ as an arc length, but it could be as well considered as the corresponding sector area (all right, up to a factor...).

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  • $\begingroup$ I have added a remark. $\endgroup$ – Jean Marie Apr 22 '17 at 9:17
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The meaning of the parameter $a$ in $\sinh(a)$ and in the case $\cosh(a)$ is the same as in the case of $\sin(a)$ and $\cos(a)$. Take a look at the figure below. enter image description here

In both cases the parameter equals the half area of the red region.

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