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I recently started thinking about what inverse tangent is. It is obvious that the definition of tangent is $\frac{\sin x}{\cos x}$, however, what is inverse tangent?

I first thought $\tan^{-1} x = \frac{\sin^{-1} x}{\cos^{-1} x}$, but it didn't seem right when I graph it out.

One interesting I find is in most of the programming language I know, there are two atan functions. Does this mean there are two definition of $\tan^{-1}$?

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up vote 3 down vote accepted

I will not comment on a relationship between arctan and arcsine and/or arccosine., other than I do not know of anything simple or commonly known.

In common programming languages, there are two arctan functions. $\phi=\text{atan}(y)$ is an angle whose tangent is $y$, such that $x \in (-\pi/2,\pi/2)$. This ignores anything in the second and third quadrant. To fix this, $\phi=\text{atan2}(x,y)$ (the arguments are sometimes reversed) is the angle whose cosine is $x$ and whose sine is $y$. This allows the second and third quadrant to be determined at the cost of having to input the sines and cosine instead of simply the tangent.

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