Valori notevoli delle funzioni trigonometriche
I valori notevoli delle funzioni trigonometriche sono particolari valori di seno, coseno, tangente e cotangente che si ottengono in corrispondenza di angoli specifici (espressi in gradi o in radianti) come \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), \(90^\circ\) e loro multipli o combinazioni significative.
A cosa servono? Questi valori mi permettono di semplificare e risolvere problemi geometrici senza dover ricorrere a calcoli e approssimazioni numeriche. Questi angoli rappresentano punti importanti nella trigonometria perché producono risultati semplici ed esatti, spesso espressi in forma frazionaria o con radici quadrate.
Ecco una tabella con i principali valori notevoli delle funzioni goniometriche.
| Gradi | Radianti | Seno | Coseno | Tangente | Cotangente |
|---|---|---|---|---|---|
| 0° | \( 0 \) | \( 0 \) | \( 1 \) | \( 0 \) | \( \pm \infty \) |
| 15° | \( \frac{\pi}{12} \) | \( \frac{\sqrt{6} - \sqrt{2}}{4} \) | \( \frac{\sqrt{6} + \sqrt{2}}{4} \) | \( 2 - \sqrt{3} \) | \( 2 + \sqrt{3} \) |
| 18° | \( \frac{\pi}{10} \) | \( \frac{\sqrt{5} - 1}{4} \) | \( \frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{25 - 10\sqrt{5}}}{5} \) | \( \sqrt{5 + 2\sqrt{5}} \) |
| 22° 30' | \( \frac{\pi}{8} \) | \( \frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( \frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( \sqrt{2} - 1 \) | \( \sqrt{2} + 1 \) |
| 30° | \( \frac{\pi}{6} \) | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{3}}{3} \) | \( \sqrt{3} \) |
| 36° | \( \frac{\pi}{5} \) | \( \frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{5} + 1}{4} \) | \( \sqrt{5 - 2\sqrt{5}} \) | \( \frac{\sqrt{25 + 10\sqrt{5}}}{5} \) |
| 45° | \( \frac{\pi}{4} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( 1 \) | \( 1 \) |
| 54° | \( \frac{3\pi}{10} \) | \( \frac{\sqrt{5} + 1}{4} \) | \( \frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{25 + 10\sqrt{5}}}{5} \) | \( \sqrt{5 - 2\sqrt{5}} \) |
| 60° | \( \frac{\pi}{3} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) | \( \frac{\sqrt{3}}{3} \) |
| 67° 30' | \( \frac{3\pi}{8} \) | \( \frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( \frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( \sqrt{2} + 1 \) | \( \sqrt{2} - 1 \) |
| 72° | \( \frac{2\pi}{5} \) | \( \frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{5} - 1}{4} \) | \( \sqrt{5 + 2\sqrt{5}} \) | \( \frac{\sqrt{25 - 10\sqrt{5}}}{5} \) |
| 75° | \( \frac{5\pi}{12} \) | \( \frac{\sqrt{6} + \sqrt{2}}{4} \) | \( \frac{\sqrt{6} - \sqrt{2}}{4} \) | \( 2 + \sqrt{3} \) | \( 2 - \sqrt{3} \) |
| 90° | \( \frac{\pi}{2} \) | \( 1 \) | \( 0 \) | \( \pm \infty \) | \( 0 \) |
| 105° | \( \frac{7\pi}{12} \) | \( \frac{\sqrt{6} + \sqrt{2}}{4} \) | \( \frac{\sqrt{2} - \sqrt{6}}{4} \) | \( -2 - \sqrt{3} \) | \( \sqrt{3} - 2 \) |
