How to Use the IMCSCH, IMSECH, and IMCOTH Functions in Microsoft Excel for Reciprocal Hyperbolics

In electrical engineering, fluid dynamics, signal processing, and advanced physics modelling, calculations frequently involve hyperbolic trigonometry evaluated over complex numbers (numbers containing both a real component and an imaginary coefficient, written as \(x + yi\) or \(x + yj\)). While Excel provides standard hyperbolic functions such as IMSINH, IMCOSH, and IMTANH, evaluating reciprocal hyperbolics manually can introduce syntax complexity and numerical rounding errors. To simplify these computations, Microsoft Excel features dedicated built-in engineering functions: IMCSCH, IMSECH, and IMCOTH.

Mathematical Foundations of the Reciprocal Functions

Each of these three functions calculates the reciprocal of a primary complex hyperbolic function:

• IMCSCH: Computes the hyperbolic cosecant of a complex number: \[ \text{csch}(z) = \frac{1}{\sinh(z)} = \frac{2}{e^z – e^{-z}} \]
• IMSECH: Computes the hyperbolic secant of a complex number: \[ \text{sech}(z) = \frac{1}{\cosh(z)} = \frac{2}{e^z + e^{-z}} \]
• IMCOTH: Computes the hyperbolic cotangent of a complex number: \[ \coth(z) = \frac{1}{\tanh(z)} = \frac{\cosh(z)}{\sinh(z)} = \frac{e^z + e^{-z}}{e^z – e^{-z}} \]

Function Syntax and Arguments

All three functions require a single argument representing the complex number to evaluate:

=IMCSCH(inumber)
=IMSECH(inumber)
=IMCOTH(inumber)

• inumber: A complex number formatted as text in the form "x+yi" or "x+yj", or a reference to a cell containing such a value. You can also construct valid complex numbers dynamically using Excel’s COMPLEX(real_num, i_num, [suffix]) function.

Step-by-Step Practical Examples

1. Evaluating Hyperbolic Cosecant with IMCSCH

Suppose you have the complex number "2+3i" in cell A2. To find its hyperbolic cosecant:

=IMCSCH(A2)

Excel returns:

-0.272549 - 0.040301i

Alternatively, using dynamic numerical inputs via the COMPLEX function:

=IMCSCH(COMPLEX(2, 3))

2. Evaluating Hyperbolic Secant with IMSECH

To evaluate the hyperbolic secant of the complex quantity "1+2i" stored in cell A3:

=IMSECH(A3)

Excel evaluates the complex exponential sums and returns:

-0.347719 - 0.741007i

3. Evaluating Hyperbolic Cotangent with IMCOTH

To compute the hyperbolic cotangent of "0.5+1.5i" in cell A4:

=IMCOTH(A4)

Excel computes the ratio of hyperbolic cosine to hyperbolic sine and outputs:

0.902964 - 0.354148i

Separating Real and Imaginary Components

Once you calculate reciprocal hyperbolic values, downstream formulas often require isolating the real amplitude from the imaginary phase shift. Use IMREAL and IMAGINARY to extract the numbers:

Target Metric	Formula	Output Type
Extract Real Part	=IMREAL(IMCSCH(A2))	Numeric Real Value
Extract Imaginary Part	=IMAGINARY(IMCSCH(A2))	Numeric Imaginary Coefficient
Calculate Modulus (Magnitude)	=IMABS(IMCSCH(A2))	Absolute Value / Vector Length
Calculate Phase Angle (Argument)	=IMARGUMENT(IMCSCH(A2))	Angle in Radians (\(-\pi\) to \(\pi\))

Common Errors and Troubleshooting

• #NUM! Error: Occurs when the input string cannot be parsed as a valid complex number (e.g., lowercase ‘j’ and ‘i’ mixed improperly, or multiple decimal points). It also occurs when IMCSCH or IMCOTH is passed zero ("0+0i"), as division by zero is undefined.
• #VALUE! Error: Occurs when a logical boolean value or invalid reference format is supplied directly to the function.
• Formatting Note: If typing complex numbers manually into Excel cells, ensure you do not add spaces between the real number, operator, and imaginary part (e.g., use "4+5i", not "4 + 5i").