# Calculating Inverse Tangent

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.

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

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.

$$\ tanx=a$$

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

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

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

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