| 108° | \( \frac{3\pi}{5} \) | \( \frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( \frac{1 - \sqrt{5}}{4} \) | \( -\sqrt{5 + 2\sqrt{5}} \) | \( -\frac{\sqrt{25 - 10\sqrt{5}}}{5} \) |
| 112° 30' | \( \frac{5\pi}{8} \) | \( \frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( -\frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( -1 - \sqrt{2} \) | \( 1 - \sqrt{2} \) |
| 120° | \( \frac{2\pi}{3} \) | \( \frac{\sqrt{3}}{2} \) | \( -\frac{1}{2} \) | \( -\sqrt{3} \) | \( -\frac{\sqrt{3}}{3} \) |
| 126° | \( \frac{7\pi}{10} \) | \( \frac{\sqrt{5} + 1}{4} \) | \( -\frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( -\frac{\sqrt{25 + 10\sqrt{5}}}{5} \) | \( -\sqrt{5 - 2\sqrt{5}} \) |
| 135° | \( \frac{3\pi}{4} \) | \( \frac{\sqrt{2}}{2} \) | \( -\frac{\sqrt{2}}{2} \) | \( -1 \) | \( -1 \) |
| 144° | \( \frac{4\pi}{5} \) | \( \frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( -\frac{\sqrt{5} + 1}{4} \) | \( -\sqrt{5 - 2\sqrt{5}} \) | \( -\frac{\sqrt{25 + 10\sqrt{5}}}{5} \) |
| 150° | \( \frac{5\pi}{6} \) | \( \frac{1}{2} \) | \( -\frac{\sqrt{3}}{2} \) | \( -\frac{\sqrt{3}}{3} \) | \( -\sqrt{3} \) |
| 157° 30' | \( \frac{7\pi}{8} \) | \( \frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( -\frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( 1 - \sqrt{2} \) | \( -\sqrt{2} - 1 \) |
| 162° | \( \frac{9\pi}{10} \) | \( \frac{\sqrt{5} - 1}{4} \) | \( -\frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( -\frac{\sqrt{25 - 10\sqrt{5}}}{5} \) | \( -\sqrt{5 + 2\sqrt{5}} \) |
| 165° | \( \frac{11\pi}{12} \) | \( \frac{\sqrt{6} - \sqrt{2}}{4} \) | \( -\frac{\sqrt{6} + \sqrt{2}}{4} \) | \( \sqrt{3} - 2 \) | \( -\sqrt{3} - 2 \) |
| 180° | \( \pi \) | \( 0 \) | \( -1 \) | \( 0 \) | \( \pm \infty \) |
| 195° | \( \frac{13\pi}{12} \) | \( \frac{\sqrt{2} - \sqrt{6}}{4} \) | \( -\frac{\sqrt{6} + \sqrt{2}}{4} \) | \( 2 - \sqrt{3} \) | \( 2 + \sqrt{3} \) |
| 198° | \( \frac{11\pi}{10} \) | \( \frac{1 - \sqrt{5}}{4} \) | \( -\frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{25 - 10\sqrt{5}}}{5} \) | \( \sqrt{5 + 2\sqrt{5}} \) |
| 202° 30' | \( \frac{9\pi}{8} \) | \( -\frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( -\frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( \sqrt{2} - 1 \) | \( \sqrt{2} + 1 \) |
| 210° | \( \frac{7\pi}{6} \) | \( -\frac{1}{2} \) | \( -\frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{3}}{3} \) | \( \sqrt{3} \) |
| 216° | \( \frac{6\pi}{5} \) | \( -\frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( -\frac{\sqrt{5} + 1}{4} \) | \( \sqrt{5 - 2\sqrt{5}} \) | \( \frac{\sqrt{25 + 10\sqrt{5}}}{5} \) |
| 225° | \( \frac{5\pi}{4} \) | \( -\frac{\sqrt{2}}{2} \) | \( -\frac{\sqrt{2}}{2} \) | \( 1 \) | \( 1 \) |
