קבועים טריגונומטריים מדויקים

הקבועים הבסיסיים על מעגל היחידה הם כפולות של 30° ו-40°

ייצוג קבועים טריגונומטריים מדויקים על ידי ביטויים מתמטיים שימושי להמרת ביטויים טריגונומטריים לביטויים רדיקלים (ביטויים מהצורה $\sqrt[n]{a}$).

ערכי סינוס, קוסינוס וטנגנס של כפולות 3° ניתנים לפיתוח על ידי זהויות טריגונומטריות מתוך הערכים הידועים של 0°, 30° ו-45°.

הרשימה כאן אינה שלמה, כיוון שניתן להמשיך ולחלק כל זווית נתונה לזוויות קטנות ממנה.

רשימת קבועים

ערכים נוספים מחוץ לטווח הרשימה ניתן לחשב בקלות על ידי זהויות טריגונומטריות.

0°:

$\begin{array}{rl}\sin (0^{\circ })& =0\\ \cos (0^{\circ })& =1\\ \tan (0^{\circ })& =0\\ \cot (0^{\circ })& =\text{undefined}\end{array}$

1.5°: מצולע בעל 120 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{120}\right)& =\sin (1.5^{\circ })=\frac{\sqrt{2+\sqrt{2}}(\sqrt{15}+\sqrt{3}-\sqrt{10-2\sqrt{5}})-\sqrt{2-\sqrt{2}}(\sqrt{30-6\sqrt{5}}+\sqrt{5}+1)}{16}\\ \cos \left(\frac{\pi }{120}\right)& =\cos (1.5^{\circ })=\frac{\sqrt{2+\sqrt{2}}(\sqrt{30-6\sqrt{5}}+\sqrt{5}+1)+\sqrt{2-\sqrt{2}}(\sqrt{15}+\sqrt{3}-\sqrt{10-2\sqrt{5}})}{16}\end{array}$

1.875°: מצולע בעל 96 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{96}\right)& =\sin (1.875^{\circ })=\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}}{2}\\ \cos \left(\frac{\pi }{96}\right)& =\cos (1.875^{\circ })=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}}{2}\end{array}$

2.25°: מצולע בעל 80 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{80}\right)& =\sin (2.25^{\circ })=\frac{(\sqrt{2+\sqrt{2}}-1)\sqrt{2(5-\sqrt{5})(2+\sqrt{2+\sqrt{2}})}-(\sqrt{5}+1)(\sqrt{2+\sqrt{2}}+1)\sqrt{2-\sqrt{2+\sqrt{2}}}}{8}\\ \cos \left(\frac{\pi }{80}\right)& =\cos (2.25^{\circ })=\frac{\sqrt{8+2\sqrt{8+2\sqrt{8+2\sqrt{10+2\sqrt{5}}}}}}{4}\end{array}$

2.8125°: מצולע בעל 64 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{64}\right)& =\sin (2.8125^{\circ })=\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}{2}\\ \cos \left(\frac{\pi }{64}\right)& =\cos (2.8125^{\circ })=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}{2}\end{array}$

3°: מצולע בעל 60 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{60}\right)& =\sin (3^{\circ })=\frac{2(1-\sqrt{3})\sqrt{5+\sqrt{5}}+\sqrt{2}(\sqrt{5}-1)(\sqrt{3}+1)}{16}\\ \cos \left(\frac{\pi }{60}\right)& =\cos (3^{\circ })=\frac{2(\sqrt{3}+1)\sqrt{5+\sqrt{5}}+\sqrt{2}(\sqrt{5}-1)(\sqrt{3}-1)}{16}\\ \tan \left(\frac{\pi }{60}\right)& =\tan (3^{\circ })=\frac{((2-\sqrt{3})(3+\sqrt{5})-2)(2-\sqrt{10-2\sqrt{5}})}{4}\\ \cot \left(\frac{\pi }{60}\right)& =\cot (3^{\circ })=\frac{((2+\sqrt{3})(3+\sqrt{5})-2)(2+\sqrt{10-2\sqrt{5}})}{4}\end{array}$

3.75°: מצולע בעל 48 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{48}\right)& =\sin (3.75^{\circ })=\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}}{2}\\ \cos \left(\frac{\pi }{48}\right)& =\cos (3.75^{\circ })=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}}{2}\end{array}$

