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Wolfram|Alpha solved a function that does not behave as expected?

Posted 1 year ago
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Dear Wolfram Community,

For my work I am using a mechanical system and have to find the relation between a linear displacement x and a certain angle alpha. To find x as a function of alpha is fairly easy and results in the following function:

Equation 1

Where H = 80mm, S = 205mm, a = 68mm, and z = 204.5mm are all system parameters. x is the linear displacement and alpha is the input angle.

I also need the inverse of this function (alpha = f(x)), but had troubles deriving this function by hand. Hence, I tried to have Wolfram Alpha find the inverse for me. This results in quite an extensive formula, which I'm ok with:

Equation 2

However, if I supply this function with a certain linear displacement, I do not get the expected (right?) answer. E.g. with an alpha of 40 degrees or 0.698 radians I get a linear displacement of 51mm (using the first equation), which seems to be correct. However, using 51mm as input for alpha = f(x) gives -3.018.

What am I doing wrong?

Thank you for your help. Matthijs

2 Replies

Thank you for your reply, Daniel. I thought about that as well and tried adding and subtracting multiples of Pi. Also, I solved the first equation for several points within a range of alpha between 45 and -45 degrees (which is the actual range in the mechanical system). Using the inverse to find the angle again results in a range that is much smaller than 90 degrees total, which also seems strange to me.

Might not be doing anything wrong but note that you have an inverse function that comes from a many-to-one function, hence loses other candidate inverse values. In particular multiples of 2*Pi might be added, and a negative of any solution is also a solution for the arccosine formula (whether it solves the original or is a "parasite" solution is another matter; this goes with the territory when inverting many-to-one functions).

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