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The question is the following:

For what numbers $y$ is it possible to solve the equation $y = \tan x$.

I know that $\tan(90)$ and $\tan(270)$ will be undefined because there is a vertical slope and $\tan(0)$ and $\tan(360)$ will be $0$ because there is a horizontal "slope".

I tried to calculate $\tan{89.999999}$ in efforts to find the closest tangent (or slope) value before it is undefined and I got a long number $572957795.1$, but when I simply do $\tan89$ I get $57.28996163$. I am not sure why this is happening. With these numbers how do I actually find the range of numbers that $y$ can be in order to be able to solve $y = \tan x$. Please note that I have worked in degree mode, but I see that in radian mode I get much smaller values that $y$ can be. It might make more sense to work with radians because the tangent graphs can be seen in its true form through radian mode. Should I be working in degree or radian mode?

Any help will be appreciated.

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    $\begingroup$ The tangent function hits all $y$ values, so the answer to your literal question is all $y$ values. However, you cannot put just any $x$ into the $\tan(x)$ function. It blows up at odd-integer multiplies of $\pi/2$. Radians are a bit more natural, but your calc should handle either. By the way, if you want to solve for $x$, you get $x=\arctan(y)$, and the arctan function is one of the nicest-behaved functions in existence. $\endgroup$ Feb 25, 2018 at 0:29
  • $\begingroup$ Thank you so much for your help! $\endgroup$
    – geo_freak
    Feb 25, 2018 at 0:32
  • $\begingroup$ math.stackexchange.com/questions/189621/… $\endgroup$ Feb 25, 2018 at 5:26

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Another version of Adrian's answer:

For $y=\tan (x)$ as $x \rightarrow \pi/2$, $y$ can take on infinitely many values, but any exact odd multiple of $\pi/2$ (i.e. $\frac {(2n+1) \pi} {2}$) is not defined as $$\frac {\sin (\pi/2)} {\cos (\pi/2)} = \frac {1}{0}\text { (division by zero)}.$$

Conversely, $\arctan y = x$ will produce as many angles as desired, but as $y \rightarrow \infty$, $x \rightarrow \pi/2$ if $y$ is positive and $x \rightarrow 3\pi/2$ if $y$ is negative.

NOTE: The same is true in degrees; substitute $90^\circ$ for $\pi/2$ and $270^\circ$ for $3\pi/2$.

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