4.5°: מצולע בעל 40 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{40}\right)& =\sin (4.5^{\circ })=\frac{\sqrt{(4-2\sqrt{2})(5+\sqrt{5})}-(\sqrt{5}-1)\sqrt{2+\sqrt{2}}}{8}\\ \cos \left(\frac{\pi }{40}\right)& =\cos (4.5^{\circ })=\frac{\sqrt{(4+2\sqrt{2})(5+\sqrt{5})}+(\sqrt{5}-1)\sqrt{2-\sqrt{2}}}{8}\end{array}$

5.625°: מצולע בעל 32 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{32}\right)& =\sin (5.625^{\circ })=\frac{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}\\ \cos \left(\frac{\pi }{32}\right)& =\cos (5.625^{\circ })=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}\end{array}$

6°: מצולע בעל 30 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{30}\right)& =\sin (6^{\circ })=\frac{\sqrt{30-6\sqrt{5}}-\sqrt{5}-1}{8}\\ \cos \left(\frac{\pi }{30}\right)& =\cos (6^{\circ })=\frac{\sqrt{10-2\sqrt{5}}+\sqrt{3}+\sqrt{15}}{8}\\ \tan \left(\frac{\pi }{30}\right)& =\tan (6^{\circ })=\frac{\sqrt{10-2\sqrt{5}}+\sqrt{3}-\sqrt{15}}{2}\\ \cot \left(\frac{\pi }{30}\right)& =\cot (6^{\circ })=\frac{3\sqrt{3}+\sqrt{15}+\sqrt{50+22\sqrt{5}}}{2}\end{array}$

7.5°: מצולע בעל 24 צלעות

$\begin{array}{rl}\sin \left(\frac{\pi }{24}\right)& =\sin (7.5^{\circ })=\frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}{2}\\ \cos \left(\frac{\pi }{24}\right)& =\cos (7.5^{\circ })=\frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}\\ \tan \left(\frac{\pi }{24}\right)& =\tan (7.5^{\circ })=(\sqrt{2}-1)(\sqrt{3}-\sqrt{2})\\ \cot \left(\frac{\pi }{24}\right)& =\cot (7.5^{\circ })=(\sqrt{2}+1)(\sqrt{3}+\sqrt{2})\end{array}$

9°: מצולע בעל 20 צלעות

$\sin \left(\frac{\pi }{20}\right)=\sin (9^{\circ })=\frac{\sqrt{10}+\sqrt{2}-2\sqrt{5-\sqrt{5}}}{8}$

$\cos \left(\frac{\pi }{20}\right)=\cos (9^{\circ })=\frac{\sqrt{10}+\sqrt{2}+2\sqrt{5-\sqrt{5}}}{8}$

$\tan \left(\frac{\pi }{20}\right)=\tan (9^{\circ })=\sqrt{5}+1-\sqrt{5+2\sqrt{5}}$

$\cot \left(\frac{\pi }{20}\right)=\cot (9^{\circ })=\sqrt{5}+1+\sqrt{5+2\sqrt{5}}$

11.25°: מצולע בעל 16 צלעות

$\sin \left(\frac{\pi }{16}\right)=\sin (11.25^{\circ })=\frac{\sqrt{2-\sqrt{2+\sqrt{2}}}}{2}$

$\cos \left(\frac{\pi }{16}\right)=\cos (11.25^{\circ })=\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}$

$\tan \left(\frac{\pi }{16}\right)=\tan (11.25^{\circ })=\sqrt{4+2\sqrt{2}}-\sqrt{2}-1$

$\cot \left(\frac{\pi }{16}\right)=\cot (11.25^{\circ })=\sqrt{4+2\sqrt{2}}+\sqrt{2}+1$

12°: מצולע בעל 15 צלעות

$\sin \left(\frac{\pi }{15}\right)=\sin (12^{\circ })=\frac{\sqrt{10+2\sqrt{5}}+\sqrt{3}-\sqrt{15}}{8}$

$\cos \left(\frac{\pi }{15}\right)=\cos (12^{\circ })=\frac{\sqrt{30+6\sqrt{5}}+\sqrt{5}-1}{8}$

$\tan \left(\frac{\pi }{15}\right)=\tan (12^{\circ })=\frac{3\sqrt{3}-\sqrt{15}-\sqrt{50-22\sqrt{5}}}{2}$

