קבועים טריגונומטריים מדויקים

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

יש להשלים ערך זה: בערך זה חסר תוכן מהותי. ייתכן שתמצאו פירוט בדף השיחה.

הקבועים הבסיסיים על מעגל היחידה הם כפולות של 30 ו45 מעלות.

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°

${\displaystyle 0}$	${\displaystyle \sin 0=}$
${\displaystyle 1}$	${\displaystyle \cos 0=}$
${\displaystyle 0}$	${\displaystyle \tan 0=}$
לא מוגדר	${\displaystyle \cot 0=}$

מצולע בעל 60 צלעות - 3°

${\displaystyle {\frac {(2-2{\sqrt {3}}){\sqrt {5+{\sqrt {5}}}}+({\sqrt {6}}+{\sqrt {2}})({\sqrt {5}}-1)}{16}}\,}$	${\displaystyle \sin {\frac {\pi }{60}}=\sin 3^{\circ }=}$
${\displaystyle {\frac {(2+2{\sqrt {3}}){\sqrt {5+{\sqrt {5}}}}+({\sqrt {6}}-{\sqrt {2}})({\sqrt {5}}-1)}{16}}\,}$	${\displaystyle \cos {\frac {\pi }{60}}=\cos 3^{\circ }=}$
${\displaystyle {\frac {\left((2-{\sqrt {3}})(3+{\sqrt {5}})-2\right)\left(2-{\sqrt {10-2{\sqrt {5}}}}\right)}{4}}\,}$	${\displaystyle \tan {\frac {\pi }{60}}=\tan 3^{\circ }=}$
${\displaystyle {\frac {\left((2+{\sqrt {3}})(3+{\sqrt {5}})-2\right)\left(2+{\sqrt {10-2{\sqrt {5}}}}\right)}{4}}\,}$	${\displaystyle \cot {\frac {\pi }{60}}=\cot 3^{\circ }=}$

מצולע בעל 30 צלעות - 6°

${\displaystyle {\frac {{\sqrt {30-6{\sqrt {5}}}}-{\sqrt {5}}-1}{8}}\,}$	${\displaystyle \sin {\frac {\pi }{30}}=\sin 6^{\circ }=}$
${\displaystyle {\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {15}}+{\sqrt {3}}}{8}}\,}$	${\displaystyle \cos {\frac {\pi }{30}}=\cos 6^{\circ }=}$
${\displaystyle {\frac {{\sqrt {10-2{\sqrt {5}}}}-{\sqrt {15}}+{\sqrt {3}}}{2}}\,}$	${\displaystyle \tan {\frac {\pi }{30}}=\tan 6^{\circ }=}$
${\displaystyle {\frac {3{\sqrt {3}}+{\sqrt {15}}+{\sqrt {50+22{\sqrt {5}}}}}{2}}\,}$	${\displaystyle \cot {\frac {\pi }{30}}=\cot 6^{\circ }=}$

מצולע בעל 20 צלעות - 9°

${\displaystyle {\frac {{\sqrt {10}}+{\sqrt {2}}-2{\sqrt {5-{\sqrt {5}}}}}{8}}\,}$	${\displaystyle \sin {\frac {\pi }{20}}=\sin 9^{\circ }=}$
${\displaystyle {\frac {{\sqrt {10}}+{\sqrt {2}}+2{\sqrt {5-{\sqrt {5}}}}}{8}}\,}$	${\displaystyle \cos {\frac {\pi }{20}}=\cos 9^{\circ }=}$
${\displaystyle {\sqrt {5}}+1-{\sqrt {5+2{\sqrt {5}}}}\,}$	${\displaystyle \tan {\frac {\pi }{20}}=\tan 9^{\circ }=}$
${\displaystyle {\sqrt {5}}+1+{\sqrt {5+2{\sqrt {5}}}}\,}$	${\displaystyle \cot {\frac {\pi }{20}}=\cot 9^{\circ }=}$

