Question

Are there any integer values of tanx apart from -1 and +1?

Answer 1

From what I can see, those are the only integer values, but I just want to make sure.

Answer 2

Does 0 count?

Answer 3

Disregarding zero, yeah, I'm looking at angles above zero.

Answer 4

Yeah pretty sure it's just -1 and 1

Answer 5

Alright thanks.

Answer 6

The range of the tangent function is \((-\infty,\infty)\), which contains all integer values.

Answer 7

Another way to think of it: the tangent of an angle is defined as the ratio of the angle's sine to its cosine, i.e. \(\tan x=\dfrac{\sin x}{\cos x}\). One of the component trig functions attains its maximum when the other attains its minimum (in this case, 1 and 0, respectively).

You have \(\tan x=1\) for \(x=\dfrac{\pi}{4}\), for example:

But if you consider some intermediate angle between \(\dfrac{\pi}{4}\) and \(\dfrac{\pi}{2}\), such as \(\dfrac{3\pi}{8}\) for example, you have something larger than 1:

and so the tangent of this angle would be
\[\frac{\text{something larger than 1}}{\text{something smaller than 1}}=\text{something larger than 1}\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{(and larger than the numerator)}\]
As the angle \(x\) approaches \(\dfrac{\pi}{2}\), \(\cos x\to0\) and \(\sin x\to1\). For angles arbitrarily close to \(\dfrac{\pi}{2}\), you have an arbitrarily large value for tangent. Since the sine and cosine functions are continuous, there's no reason to think that any integer values other than \(\pm1,0\) are attained.