$\cot \left(\frac{\pi }{15}\right)=\cot (12^{\circ })=\frac{\sqrt{15}+\sqrt{3}+\sqrt{10+2\sqrt{5}}}{2}$

15°: מתורסר, מצולע בעל 12 צלעות

$\sin \left(\frac{\pi }{12}\right)=\sin (15^{\circ })=\frac{\sqrt{6}-\sqrt{2}}{4}$

$\cos \left(\frac{\pi }{12}\right)=\cos (15^{\circ })=\frac{\sqrt{6}+\sqrt{2}}{4}$

$\tan \left(\frac{\pi }{12}\right)=\tan (15^{\circ })=2-\sqrt{3}$

$\cot \left(\frac{\pi }{12}\right)=\cot (15^{\circ })=2+\sqrt{3}$

18°: מעושר, מצולע בעל 10 צלעות

$\sin \left(\frac{\pi }{10}\right)=\sin (18^{\circ })=\frac{\sqrt{5}-1}{4}$

$\cos \left(\frac{\pi }{10}\right)=\cos (18^{\circ })=\frac{\sqrt{10+2\sqrt{5}}}{4}$

$\tan \left(\frac{\pi }{10}\right)=\tan (18^{\circ })=\frac{\sqrt{25-10\sqrt{5}}}{5}$

$\cot \left(\frac{\pi }{10}\right)=\cot (18^{\circ })=\sqrt{5+2\sqrt{5}}$

סכום זויות: 21° = 9° + 12°

$\sin \left(\frac{7\pi }{60}\right)=\sin (21^{\circ })=\frac{2(\sqrt{3}+1)\sqrt{5-\sqrt{5}}-\sqrt{2}(\sqrt{3}-1)(\sqrt{5}+1)}{16}$

$\cos \left(\frac{7\pi }{60}\right)=\cos (21^{\circ })=\frac{2(\sqrt{3}-1)\sqrt{5-\sqrt{5}}+\sqrt{2}(\sqrt{3}+1)(\sqrt{5}+1)}{16}$

$\tan \left(\frac{7\pi }{60}\right)=\tan (21^{\circ })=\frac{(2-(2+\sqrt{3})(3-\sqrt{5}))(2-\sqrt{10+2\sqrt{5}})}{4}$

$\cot \left(\frac{7\pi }{60}\right)=\cot (21^{\circ })=\frac{(2-(2-\sqrt{3})(3-\sqrt{5}))(2+\sqrt{10+2\sqrt{5}})}{4}$

22.5°: מתומן

$\sin \left(\frac{\pi }{8}\right)=\sin (22.5^{\circ })=\frac{\sqrt{2-\sqrt{2}}}{2}$

$\cos \left(\frac{\pi }{8}\right)=\cos (22.5^{\circ })=\frac{\sqrt{2+\sqrt{2}}}{2}$

$\tan \left(\frac{\pi }{8}\right)=\tan (22.5^{\circ })=\sqrt{2}-1$

$\cot \left(\frac{\pi }{8}\right)=\cot (22.5^{\circ })=\sqrt{2}+1$

סכום זויות: 24° = 12° + 12°

$\sin \left(\frac{2\pi }{15}\right)=\sin (24^{\circ })=\frac{\sqrt{15}+\sqrt{3}-\sqrt{10-2\sqrt{5}}}{8}$

$\cos \left(\frac{2\pi }{15}\right)=\cos (24^{\circ })=\frac{\sqrt{30-6\sqrt{5}}+\sqrt{5}+1}{8}$

$\tan \left(\frac{2\pi }{15}\right)=\tan (24^{\circ })=\frac{\sqrt{50+22\sqrt{5}}-3\sqrt{3}-\sqrt{15}}{2}$

$\cot \left(\frac{2\pi }{15}\right)=\cot (24^{\circ })=\frac{\sqrt{15}-\sqrt{3}+\sqrt{10-2\sqrt{5}}}{2}$

סכום זויות: 27° = 12° + 15°

$\sin \left(\frac{3\pi }{20}\right)=\sin (27^{\circ })=\frac{2\sqrt{5+\sqrt{5}}-\sqrt{2}(\sqrt{5}-1)}{8}$

