Purpose
Return value
Syntax
=SINH(number)
- number - The hyperbolic angle.
Using the SINH function
The SINH function returns the hyperbolic sine of a number. Given the input 1, the function returns the value 1.175201194.
=SINH(0) // returns 1.175201194
Explanation
The hyperbolic sine function returns the vertical component of the point on the right branch (x ≥ 1) of the unit hyperbola corresponding to the hyperbolic angle given as input.
Together with hyperbolic cosine , the functions parameterize the right branch of the unit hyperbola given by the equation x² - y² = 1. In plain language, this means that the two functions trace out the shape of the right branch of the unit hyperbola.
Given a hyperbolic angle corresponding to a point on the hyperbola’s curve, SINH returns the vertical component of the point, while COSH returns the horizontal component of the point. For example, given the hyperbolic angle -1, hyperbolic cosine returns the x-component 1.543080635, and hyperbolic sine returns the y-component -1.175201194.
Here is a table that shows some points on the unit hyperbola formed by COSH and SINH.
As the hyperbolic angle increases positively, the vertical component goes to infinity, and as the hyperbolic angle increases negatively, the vertical component goes to negative infinity. This is different from the sine function , which is periodic and repeats values.
Images courtesy of wumbo.net .
Purpose
Return value
Syntax
=TAN(number)
- number - The angle in radians for which you want the tangent.
Using the TAN function
The TAN function returns the tangent of an angle provided in radians . For example, given the π/4 as input, the function returns 1.0 as output.
=TAN(PI()/4) // Returns 1
To supply an angle in degrees, use the radians function.
=TAN(RADIANS(45)) // Returns 1
Explanation
In math, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side of a right triangle.
The definition of tangent is extended to all angles by defining the function in terms of the angle formed by a point on the unit circle.
The plot below visualizes the output of tangent for angles between -π and π. The red lines represent vertical asymptotes where the function diverges to positive and negative infinity.
Even though the function is undefined at its vertical asymptotes, Excel does not return an error for input at or near these vertical asymptotes. For example, see the following table for input around and including the location of the asymptote at the angle π/2.
This is because the tangent function is likely defined in terms of the functions sine and cosine like this:
=SIN(a)/COS(a) // equivalent to TAN(a)
It’s reasonable to expect that the output of cosine for the angle π/2 should be zero, meaning you would get a division by zero error for tangent at the same input. However, Excel’s formula engine calculates this value as a very small value that is close to, but not quite equal to, zero.
=COS(PI()/2) // returns 6.12323E-17 instead of zero
As a result, the tangent of π/2 does not return an error and instead returns a very large value.
=TAN(PI()/2) // returns 1.63312E+16
Images courtesy of wumbo.net .
