Hyperbolic cosine online calculator (cosh)
Input cosh and numbers in the above form to calculate the hyperbolic cosine as follows.
The cosh is a command to calculate the hyperbolic cosine and all numbers are separated by one space. The value that ends with d means a degree angle so 180d is about 3.141592... (pi).
- 90d ... 90°
- 180d ... 180°
The calculator takes radians, not degrees. So the degree numbers should have the suffix d and the calculator converts them to radians.
The definition of hyperbolic cosine
\[ \textnormal{cosh} \ x = \dfrac{ e^x + e^{-x} }{2} \]
The hyperbolic cosine function is defined as the average of two exponential functions. If substituting x for -x in the function, you get the same function:
\[ \begin{aligned} \textnormal{cosh} (-x) &= \dfrac{ e^{-x} + e^{-(-x)} }{2} \\ &= \dfrac{ e^{-x} + e^x }{2} \\ &= \textnormal{cosh} \ x \end{aligned} \]
So the graph of the hyperbolic cosine function is symmetric about y-axis.
Hyperbolic cosine formula
\[ \textnormal{cosh} (x+y) = \textnormal{cosh} \ x \ \textnormal{cosh} \ y + \textnormal{sinh} \ x \ \textnormal{sinh} \ y \\~\\ \textnormal{cosh} (x-y) = \textnormal{cosh} \ x \ \textnormal{cosh} \ y - \textnormal{sinh} \ x \ \textnormal{sinh} \ y \]
\[ \begin{aligned} \textnormal{cosh} \ 2x &= \textnormal{cosh}^2 \ x + \textnormal{sinh}^2 \ x \\ &= 2 \textnormal{cosh}^2 \ x - 1 \\ &= 2 \textnormal{sinh}^2 \ x + 1 \end{aligned} \]
\[ \textnormal{cosh} \left( \dfrac{x}{2} \right) = \sqrt{ \dfrac{ \textnormal{cosh} \ x + 1 }{ 2 } } \]
\[ \textnormal{cosh}^2 x = \dfrac{ \textnormal{cosh} \ 2x + 1 }{ 2 } \]