Le funzioni trigonometriche seno, coseno, tangente e cotangente possono valere quantità differenti a seconda dell’angolo in cui vengono calcolate; in generale non esiste un modo per scrivere queste quantità, se non come numeri con la virgola (e parte decimale potenzialmente infinita non periodica). Questo è in effetti il risultato che si ottiene se si svolgono conti di questo tipo con una calcolatrice scientifica.
Per alcuni speciali valori di $\alpha$, tuttavia, i valori di $\sin(\alpha), \cos(\alpha), \tan(\alpha), \cot(\alpha)$ possono essere scritti con un’espressione più stringata e soprattutto più precisa, che a volte può essere molto più utile del valore fornito dalla calcolatrice. Riportiamo qua sotto una tabella con l’elenco di questi valori notevoli. Dato che le funzioni seno e coseno sono periodiche di periodo $2\pi$ radianti, possiamo limitarci a elencare questi valori per angoli di ampiezza compresa tra $0$ e $2\pi$ radianti, cioè tra $0$ e $360^\circ$.
| $$\color{red}{ \large \alpha}$$ | $$\color{red}{ \large \sin(\alpha)}$$ | $$\color{red}{ \large \cos(\alpha)}$$ | $$\color{red}{ \large \tan(\alpha)}$$ | $$\color{red}{ \large \cot(\alpha)}$$ |
| $$ \color{blue}{ \large 0 \text{ rad}; \ 0^\circ}$$ | $ \color{blue}{ \large 0}$ | $ \color{blue}{ \large 1}$ | $ \color{blue}{ \large 0}$ | $\color{blue}{ \large \text{non definito}}$ |
| $$\frac{\pi}{12} \text{ rad}; \ 15^\circ$$ | $\frac{\sqrt{6} - \sqrt{2}}{4}$ | $\frac{\sqrt{6} + \sqrt{2}}{4}$ | $2 - \sqrt{3}$ | $2 + \sqrt{3}$ |
| $$\frac{\pi}{10} \text{ rad}; \ 18^\circ$$ | $\frac{\sqrt{5} - 1}{4}$ | $\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$ | $\sqrt{1 - \frac{2\sqrt{5}}{5}}$ | $\sqrt{5 +2\sqrt{5}}$ |
| $$\frac{\pi}{8} \text{ rad}; \ 22^\circ 30’$$ | $\frac{\sqrt{2 -\sqrt{2}}}{2}$ | $\frac{\sqrt{2 +\sqrt{2}}}{2}$ | $\sqrt{2} - 1$ | $\sqrt{2} + 1$ |
| $$\frac{\pi}{6} \text{ rad}; \ 30^\circ$$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ | $\sqrt{3}$ |
| $$\frac{\pi}{5} \text{ rad}; \ 36^\circ$$ | $\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $\frac{\sqrt{5}+1}{4}$ | $\sqrt{5-2\sqrt{5}}$ | $\frac{\sqrt{25+10\sqrt{5}}}{5}$ |
| $$\frac{\pi}{4} \text{ rad}; \ 45^\circ$$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ | $1$ |
| $$\frac{3\pi}{10} \text{ rad}; \ 54^\circ$$ | $\frac{\sqrt{5}+1}{4}$ | $\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $\frac{\sqrt{25+10\sqrt{5}}}{5}$ | $\sqrt{5-2\sqrt{5}}$ |
| $$\frac{\pi}{3} \text{ rad}; \ 60^\circ$$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ | $\frac{\sqrt{3}}{3}$ |
| $$\frac{3\pi}{8} \text{ rad}; \ 67^\circ 30'$$ | $\frac{\sqrt{2+\sqrt{2}}}{2}$ | $\frac{\sqrt{2-\sqrt{2}}}{2}$ | $\sqrt{2}+1$ | $\sqrt{2}-1$ |
| $$\frac{2\pi}{5} \text{ rad}; \ 72^\circ$$ | $\frac{\sqrt{10+2 \sqrt{5}}}{4}$ | $\frac{\sqrt{5}-1}{4}$ | $\sqrt{5+2\sqrt{5}}$ | $\frac{\sqrt{25-10\sqrt{5}}}{5}$ |
| $$\frac{5\pi}{12} \text{ rad}; \ 75^\circ$$ | $\frac{\sqrt{6} + \sqrt{2}}{4}$ | $\frac{\sqrt{6} - \sqrt{2}}{4}$ | $2 + \sqrt{3}$ | $2 - \sqrt{3}$ |
