How to use the IMCOS() function in Excel

Its calculates the cosine of a complex number in rectangular form (x + yi or x + yj).

Syntax
IMCOS(complex_number)

Argument

• complex_number (required)
  A text string representing a complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

• Complex cosine is calculated as:
  cos(z) = cos(x)cosh(y) – i·sin(x)sinh(y)
• Uses Euler’s formula extension to complex plane
• Combines trigonometric and hyperbolic functions

Example

=IMCOS(« 3-4i »)  // Returns « -27.0349456+3.85115334i »

Additional Examples

=IMCOS(« 1+i »)     // Returns « 0.833730025-0.988897706i »

=IMCOS(« 0+πi »)    // Returns « 11.59195328 » (real number)

=IMCOS(« π/2-i »)   // Returns « 1.543080635i » (pure imaginary)

Key Features

• Returns result in standard complex form
• Preserves input’s imaginary unit convention
• Handles all complex number formats
• Maintains periodicity properties in complex plane

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Missing imaginary unit when needed

Usage Notes

1. For real numbers (y=0), reduces to standard cosine
2. For pure imaginary numbers (x=0), becomes hyperbolic cosine:
   cos(iy) = cosh(y)
3. Essential for:
   • Complex analysis
   • Wave propagation modeling
   • Quantum mechanical calculations