IMSINH Function in Excel

The IMSINH function in Excel returns the hyperbolic sine of a complex number, such as 3+4i or 2-0.5j. You’ll mostly need it in math, physics, and electrical engineering work, where complex numbers show up all the time.

IMSINH reads the complex number as text and returns its answer as text too. That’s why adding up its results gives 0 and rounding them gives an error, and a couple of the examples below show how to work around that.

In this article, I’ll show you six examples, including how to use IMSINH on a whole column and how to clean up the tiny leftover values it sometimes returns.

IMSINH Function Syntax

Here is the syntax of the IMSINH function:

=IMSINH(inumber)

• inumber – The complex number you want the hyperbolic sine of.

The complex number has to be text in the x+yi or x+yj form. When you type it straight into the formula, put it in double quotes, so =IMSINH("4+3i") returns -27.0168132580039+3.85373803791938i.

You can also give it a regular number. IMSINH treats it as a complex number with no imaginary part, so =IMSINH(2) returns 3.62686040784702.

If Excel can’t read the text as a complex number, IMSINH returns the #NUM! error. A TRUE or FALSE value returns the #VALUE! error.

The IMSINH function works in Excel for Microsoft 365, Excel 2024, 2021, 2019, 2016, and Excel for Mac.

When to Use the IMSINH Function

Use this function when you need to:

• Find the hyperbolic sine of complex numbers in a math or engineering worksheet
• Work with complex values written in j notation, as electrical engineers usually write them
• Build other complex-number calculations Excel has no function for, such as the hyperbolic tangent
• Check homework or textbook answers that involve the hyperbolic sine of a complex number

Let me show you a few practical examples of how to use the IMSINH function.

Example 1: Get the Hyperbolic Sine of a List of Complex Numbers

Let’s start with a simple example.

Below is a list of complex numbers, and I want to get the hyperbolic sine of each one in column B.

Here is the formula:

=IMSINH(+A2:A11)

Enter this in cell B2, and it spills the results down to B11. For 3+4i, you get -6.548120040911-7.61923172032141i, so the hyperbolic sine of a complex number is also a complex number.

The plus sign in front of the range is what makes this work. IMSINH can’t take a multi-cell range reference, so =IMSINH(A2:A11) returns a single #VALUE! error instead of ten results.

With the plus sign, Excel hands IMSINH the values as an array (a list of values held in memory), and IMSINH works through each one. The same trick works for other functions that won’t spill when you give them a range.

You’ll also notice the results sit on the left side of the cells. That’s because IMSINH always returns text, even when the answer has no imaginary part (like 2.12927945509482 for 1.5+0i).

Spilling needs Excel 365 or Excel 2021 or later. If you are using Excel 2019 or earlier, enter the below formula in cell B2 and then copy it down the range:

=IMSINH(A2)

Example 2: IMSINH vs SINH for Regular Numbers

IMSINH works on numbers with no imaginary part too, but Excel has a better function for those.

Below is a list of regular numbers, and I want to get the hyperbolic sine of each one in column B.

Here is the IMSINH formula:

=IMSINH(+A2:A11)

Each number is treated as a complex number with no imaginary part, so the results have no i in them. The sign carries through, so 2 returns 3.62686040784702 and -2 returns -3.62686040784702.

But these results are still text, which is why they’re left-aligned. If you try to total them with the SUM function, you get 0.

Now here is the SINH function for comparison. Enter this formula in cell C2:

=SINH(A2:A11)

The SINH function takes the range directly (no plus sign needed) and returns the same values as real numbers. You can total, format, or chart them like any other number.

Column C shows fewer decimals because of its General format, but the stored values match the IMSINH results.

So when your numbers have no imaginary part, use the SINH function. Keep IMSINH for numbers that do.

If you are using Excel 2019 or earlier, enter the below formula in cell B2 and then copy it down the range:

=IMSINH(A2)

And for the SINH version, enter the below formula in cell C2 and then copy it down the range:

=SINH(A2)

Example 3: IMSINH with Real and Imaginary Parts in Separate Columns

Sometimes the complex number isn’t stored as one value. The real part sits in one column and the imaginary part in another.

Below is a table with the real part of each number in column A and the imaginary part in column B, and I want to get the hyperbolic sine of each complex number, written with j, in column C.

Here is the formula:

=IMSINH(COMPLEX(+A2:A11,+B2:B11,"j"))

How this formula works:

• COMPLEX(+A2:A11,+B2:B11,”j”) joins each real part with its imaginary part, so 2 and 1 become 2+j. The COMPLEX function can’t take range references either, so both ranges get a plus sign.
• IMSINH then returns the hyperbolic sine of each complex number. Since the input uses j, the result uses j too, and the results spill down to C11.

Look at row 6, where the real part is 0. The result is 0.841470984807897j, which is the regular sine of 1 (the same value the SIN function returns), with a j after it.

Now look at row 7, where the imaginary part is 0. You get 1.50946135541217, which is simply the hyperbolic sine of 1.2.

This comes from how IMSINH does the math. For a number x+yj, it calculates SINH(x)*COS(y) for the real part and COSH(x)*SIN(y) for the imaginary part.

When x is 0, SINH(x) is 0 and COSH(x) is 1, so only the sine of y is left in the imaginary part. And when y is 0, COS(y) is 1 and SIN(y) is 0, so you get just the hyperbolic sine of x.

If you are using Excel 2019 or earlier, enter the below formula in cell C2 and then copy it down the range:

=IMSINH(COMPLEX(A2,B2,"j"))

Example 4: Calculate the Hyperbolic Tangent of a Complex Number

Excel has IMSINH and IMCOSH, but there’s no IMTANH function. If you type =IMTANH("1+i"), you get a #NAME? error.

