Angoli notevoli: tabella delle funzioni goniometriche per angoli noti - WeSchool


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.





















Videolezione "Studio di funzione: il procedimento"

























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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}}$



