In Excel, the hyperbolic functions COSH , SINH , and TANH all take a number representing a hyperbolic angle as input. A hyperbolic angle is defined by the area of the sector on the right branch of the unit hyperbola x² - y² = 1, formed by the origin, the point (1,0), and a point on the unit hyperbola. For example, a hyperbolic angle of 1 corresponds to the sector formed on the unit hyperbola with an area of one-half .
In general, you can think about a hyperbolic angle forming a point on the unit hyperbola, where COSH and SINH give the coordinates of the point:
A negative hyperbolic angle corresponds to a point with a negative y -coordinate.
The area of the sector is half the angle’s value to align the hyperbolic functions with their circular counterparts: cosine and sine . This is because the area formed by a circular angle on the unit circle is one-half the angle’s value.
Dividing by two makes the area of a hyperbolic angle equal to that of a circular angle. For example, the geometry of the hyperbolic and circular functions for the angle a=1 is shown below.
Unlike a circular angle, whose point rotates periodically around the circle as the angle grows, a hyperbolic angle diverges toward infinity as it increases positively and toward negative infinity as it increases negatively.
Images courtesy of wumbo.net .
Implicit Intersection describes a formula behavior in Excel where many values are reduced to a single value. Typically, it happens when a range or array is passed into a formula meant to display a single result in a single cell. In this situation, Excel will resolve the formula to a single value following the steps below:
- If the result is already single value, return the value.
- If the result is an array , try to return a value from the same row or column as the formula.
- Otherwise, return the top-left value from the array
Sometimes, when the result is an array, Excel won’t return the top-left value in array (step #3) unless the formula is entered an array formula with control + shift + enter. In other words, entering a formula with control + shift + enter disables implicit intersection.
Example
Implicit intersection can occur when a formula is entered next to vertical data, or above or below horizontal data. For example, the formula in D6 in the example shown is:
=B4:B8+1
In this case, Excel resolves the range B4:B8 to the value in B6 (3) and returns a result of 4. When Excel can’t determine a single reference in a formula where a range is passed, but a single value is expected, a #VALUE error is returned. For example, if same formula above is entered in cell D9, the result is #VALUE.
Array formulas
Entering an array formula with control + shift + enter (CSE) explicitly disables the implicit intersection behavior. This makes it possible to create formulas that manipulate multiple values input as ranges. If the formula in D6 is wrapped in SUM, then entered with control + shift + enter:
{=SUM(B4:B8+1)}
All values in the range are processed, and the formula returns 27.
Excel Tables
Implicit intersections can be useful in Excel Tables , where the same formula can be used in multiple cells (for consistency) but continue to resolve to a single cell at the row label. For example, this formula would add 7 days to the value in a “date” column in table:
=table[date]+7
Even though the formula refers to the entire “date” column, the formula will operate on a single value in the date column at the row level.
Dynamic Array Excel and @ operator
In Excel 365 , there is no need to enter a formula with control + shift + enter to enable array behavior, because all formulas are treated as array formulas by default. However, for compatibility reasons, you will sometimes see the @ symbol inserted in a formula created in an older (pre dynamic array) version of Excel. This @ symbol is called the implicit intersection operator, and it disables array behavior. In other words, it tells Excel you want a single value.
This is done to ensure that older formulas continue to return the same (single) result when they might otherwise spill multiple values onto the worksheet. In general, functions that return multi-cell ranges or arrays will be automatically prefixed with @ if they were created in an earlier version. This behavior only occurs in dynamic array versions of Excel.
