In my textbook a problem is given:
For $x\in [1,\,2]$ find the range of $f(x) = \tan{x}$
I knew that lowest and highest bound is $-\infty$ and $\infty$ since $\frac{\pi}{2}$ lies in the interval $[1,\,2]$ but the interval exclude some value between $[0,\,\tan{1})$ $\rightarrow$ positive and $[0,\,\tan{2})$ $\rightarrow$ negative.
But I can't evauluate $\tan 1$ and $\tan 2$. I've tried many trigonometric substitutions, identities, etc..
At last when I resolved to the answers page the answer is ($-\infty,\,\tan2] \cup (-\infty,\,\tan2]$
And this I coerced me into slapping myself. I've wasted an hour, not expecting this kind of solution. So, can anybody can solve $\tan 1$ and $\tan 2$ analytically (not by using graphs, etc... cause they won't be provided in an exam no matter how many time books solutions reference them freely.) :(
You can't use any mathematical gizmos like Taylor series, etc... :(
^I've already googled but couldn't find any solutions.
Using Euler's formula we can get \begin{align*}\sin 1=\frac{e^i-e^{-i}}{2i} && \cos 1=\frac{e^i+e^{-i}}{2} && \tan 1=\frac{i(e^{-i}-e^i)}{e^i+e^{-i}}\end{align*}
There are no simpler closed forms for these.