| $$ \color{blue}{ \large \frac{\pi}{2} \text{ rad}; \ 90^\circ}$$ | $ \color{blue}{ \large 1}$ | $ \color{blue}{ \large 0}$ | $ \color{blue}{ \large\text{non definito}}$ | $ \color{blue}{ \large 0}$ |
| $$\frac{7\pi}{12} \text{ rad}; \ 105^\circ$$ | $\frac{\sqrt{6} + \sqrt{2}}{4}$ | $\frac{\sqrt{2} - \sqrt{6}}{4}$ | $-2 - \sqrt{3}$ | $\sqrt{3}-2$ |
| $$\frac{3\pi}{5} \text{ rad}; \ 108^\circ$$ | $\frac{\sqrt{10+2 \sqrt{5}}}{4}$ | $\frac{1-\sqrt{5}}{4}$ | $-\sqrt{5+2\sqrt{5}}$ | $-\frac{\sqrt{25-10\sqrt{5}}}{5}$ |
| $$\frac{5\pi}{8} \text{ rad}; \ 112^\circ 30'$$ | $\frac{\sqrt{2+\sqrt{2}}}{2}$ | $-\frac{\sqrt{2-\sqrt{2}}}{2}$ | $-\sqrt{2}-1$ | $1-\sqrt{2}$ |
| $$\frac{2\pi}{3} \text{ rad}; \ 120^\circ$$ | $\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ | $-\sqrt{3}$ | $-\frac{\sqrt{3}}{3}$ |
| $$\frac{7\pi}{10} \text{ rad}; \ 126^\circ$$ | $\frac{\sqrt{5}+1}{4}$ | $-\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $-\frac{\sqrt{25+10\sqrt{5}}}{5}$ | $-\sqrt{5-2\sqrt{5}}$ |
| $$\frac{3\pi}{4} \text{ rad}; \ 135^\circ$$ | $\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-1$ | $-1$ |
| $$\frac{4\pi}{5} \text{ rad}; \ 144^\circ$$ | $\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $-\frac{\sqrt{5}+1}{4}$ | $-\sqrt{5-2\sqrt{5}}$ | $-\frac{\sqrt{25+10\sqrt{5}}}{5}$ |
| $$\frac{5\pi}{6} \text{ rad}; \ 150^\circ$$ | $\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{\sqrt{3}}{3}$ | $-\sqrt{3}$ |
| $$\frac{7\pi}{8} \text{ rad}; \ 157^\circ 30'$$ | $\frac{\sqrt{2 -\sqrt{2}}}{2}$ | $-\frac{\sqrt{2 +\sqrt{2}}}{2}$ | $1-\sqrt{2}$ | $-\sqrt{2} - 1$ |
| $$\frac{9\pi}{10} \text{ rad}; \ 162^\circ$$ | $\frac{\sqrt{5} - 1}{4}$ | $-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$ | $-\sqrt{1 - \frac{2\sqrt{5}}{5}}$ | $-\sqrt{5 +2\sqrt{5}}$ |
| $$\frac{11\pi}{12} \text{ rad}; \ 165^\circ$$ | $\frac{\sqrt{6} - \sqrt{2}}{4}$ | $-\frac{\sqrt{6} + \sqrt{2}}{4}$ | $\sqrt{3}-2$ | $-2 - \sqrt{3}$ |
| $$ \color{blue}{ \large \pi \text{ rad}; \ 180^\circ}$$ | $ \color{blue}{\large 0}$ | $ \color{blue}{ \large -1}$ | $\color{blue}{\large 0}$ | $\color{blue}{\large \text{non definito}}$ |
| $$\frac{13\pi}{12} \text{ rad}; \ 195^\circ$$ | $\frac{\sqrt{2} - \sqrt{6}}{4}$ | $-\frac{\sqrt{6} + \sqrt{2}}{4}$ | $2 - \sqrt{3}$ | $2 + \sqrt{3}$ |
| $$\frac{11\pi}{10} \text{ rad}; \ 198^\circ$$ | $\frac{1-\sqrt{5}}{4}$ | $-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$ | $\sqrt{1 - \frac{2\sqrt{5}}{5}}$ | $\sqrt{5 +2\sqrt{5}}$ |
| $$\frac{9\pi}{8} \text{ rad}; \ 202^\circ 30'$$ | $-\frac{\sqrt{2-\sqrt{2}}}{2}$ | $-\frac{\sqrt{2 +\sqrt{2}}}{2}$ | $\sqrt{2} - 1$ | $\sqrt{2} + 1$ |
| $$\frac{7\pi}{6} \text{ rad}; \ 210^\circ$$ | $-\frac{1}{2}$ | $-\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ | $\sqrt{3}$ |
| $$\frac{6\pi}{5} \text{ rad}; \ 216^\circ$$ | $-\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $-\frac{\sqrt{5}+1}{4}$ | $\sqrt{5-2\sqrt{5}}$ | $\frac{\sqrt{25+10\sqrt{5}}}{5}$ |
| $$\frac{5\pi}{4} \text{ rad}; \ 225^\circ$$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $1$ | $1$ |
