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When calculating the inverse tangent of a degree, the calculator will always give an angle between 90 degrees and –90 degrees. But I want to find the positive value of the negative angle.

Do I add 180 or 360 to the angle?

How do I know if I should add 180 or 360 to the negative angle.

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  • $\begingroup$ The period of the tangent is $180°$. $\endgroup$
    – user65203
    Oct 14, 2020 at 20:30
  • $\begingroup$ I dont understand. Could you explain more $\endgroup$ Oct 14, 2020 at 20:33

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I'm not sure I understand what you mean by "positive value of the negative angle"...

A periodic function $f(x)$ has period $p$ if for all integers $n$, $f(x+np)=f(x)$. As Yves pointed out, $\tan x$ has a period of $180^\circ$, meaning that for all real $x$,

$$\tan x=\tan(x\pm180^\circ)=\tan(x\pm360^\circ)=\cdots$$

In particular, if $\tan x=y$ for some given $y$, then $x=\tan^{-1}y+180^\circ n$. By definition of the inverse tangent, $\tan^{-1}y$ is some angle between $-90^\circ$ and $90^\circ$, but it's not the only one that satisfies $\tan x=y$. To get other solutions, you have to add or subtract a multiple of $180^\circ$.

As an example, consider the equation

$$\tan x=1$$

A calculator will tell you that $x=\tan^{-1}1=45^\circ$, but it's also true that $\tan(225^\circ)=\tan(45^\circ+180^\circ)=1$, so $x=225^\circ$ is also a valid solution. Whichever multiple of $180^\circ$ you need to add depends on which domain you expect to find $x$. If $x\in(-90^\circ,90^\circ)$, then $x=45^\circ$; if $x\in(90^\circ,270^\circ)$, then $x=225^\circ$; and so on.

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$\ tanx=a$

$x=\ arctan(a) + k{\pi}$, $k\in Z$

Period of tangent function is ${\pi}$, it could be added $k{\pi}$, $k\in Z$

Since $f(x)=\tan(x)$ is bijective when $x\in(\dfrac{-{\pi}}{2},\dfrac{{\pi}}{2})$, then, $f(x)=\ arctanx$ is defined as

$f:(\dfrac{-{\pi}}{2},\dfrac{{\pi}}{2}) \rightarrow R$,

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  • $\begingroup$ How do I know what the value of k is? $\endgroup$ Oct 14, 2020 at 20:38
  • $\begingroup$ $k$ is any integer, $k=....,-2,-1,0,1,2,...$ $\endgroup$
    – Lion Heart
    Oct 14, 2020 at 20:49

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