But the hyperbolic tangent is just the hyperbolic sine divided by the hyperbolic cosine. So you can build it from IMSINH, IMCOSH, and IMDIV.

Below is a list of complex numbers, and I want to get the hyperbolic tangent of each one in column B.

Here is the formula:

=IMDIV(IMSINH(+A2:A11),IMCOSH(+A2:A11))

How this formula works:

• IMSINH(+A2:A11) returns the hyperbolic sine of each complex number.
• IMCOSH(+A2:A11) returns the hyperbolic cosine of each one.
• IMDIV divides the first by the second, so you get the hyperbolic tangent, and the results spill down to B11.

For 1+i, the result is 1.0839233273387+0.271752585319513i.

You can check the formula with row 6. There the input is 1, with no imaginary part, and the result is 0.761594155955766, which is exactly what the regular TANH function returns for 1.

If you are using Excel 2019 or earlier, enter the below formula in cell B2 and then copy it down the range:

=IMDIV(IMSINH(A2),IMCOSH(A2))

Example 5: Round Off Tiny Leftover Values in IMSINH Results

When the imaginary part of your number is a multiple of pi, some results come back with strange tiny values like 4.98590850450457E-15i. These are leftovers from the calculation, and they’re really meant to be 0.

Below is a table with the real part in column A and the imaginary part as a multiple of pi in column B (so 1 means π), and I want to get the hyperbolic sine of each number in column C.

Here is the formula:

=IMSINH(COMPLEX(+A2:A11,PI()*B2:B11))

How this formula works:

• PI()*B2:B11 multiplies each value in column B by pi, so 1 becomes π and 0.5 becomes π/2. Since this already does math on the range, no plus sign is needed here.
• COMPLEX(+A2:A11,…) joins the real part with that imaginary part.
• IMSINH returns the hyperbolic sine of each one.

Now look at row 2, where the number is 1+πi. The exact answer is -1.1752011936438, but Excel shows -1.1752011936438+4.98590850450457E-15i.

That extra bit is 0.00000000000000498, which is basically zero.

It shows up because Excel works with about 15 significant digits. Pi is stored slightly off, and each step of the calculation adds a tiny rounding error, so the imaginary part never lands on exactly 0.

Row 3 is the extreme case. The hyperbolic sine of πi should be 0, but you get 3.2311393144413E-15i.

You can’t fix this with the ROUND function directly, because the result is text and the ROUND function returns a #VALUE! error.

Instead, round the real and imaginary parts separately and join them back. Enter this formula in cell D2:

=LET(a,IMSINH(COMPLEX(+A2:A11,PI()*B2:B11)),COMPLEX(ROUND(IMREAL(a),6),ROUND(IMAGINARY(a),6)))

How this formula works:

• IMSINH(COMPLEX(…)) is the same formula as before, and the LET function stores its results as a, so IMSINH runs only once.
• IMREAL(a) and IMAGINARY(a) pull out the real and imaginary parts as regular numbers.
• ROUND(…,6) rounds each part to six decimal places, which turns the tiny leftovers into 0.
• COMPLEX(…) joins the rounded parts back into a complex number.

Row 2 now shows -1.175201, and row 3 shows 0. The COMPLEX function leaves out a part that is 0, so you don’t see +0i at the end.

Both of these formulas spill, so they need Excel 365 or Excel 2021 or later (the LET function also needs Excel 2021 or later).

If you are using Excel 2019 or earlier, enter the below formula in cell C2 and then copy it down the range:

=IMSINH(COMPLEX(A2,PI()*B2))

And for the rounded result, enter the below formula in cell D2 and then copy it down the range:

=COMPLEX(ROUND(IMREAL(C2),6),ROUND(IMAGINARY(C2),6))

Example 6: Why IMSINH Returns the #NUM! Error (and 0 for Blanks)

IMSINH is strict about how a complex number is written. This example shows what happens with badly written entries and empty cells.

Below is a list of complex numbers typed in by different people, and I want to get the hyperbolic sine of each entry in column C.

Here is the formula:

=IMSINH(+B2:B11)

Four entries return the #NUM! error, because IMSINH can’t read them as complex numbers:

• 3+4*i (Judith) has a multiplication sign. The i goes right after the number, as in 3+4i.
• 2-3I (Stephanie) uses an uppercase I. Only a lowercase i or j works.
• (1+2i) (Harold) has parentheses around the number.
• 2+i3 (Brooke) puts the i before the number. It needs to be 2+3i.

Fix the entry, and the error goes away. Spaces are fine, though, which is why Roger’s 1.5 – 2i works.

Now look at Todd’s row. His entry is blank, but IMSINH returns 0 instead of an error.

That’s because IMSINH treats an empty cell as 0, and the hyperbolic sine of 0 is 0.

Important: A blank entry returns 0, which looks like a real answer. Check for empty inputs before you use IMSINH results in other calculations.

Here is the formula that skips the blank entries. Enter it in cell D2:

=IF(B2:B11="","",IMSINH(+B2:B11))

The IF function checks each entry first. If it’s empty, the formula returns an empty string, and otherwise it calculates the hyperbolic sine, so Todd’s row now stays blank.

The #NUM! errors are still there on purpose. Hiding them with the IFERROR function would also hide the entries that need fixing.

If you are using Excel 2019 or earlier, enter the below formula in cell D2 and then copy it down the range:

=IF(B2="","",IMSINH(B2))

So these are some practical ways to use the IMSINH function in Excel to get the hyperbolic sine of complex numbers.

Just remember to write the numbers with a lowercase i or j, and put a plus sign in front of a range when you want one formula for the whole column. I hope you found this article helpful.

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