$\cos \left(\frac{3\pi }{20}\right)=\cos (27^{\circ })=\frac{2\sqrt{5+\sqrt{5}}+\sqrt{2}(\sqrt{5}-1)}{8}$

$\tan \left(\frac{3\pi }{20}\right)=\tan (27^{\circ })=\sqrt{5}-1-\sqrt{5-2\sqrt{5}}$

$\cot \left(\frac{3\pi }{20}\right)=\cot (27^{\circ })=\sqrt{5}-1+\sqrt{5-2\sqrt{5}}$

30°: משושה

$\sin \left(\frac{\pi }{6}\right)=\sin (30^{\circ })=\frac{1}{2}$

$\cos \left(\frac{\pi }{6}\right)=\cos (30^{\circ })=\frac{\sqrt{3}}{2}$

$\tan \left(\frac{\pi }{6}\right)=\tan (30^{\circ })=\frac{\sqrt{3}}{3}=\frac{\sqrt{3}}{3}$

$\cot \left(\frac{\pi }{6}\right)=\cot (30^{\circ })=\sqrt{3}$

סכום זויות: 33° = 15° + 18°

$\sin \left(\frac{11\pi }{60}\right)=\sin (33^{\circ })=\frac{2(\sqrt{3}-1)\sqrt{5+\sqrt{5}}+\sqrt{2}(\sqrt{3}+1)(\sqrt{5}-1)}{16}$

$\cos \left(\frac{11\pi }{60}\right)=\cos (33^{\circ })=\frac{2(\sqrt{3}+1)\sqrt{5+\sqrt{5}}-\sqrt{2}(\sqrt{3}-1)(\sqrt{5}-1)}{16}$

$\tan \left(\frac{11\pi }{60}\right)=\tan (33^{\circ })=\frac{(2-(2-\sqrt{3})(3+\sqrt{5}))(2+\sqrt{10-2\sqrt{5}})}{4}$

$\cot \left(\frac{11\pi }{60}\right)=\cot (33^{\circ })=\frac{(2-(2+\sqrt{3})(3+\sqrt{5}))(2-\sqrt{10-2\sqrt{5}})}{4}$

36°: מחומש

$\sin \left(\frac{\pi }{5}\right)=\sin (36^{\circ })=\frac{\sqrt{10-2\sqrt{5}}}{4}$

$\cos \left(\frac{\pi }{5}\right)=\cos (36^{\circ })=\frac{1+\sqrt{5}}{4}$

$\tan \left(\frac{\pi }{5}\right)=\tan (36^{\circ })=\sqrt{5-2\sqrt{5}}$

$\cot \left(\frac{\pi }{5}\right)=\cot (36^{\circ })=\frac{\sqrt{25+10\sqrt{5}}}{5}$

סכום זויות: 39° = 18° + 21°

$\sin \left(\frac{13\pi }{60}\right)=\sin (39^{\circ })=\frac{\sqrt{2}(\sqrt{3}+1)(\sqrt{5}+1)-2(\sqrt{3}-1)\sqrt{5-\sqrt{5}}}{16}$

$\cos \left(\frac{13\pi }{60}\right)=\cos (39^{\circ })=\frac{2(\sqrt{3}+1)\sqrt{5-\sqrt{5}}+\sqrt{2}(\sqrt{3}-1)(\sqrt{5}+1)}{16}$

$\tan \left(\frac{13\pi }{60}\right)=\tan (39^{\circ })=\frac{((2-\sqrt{3})(3-\sqrt{5})-2)(2-\sqrt{10+2\sqrt{5}})}{4}$

$\cot \left(\frac{13\pi }{60}\right)=\cot (39^{\circ })=\frac{((2+\sqrt{3})(3-\sqrt{5})-2)(2+\sqrt{10+2\sqrt{5}})}{4}$

סכום זויות: 42° = 21° + 21°

$\sin \left(\frac{7\pi }{30}\right)=\sin (42^{\circ })=\frac{\sqrt{30+6\sqrt{5}}-\sqrt{5}+1}{8}$

$\cos \left(\frac{7\pi }{30}\right)=\cos (42^{\circ })=\frac{\sqrt{15}-\sqrt{3}+\sqrt{10+2\sqrt{5}}}{8}$

$\tan \left(\frac{7\pi }{30}\right)=\tan (42^{\circ })=\frac{\sqrt{15}+\sqrt{3}-\sqrt{10+2\sqrt{5}}}{2}$