| 234° | \( \frac{13\pi}{10} \) | \( -\frac{\sqrt{5} + 1}{4} \) | \( -\frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{25 + 10\sqrt{5}}}{5} \) | \( \sqrt{5 - 2\sqrt{5}} \) |
| 240° | \( \frac{4\pi}{3} \) | \( -\frac{\sqrt{3}}{2} \) | \( -\frac{1}{2} \) | \( \sqrt{3} \) | \( \frac{\sqrt{3}}{3} \) |
| 247° 30' | \( \frac{11\pi}{8} \) | \( -\frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( -\frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( \sqrt{2} + 1 \) | \( \sqrt{2} - 1 \) |
| 252° | \( \frac{7\pi}{5} \) | \( -\frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( \frac{1 - \sqrt{5}}{4} \) | \( \sqrt{5 + 2\sqrt{5}} \) | \( \frac{\sqrt{25 - 10\sqrt{5}}}{5} \) |
| 255° | \( \frac{17\pi}{12} \) | \( -\frac{\sqrt{6} + \sqrt{2}}{4} \) | \( \frac{\sqrt{2} - \sqrt{6}}{4} \) | \( 2 + \sqrt{3} \) | \( 2 - \sqrt{3} \) |
| 270° | \( \frac{3\pi}{2} \) | \( -1 \) | \( 0 \) | \( \pm \infty \) | \( 0 \) |
| 285° | \( \frac{19\pi}{12} \) | \( -\frac{\sqrt{6} + \sqrt{2}}{4} \) | \( \frac{\sqrt{6} - \sqrt{2}}{4} \) | \( -2 - \sqrt{3} \) | \( \sqrt{3} - 2 \) |
| 288° | \( \frac{8\pi}{5} \) | \( -\frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{5} - 1}{4} \) | \( -\sqrt{5 + 2\sqrt{5}} \) | \( -\frac{\sqrt{25 - 10\sqrt{5}}}{5} \) |
| 292° 30' | \( \frac{13\pi}{8} \) | \( -\frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( \frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( -1 - \sqrt{2} \) | \( 1 - \sqrt{2} \) |
| 300° | \( \frac{5\pi}{3} \) | \( -\frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( -\sqrt{3} \) | \( -\frac{\sqrt{3}}{3} \) |
| 306° | \( \frac{17\pi}{10} \) | \( -\frac{\sqrt{5} + 1}{4} \) | \( \frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( -\frac{\sqrt{25 + 10\sqrt{5}}}{5} \) | \( -\sqrt{5 - 2\sqrt{5}} \) |
| 315° | \( \frac{7\pi}{4} \) | \( -\frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( -1 \) | \( -1 \) |
| 324° | \( \frac{9\pi}{5} \) | \( -\frac{\sqrt{10 - 2\sqrt{5}}}{4} \) | \( \frac{\sqrt{5} + 1}{4} \) | \( -\sqrt{5 - 2\sqrt{5}} \) | \( -\frac{\sqrt{25 + 10\sqrt{5}}}{5} \) |
| 330° | \( \frac{11\pi}{6} \) | \( -\frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( -\frac{\sqrt{3}}{3} \) | \( -\sqrt{3} \) |
| 337° 30' | \( \frac{15\pi}{8} \) | \( -\frac{\sqrt{2 - \sqrt{2}}}{2} \) | \( \frac{\sqrt{2 + \sqrt{2}}}{2} \) | \( 1 - \sqrt{2} \) | \( -1 - \sqrt{2} \) |
| 342° | \( \frac{19\pi}{10} \) | \( \frac{1 - \sqrt{5}}{4} \) | \( \frac{\sqrt{10 + 2\sqrt{5}}}{4} \) | \( -\frac{\sqrt{25 - 10\sqrt{5}}}{5} \) | \( -\sqrt{5 + 2\sqrt{5}} \) |
| 345° | \( \frac{23\pi}{12} \) | \( \frac{\sqrt{2} - \sqrt{6}}{4} \) | \( \frac{\sqrt{2} + \sqrt{6}}{4} \) | \( \sqrt{3} - 2 \) | \( -2 - \sqrt{3} \) |
| 360° | \( 2\pi \) | \( 0 \) | \( 1 \) | \( 0 \) | \( \pm \infty \) |
