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# A to Z of Excel Functions: The IMCOSH Function

27 July 2020

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

The IMCOSH function

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit (sometimes denoted j) which is defined by its property i2 = −1.  In general, the square of an imaginary number bi is −b2.  For example, 9i is an imaginary number, and its square is −81.  Zero is considered to be both real and imaginary.

An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.

The polar form of a complex number is another way to represent the number.  The form z = a + bi is called the rectangular form of a complex number.

The horizontal axis is the real axis and the vertical axis is the imaginary axis.  You can find the real and imaginary components in terms of and θ, where is the length of the vector and θ is the angle made with the real axis.

From the Pythagorean Theorem,

r2 = a2 + b2

By using the basic trigonometric ratios,

cos θ = a / r and sin θ = b / r

Therefore, multiplying each side by r:

r cos θ = a and sin θ = b

Therefore,

z = a + bi

z = r cos θ + (r sin θ)i

z = r(cos θ + i sin θ)

In the case of a complex number, represents the absolute value, or modulus (where r = |z| = ), and the angle θ is called the argument of the complex number ( for > 0 and  for a < 0).

Using Euler’s Formula,

Given

then

It then follows

you eventually get:

cosh(x + yi) = cosh x cos y – i sinh x sin y

The IMCOSH function returns the hyperbolic cosine of a complex number in x + yi or x + yj text format.

The IMCOSH function employs the following syntax to operate:

IMCOSH(inumber)

The IMCOSH function has the following argument:

• inumber: this is required and represents the complex number for which you want to calculate the hyperbolic cosine.

It should be further noted that:

• you should use COMPLEX to convert real and imaginary coefficients into a complex number
• IMCOSH recognises either the i or j notation
• if inumber is a value that is not in the x + yi or x + yj text format, IMCOSH returns the #NUM! error value
• if inumber is a logical value, IMCOSH returns the #VALUE! error value
• if the complex number ends in +i or -i (or j), i.e. there is no coefficient between the operator and the imaginary unit, there must be no space, otherwise IMCOSH will return an #NUM! error.