How to Use the IMSINH, IMCOSH, and IMTANH Functions in Microsoft Excel for Complex Hyperbolics

Analyzing alternating current transmission networks, electromagnetic wave absorption, relativistic velocity additions, and heat conduction in composite materials frequently requires evaluating hyperbolic functions on numbers with both real and imaginary components. Standard Excel hyperbolic functions like SINH, COSH, and TANH are restricted solely to real numbers. To perform complex mathematical modeling, Microsoft Excel provides three specialised engineering functions: IMSINH, IMCOSH, and IMTANH.

These functions evaluate the hyperbolic sine, hyperbolic cosine, and hyperbolic tangent of complex numbers expressed in the standard x + yi or x + yj format. By taking advantage of Euler’s formula and identities relating hyperbolic and circular trigonometric components, Excel outputs exact complex values in a single calculation step. In this tutorial, you will learn the mathematical foundations, syntax, input formatting rules, error handling methods, and practical electrical engineering applications for IMSINH, IMCOSH, and IMTANH.

Mathematical Foundations of Complex Hyperbolic Functions

When a variable possesses both real and imaginary parts (z = x + yi), hyperbolic functions expand into trigonometric products:

• Complex Hyperbolic Sine (IMSINH):
  sinh(x + yi) = sinh(x)cos(y) + i·cosh(x)sin(y)
• Complex Hyperbolic Cosine (IMCOSH):
  cosh(x + yi) = cosh(x)cos(y) + i·sinh(x)sin(y)
• Complex Hyperbolic Tangent (IMTANH):
  tanh(x + yi) = sinh(x + yi) / cosh(x + yi) = (sinh(2x) + i·sin(2y)) / (cosh(2x) + cos(2y))

Input Formatting Rules for Excel Complex Functions

To avoid formula errors when working with engineering functions in Excel:

• The complex number must be passed as a text string, such as "2+3i", "-4-1.5j", or "5i".
• Coefficients can be generated dynamically using the COMPLEX(real_num, i_num, [suffix]) function.
• Both lowercase i and j are valid imaginary suffixes; however, do not mix different suffixes in the same mathematical pipeline.
• Results are returned as complex text strings. Use IMREAL() to extract the real portion and IMAGINARY() to extract the imaginary coefficient.

1. The IMSINH Function (Complex Hyperbolic Sine)

The IMSINH function returns the hyperbolic sine of a complex number.

Syntax

=IMSINH(inumber)

• inumber: A complex number formatted as text for which you want the hyperbolic sine.

Example Calculation

To evaluate the hyperbolic sine of "1+2i":

=IMSINH("1+2i")

Excel calculates sinh(1)cos(2) + i·cosh(1)sin(2) and returns approximately "-0.489056259041294+1.40311925062204i".

2. The IMCOSH Function (Complex Hyperbolic Cosine)

The IMCOSH function calculates the hyperbolic cosine of a complex number.

Syntax

=IMCOSH(inumber)

• inumber: A complex number for which you want the hyperbolic cosine.

Example Calculation

To evaluate the hyperbolic cosine of "1+2i":

=IMCOSH("1+2i")

Excel calculates cosh(1)cos(2) + i·sinh(1)sin(2) and returns approximately "-0.642148123540634+1.06860742138278i".

3. The IMTANH Function (Complex Hyperbolic Tangent)

The IMTANH function returns the hyperbolic tangent of a complex number.

Syntax

=IMTANH(inumber)

• inumber: A complex number for which you want the hyperbolic tangent.

Example Calculation

To evaluate the hyperbolic tangent of "1+2i":

=IMTANH("1+2i")

Excel computes the ratio of IMSINH over IMCOSH and returns approximately "1.16673625724092-0.243458201185725i".

Practical Engineering Example: Lossy Transmission Lines

In high-frequency electrical engineering, the input impedance (Z_in) of a transmission line with characteristic impedance Z_0 = 50 Ω, propagation constant γ = 0.2 + 0.8i, and physical length L = 2.5 m is calculated using hyperbolic tangent:

Z_in = Z_0 · tanh(γ · L)

You can model this directly in Excel:

1. In cell A2, input real attenuation: 0.2.
2. In cell B2, input phase constant: 0.8.
3. In cell C2, build the propagation constant:

   =COMPLEX(A2, B2)

   (Outputs "0.2+0.8i")
4. In cell D2, multiply by length 2.5 using IMPRODUCT:

   =IMPRODUCT(C2, 2.5)

   (Outputs "0.5+2i")
5. In cell E2, compute the complex hyperbolic tangent:

   =IMTANH(D2)

   (Outputs "0.7937-0.4578i")
6. In cell F2, multiply by characteristic impedance 50:

   =IMPRODUCT(50, E2)

   (Returns input impedance "39.685-22.890i" Ω)

Common Error Troubleshooting

Error Code	Root Cause	Resolution
#NUM!	Malformed complex string (missing operator, misplaced i or j, or conflicting signs like "2+-3i").	Use COMPLEX(real, imag) to construct properly formatted text representations.
#VALUE!	Logical values or non-text boolean inputs supplied as the argument.	Verify that inputs reference valid text cells or string formulas.
#NUM! (Overflow)	Real component |x| > 709, causing standard floating-point exponential overflow in cosh(x) or sinh(x).	Normalise data units or scale mathematical parameters within valid IEEE limits.

Summary Best Practices

• Combine with Engineering Tools: Use IMABS to determine the total signal magnitude and IMARGUMENT to extract phase angles in radians.
• Format Output Appropriately: Because outputs are text strings, format numbers cleanly before building strings rather than applying spreadsheet cell formatting after computation.