# How to calculate $\sin 1$, $\cos 1$, $\tan 1$?

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... :(

• There are no closed form analytic solutions for non-zero integer arguments for the trig functions, though you might want to use a series expansion. Only rational multiples of $\pi$ as arguments give algebraic results. Commented Oct 22, 2016 at 19:00
• It is not totally clear what you are asking for. Tan 1 is an irrational number and you cannot write all decimal places. The shortest form to refer to it is just to write $\tan 1$. It's similar to $\sqrt{2}$ or $\pi$; you mentioned $\pi$ in your text and somehow you don't feel a need to rewrite or rename it. It's the same with $\tan 1$. Commented Oct 22, 2016 at 19:09
• @PeterFranek:A clsoe approximation will be sufficient..I don't knwo whether they will accept the solution given in book in exams or not :((in many cases they don't ) Commented Oct 22, 2016 at 19:20
• I guess they will accept the answer $[\tan 1, \infty)\cup (-\infty, \tan 2]$. Commented Oct 22, 2016 at 19:22
• Well lets hope so @PeterFranek Commented Oct 22, 2016 at 19:32