$\cot \left(\frac{7\pi }{30}\right)=\cot (42^{\circ })=\frac{\sqrt{50-22\sqrt{5}}+3\sqrt{3}-\sqrt{15}}{2}$

45°: ריבוע

$\sin \left(\frac{\pi }{4}\right)=\sin (45^{\circ })=\frac{\sqrt{2}}{2}$

$\cos \left(\frac{\pi }{4}\right)=\cos (45^{\circ })=\frac{\sqrt{2}}{2}$

$\tan \left(\frac{\pi }{4}\right)=\tan (45^{\circ })=1$

$\cot \left(\frac{\pi }{4}\right)=\cot (45^{\circ })=1$

סכום זויות: 54° = 27° + 27°

$\sin \left(\frac{3\pi }{10}\right)=\sin (54^{\circ })=\frac{1+\sqrt{5}}{4}$

$\cos \left(\frac{3\pi }{10}\right)=\cos (54^{\circ })=\frac{\sqrt{10-2\sqrt{5}}}{4}$

$\tan \left(\frac{3\pi }{10}\right)=\tan (54^{\circ })=\frac{\sqrt{25+10\sqrt{5}}}{5}$

$\cot \left(\frac{3\pi }{10}\right)=\cot (54^{\circ })=\sqrt{5-2\sqrt{5}}$

60°: משולש שוה-צלעות

$\sin \left(\frac{\pi }{3}\right)=\sin (60^{\circ })=\frac{\sqrt{3}}{2}$

$\cos \left(\frac{\pi }{3}\right)=\cos (60^{\circ })=\frac{1}{2}$

$\tan \left(\frac{\pi }{3}\right)=\tan (60^{\circ })=\sqrt{3}$

$\cot \left(\frac{\pi }{3}\right)=\cot (60^{\circ })=\frac{\sqrt{3}}{3}$

סכום זויות: 67.5° = 7.5° + 60°

$\sin \left(\frac{3\pi }{8}\right)=\sin (67.5^{\circ })=\frac{\sqrt{2+\sqrt{2}}}{2}$

$\cos \left(\frac{3\pi }{8}\right)=\cos (67.5^{\circ })=\frac{\sqrt{2-\sqrt{2}}}{2}$

$\tan \left(\frac{3\pi }{8}\right)=\tan (67.5^{\circ })=\sqrt{2}+1$

$\cot \left(\frac{3\pi }{8}\right)=\cot (67.5^{\circ })=\sqrt{2}-1$

סכום זויות: 72° = 36° + 36°

$\sin \left(\frac{2\pi }{5}\right)=\sin (72^{\circ })=\frac{\sqrt{10+2\sqrt{5}}}{4}$

$\cos \left(\frac{2\pi }{5}\right)=\cos (72^{\circ })=\frac{\sqrt{5}-1}{4}$

$\tan \left(\frac{2\pi }{5}\right)=\tan (72^{\circ })=\sqrt{5+2\sqrt{5}}$

$\cot \left(\frac{2\pi }{5}\right)=\cot (72^{\circ })=\frac{\sqrt{25-10\sqrt{5}}}{5}$

סכום זויות: 75° = 30° + 45°

$\sin \left(\frac{5\pi }{12}\right)=\sin (75^{\circ })=\frac{\sqrt{6}+\sqrt{2}}{4}$

$\cos \left(\frac{5\pi }{12}\right)=\cos (75^{\circ })=\frac{\sqrt{6}-\sqrt{2}}{4}$

$\tan \left(\frac{5\pi }{12}\right)=\tan (75^{\circ })=2+\sqrt{3}$

$\cot \left(\frac{5\pi }{12}\right)=\cot (75^{\circ })=2-\sqrt{3}$

90°: זוית ישרה

$\sin \left(\frac{\pi }{2}\right)=\sin (90^{\circ })=1$

$\cos \left(\frac{\pi }{2}\right)=\cos (90^{\circ })=0$

$\tan \left(\frac{\pi }{2}\right)=\tan (90^{\circ })=\text{ undefined}$

$\cot \left(\frac{\pi }{2}\right)=\cot (90^{\circ })=0$

קישורים חיצוניים

• Constructible Regular Polygons
• Naming polygons