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# A to Z of Excel Functions: the IMCOS Function

20 July 2020

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

The IMCOS function

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i (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 r and θ, where r 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 r 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, r represents the absolute value, or modulus (where r = |z| = ), and the angle θ is called the argument of the complex number ( for a > 0 and  for a < 0).

Using Euler’s Formula,

Given

by doing more mathematics than you would probably ever wish to read,

you eventually get:

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

The IMCOS function employs the following syntax to operate:

IMCOS(inumber)

The IMCOS function has the following argument:

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

It should be further noted that:

• you should use COMPLEX to convert real and imaginary coefficients into a complex number
• IMCOS recognises either the i or j notation
• if inumber is a value that is not in the x + yi or x + yj text format, IMCOS returns the #NUM! error value
• if inumber is a logical value, IMCOS 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 IMCOS will return an #NUM! error.