The Excel COSH function returns the hyperbolic cosine of a number. In geometric terms, the function returns the x-component of the point on the unit hyperbola defined by a hyperbolic angle. For example, the hyperbolic sine of 0 is 1, since the point corresponding to the hyperbolic angle of 0 is P=(1,0), where P=(x,y).
A hyperbolic angle is defined as a ray from the origin of the coordinate system that passes through a point on the hyperbola. The area formed by the hyperbolic angle is equal to one-half the hyperbolic angle. Area above the x-axis is considered positive and area below the x-axis is negative.
Shown below are some hyperbolic angles on the unit hyperbola and their corresponding points. As the angle approaches positive infinity, the angle converges to the diagonal asymptote in the first quadrant of the coordinate system. As the angle approaches negative infinity the angle converges to the diagonal asymptote in the fourth quadrant of the coordinate system.
Together the functions COSH and SINH parameterize the unit hyperbola. A point P = (x,y) along the curve is given in the form P = (COSH(a), SINH(a)).