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# A to Z of Excel Functions: the COTH Function

4 December 2017

Welcome back to our regular A to Z of Excel Functions blog.  Today we look at the COTH function.

The COTH function

This function returns the hyperbolic cotangent of a hyperbolic angle.  Clear as mud?  Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh z, sinh z) form the right half of the equilateral hyperbola (please see the figure below).  The hyperbolic functions take a real argument called a hyperbolic angle.  The size of a hyperbolic angle is twice the area of its hyperbolic sector.  The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.

Hyperbolic functions occur in the solutions of many linear differential equations, such as some cubic equations.  Further, in complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine – but that’s a story for another day.

Essentially, COTH(N) is equal to COSH(N) divided by SINH(N).  The COTH function employs the following syntax to operate:

COTH(number)

The COTH function has the following arguments:

• number: this is required.

It should be further noted that:

• the hyperbolic cotangent is analogous to the ordinary (circular) cotangent
• the absolute value of number must be less than 2^27
• if number exceeds its constraints, COTH returns the #NUM! error value
• if number is a non-numeric value, COTH returns the #VALUE! error value.

The following equation is used: