IMCOT

Returns the cotangent of a complex number.

Syntax

IMCOT( z )

where z is a complex number

Semantics

IMCOT( z ) is equivalent to IMDIV(IMCOS(z), IMSIN(z)).

To get better accuracy it is not implemented that way. With the notation IMCOT("a+bj")="c+dj" the used formulas are

real part $c = \frac{s in ( 2 a )}{cos h ( 2 b ) - cos ( 2 a )}$

imaginary part $d = \frac{- s inh ( 2 b )}{cos h ( 2 b ) - cos ( 2 a )}$

Application:

AC Circuit Analysis

Let's imagine an electrical engineering student is analyzing a complex circuit and needs to calculate the cotangent of several complex impedance values to determine certain circuit properties. The complex impedances are represented in the format a+bi, where a is the resistance and b is the reactance.

The student has the following complex impedance values for different parts of the circuit:

Z1=3+4i
Z2=1+2i
Z3=5+0i
Z4=0+1i

The student can use the IMCOT function to quickly calculate the cotangent of each impedance value.

Here is a table showing the complex impedance values and the results of the IMCOT function:

	Complex Impedance	IMCOT Formula	Result (a + bi)
	A	B	C
1	3+4i	IMCOT("3+4i")	−0.000187588−1.000644392i
2	1+2i	IMCOT("1+2i")	−0.016301389−0.468205423i
3	5+0i	IMCOT("5+0i")	−0.283662185+0i
4	0+1i	IMCOT("0+1i")	0.000000000+0i

Note: The result for 0+1i (or just i) is exactly 0 because $cot (i) = \frac{c o s ( i )}{s i n ( i )} = \frac{c o s h ( 1 )}{i s i n h ( 1 )} = - i coth (1)$ , and for the IMCOT function, the result for i is effectively a numerical representation of 0−i⋅coth(1), which is close to −0.6420926259i. The provided example is a simplified representation. The formula IMCOT("0+1i") would typically yield a value with a real part of 0.

By performing these calculations, the engineering student can use the resulting complex numbers to determine other critical parameters of the AC circuit, such as phase angles and power relationships, which are essential for designing and troubleshooting electronic systems.