| $$\frac{13\pi}{10} \text{ rad}; \ 234^\circ$$ | $-\frac{\sqrt{5}+1}{4}$ | $-\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $\frac{\sqrt{25+10\sqrt{5}}}{5}$ | $\sqrt{5-2\sqrt{5}}$ |
| $$\frac{4\pi}{3} \text{ rad}; \ 240^\circ$$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{2}$ | $\sqrt{3}$ | $\frac{\sqrt{3}}{3}$ |
| $$\frac{11\pi}{8} \text{ rad}; \ 247^\circ 30'$$ | $-\frac{\sqrt{2 +\sqrt{2}}}{2}$ | $-\frac{\sqrt{2-\sqrt{2}}}{2}$ | $\sqrt{2}+1$ | $\sqrt{2}-1$ |
| $$\frac{7\pi}{5} \text{ rad}; \ 252^\circ$$ | $-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$ | $\frac{1-\sqrt{5}}{4}$ | $\sqrt{5+2\sqrt{5}}$ | $\frac{\sqrt{25-10\sqrt{5}}}{5}$ |
| $$\frac{17\pi}{12} \text{ rad}; \ 255^\circ$$ | $-\frac{\sqrt{6} + \sqrt{2}}{4}$ | $\frac{\sqrt{2} - \sqrt{6}}{4}$ | $2 + \sqrt{3}$ | $2 - \sqrt{3}$ |
| $$ \color{blue}{ \large \frac{3\pi}{2} \text{ rad}; \ 270^\circ}$$ | $ \color{blue}{ \large -1}$ | $ \color{blue}{ \large 0}$ | $\color{blue}{ \large \text{non definito}}$ | $ \color{blue}{ \large 0}$ |
| $$\frac{19\pi}{12} \text{ rad}; \ 285^\circ$$ | $-\frac{\sqrt{6} + \sqrt{2}}{4}$ | $\frac{\sqrt{6} - \sqrt{2}}{4}$ | $-2 - \sqrt{3}$ | $\sqrt{3}-2$ |
| $$\frac{8\pi}{5} \text{ rad}; \ 288^\circ$$ | $-\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$ | $\frac{\sqrt{5}-1}{4}$ | $-\sqrt{5+2\sqrt{5}}$ | $-\frac{\sqrt{25-10\sqrt{5}}}{5}$ |
| $$\frac{13\pi}{8} \text{ rad}; \ 292^\circ 30'$$ | $-\frac{\sqrt{2 +\sqrt{2}}}{2}$ | $\frac{\sqrt{2-\sqrt{2}}}{2}$ | $-\sqrt{2}-1$ | $1-\sqrt{2}$ |
| $$\frac{5\pi}{3} \text{ rad}; \ 300^\circ$$ | $-\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $-\sqrt{3}$ | $-\frac{\sqrt{3}}{3}$ |
| $$\frac{17\pi}{10} \text{ rad}; \ 306^\circ$$ | $-\frac{\sqrt{5}+1}{4}$ | $\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $-\frac{\sqrt{25+10\sqrt{5}}}{5}$ | $-\sqrt{5-2\sqrt{5}}$ |
| $$\frac{7\pi}{4} \text{ rad}; \ 315^\circ$$ | $-\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $-1$ | $-1$ |
| $$\frac{9\pi}{5} \text{ rad}; \ 324^\circ$$ | $-\frac{\sqrt{10-2 \sqrt{5}}}{4}$ | $\frac{\sqrt{5}+1}{4}$ | $-\sqrt{5-2\sqrt{5}}$ | $-\frac{\sqrt{25+10\sqrt{5}}}{5}$ |
| $$\frac{11\pi}{6} \text{ rad}; \ 330^\circ$$ | $-\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $-\frac{\sqrt{3}}{3}$ | $-\sqrt{3}$ |
| $$\frac{15\pi}{8} \text{ rad}; \ 337^\circ 30'$$ | $-\frac{\sqrt{2-\sqrt{2}}}{2}$ | $\frac{\sqrt{2 +\sqrt{2}}}{2}$ | $1-\sqrt{2}$ | $-\sqrt{2} - 1$ |
| $$\frac{19\pi}{10} \text{ rad}; \ 342^\circ$$ | $\frac{1-\sqrt{5}}{4}$ | $\sqrt{\frac{5}{8} +\frac{\sqrt{5}}{8}}$ | $-\sqrt{1 - \frac{2\sqrt{5}}{5}}$ | $-\sqrt{5 +2\sqrt{5}}$ |
| $$\frac{23\pi}{12} \text{ rad}; \ 345^\circ$$ | $\frac{\sqrt{2} - \sqrt{6}}{4}$ | $\frac{\sqrt{6} + \sqrt{2}}{4}$ | $\sqrt{3}-2$ | $-2 - \sqrt{3}$ |
| $$ \color{blue}{ \large 2 \pi \text{ rad}; \ 360^\circ}$$ | $ \color{blue}{ \large 0}$ | $ \color{blue}{ \large 1}$ | $ \color{blue}{ \large 0}$ | $ \color{blue}{ \large \text{non definito}}$ |