atan vs tan^-1

[Sergey]

Hello.

I stumbled randomly on the following:

It seems that Mathcad defines atan and tan^-1 differently. Look at the attached picture. 

I thought that atan is the same as the inverse tan but it seems that it is not the case. Please, clarify this for me. Shouldn't atan and inverse tangens be the same definition?

[Werner_E · Accepted solution]

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@Sergey wrote:

It seems that Mathcad defines atan and tan^-1 differently. Look at the attached picture. 

I thought that atan is the same as the inverse tan

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Yes, "atan" is "arctan", the (limited) inverse function of "tan" and tan^-1 is 1/tan, the reciprocal of "tan".

This is exactly as it should be (apart from "atan" should be named "arctan" to adhere to the standards).

Thankfully, Prime adheres to the existing standard!!

If a function like tan has a named inverse (arctan), then you must also use this name and must NOT use the notation tan^-1 for it.
However, you can then also use abbreviations such as tan^3 x instead of (tan(x))^3 and, logically, tan^-2 x = 1/tan^2 x and also tan^-1 x must be 1/tan x.

It is unfortunate that on most common calculators sin^-1 is written on the key that stands for the arcsin function.
It is not surprising that Mathematica and WolframAlpha do not adhere to the standard in this respect either - unfortunately, they do not do so in many respects. Wolfram is arrogant enough to know everything better and would like to establish its own 'standard'.

It is very unfortunate that even in serious publications you sometimes see the spelling sin^-1, although arcsin is actually meant.

Writing asin, acos and atan for arcsin, arccos and arctan, which is widespread on the other side of the pond (from my point of view) and used by Prime , is basically not standardized either.

The notation sin^-1 was introduced by John Herschel in 1913, but it often was advised against using it to avoid confusion.
And since 2009 the ISO 80000-2 standard solely specified the "arc" prefix for these functions. So sin^-1 never should be used for arcsin but I have the impression that this wrong notation nowadays is used even more often than in the last decades 😞

One additional remark:

I don't suppose anyone would think that (e^x)^-1 or exp^-1(x) stands for the inverse function of the exponential function - after all, this has its own name (ln, natural logarithm). Of course, this is understood as e^-x or 1/exp(x).
So why should arctan vs. tan^-1 be any different?

[Sergey]

Thank you for the clarification.

So, what Mathcad is basically doing to calculate tan(t)^-1 is that it takes the value of tan(t) and then calculates 1/tan(t). 

Så arctan (or atan) is the inverse as it should be and is the same as tan^-1(t).

[Werner_E]

> Så arctan (or atan) is the inverse as it should be and is the same as tan^-1(t).

Not if you adhere to the standards! But atan(t) may be what you think that tan^-1(t) should be.

The reciprocal of tan(t) could be written as tan^-1(t) or tan(t)^-1 or (tan(t))^-1.

The second one (tan(t)^-1) is perfectly 'legal' but its often advised to avoid it because it may be confused with tan(t^-1)).
The first way to write it is not supported by Prime.

For the inverse function of tan(x) only arctan(x) should be used, Prime uses the name atan(x) instead.

[LucMeekes]

And if you  -  desperately  -  want Prime to recognise tan⁻¹ as arctan, you can define it:

then with:

you can show the inverse function:

But note that:

(where 0.667 is the reciprocal of 1.5)

Success!
Luc