מצולע בעל 15 צלעות - 12°

${\displaystyle {\frac {{\sqrt {10+2{\sqrt {5}}}}-{\sqrt {15}}+{\sqrt {3}}}{8}}\,}$	${\displaystyle \sin {\frac {\pi }{15}}=\sin 12^{\circ }=}$
${\displaystyle {\frac {{\sqrt {30+6{\sqrt {5}}}}+{\sqrt {5}}-1}{8}}\,}$	${\displaystyle \cos {\frac {\pi }{15}}=\cos 12^{\circ }=}$
${\displaystyle {\frac {3{\sqrt {3}}-{\sqrt {15}}-{\sqrt {50-22{\sqrt {5}}}}}{2}}\,}$	${\displaystyle \tan {\frac {\pi }{15}}=\tan 12^{\circ }=}$
${\displaystyle {\frac {{\sqrt {15}}+{\sqrt {3}}+{\sqrt {10+2{\sqrt {5}}}}}{2}}\,}$	${\displaystyle \cot {\frac {\pi }{15}}=\cot 12^{\circ }=}$

מתורסר, מצולע בעל 12 צלעות - 15°

${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}\,}$	${\displaystyle \sin {\frac {\pi }{12}}=\sin 15^{\circ }=}$
${\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}\,}$	${\displaystyle \cos {\frac {\pi }{12}}=\cos 15^{\circ }=}$
${\displaystyle 2-{\sqrt {3}}\,}$	${\displaystyle \tan {\frac {\pi }{12}}=\tan 15^{\circ }=}$
${\displaystyle 2+{\sqrt {3}}\,}$	${\displaystyle \cot {\frac {\pi }{12}}=\cot 15^{\circ }=}$

מעושר, מצולע בעל 10 צלעות - 18°

${\displaystyle {\frac {{\sqrt {5}}-1}{4}}={\frac {\varphi -1}{2}}={\frac {1}{2\varphi }}\,}$	${\displaystyle \sin {\frac {\pi }{10}}=\sin 18^{\circ }=}$
${\displaystyle {\frac {\sqrt {10+2{\sqrt {5}}}}{4}}\,}$	${\displaystyle \cos {\frac {\pi }{10}}=\cos 18^{\circ }=}$
${\displaystyle {\frac {\sqrt {25-10{\sqrt {5}}}}{5}}\,}$	${\displaystyle \tan {\frac {\pi }{10}}=\tan 18^{\circ }=}$
${\displaystyle {\sqrt {5+2{\sqrt {5}}}}\,}$	${\displaystyle \cot {\frac {\pi }{10}}=\cot 18^{\circ }=}$

סכום זוויות: 9° + 12° = 21°

${\displaystyle {\frac {(2{\sqrt {3}}+2){\sqrt {5-{\sqrt {5}}}}-({\sqrt {6}}-{\sqrt {2}})(1+{\sqrt {5}})}{16}}\,}$	${\displaystyle \sin {\frac {7\pi }{60}}=\sin 21^{\circ }=}$
${\displaystyle {\frac {(2{\sqrt {3}}-2){\sqrt {5-{\sqrt {5}}}}+({\sqrt {6}}+{\sqrt {2}})(1+{\sqrt {5}})}{16}}\,}$	${\displaystyle \cos {\frac {7\pi }{60}}=\cos 21^{\circ }=}$
${\displaystyle {\frac {\left(2-(2+{\sqrt {3}})(3-{\sqrt {5}})\right)\left(2-{\sqrt {2(5+{\sqrt {5}})}}\right)}{4}}\,}$	${\displaystyle \tan {\frac {7\pi }{60}}=\tan 21^{\circ }=}$
${\displaystyle {\frac {\left(2-(2-{\sqrt {3}})(3-{\sqrt {5}})\right)\left(2+{\sqrt {10{\sqrt {5}}}}\right)}{4}}\,}$	${\displaystyle \cot {\frac {7\pi }{60}}=\cot 21^{\circ }=}$

מתומן, מצולע בעל שמונה צלעות - 22.5°

${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}\,}$	${\displaystyle \sin {\frac {\pi }{8}}=\sin 22.5^{\circ }=}$
${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\,}$	${\displaystyle \cos {\frac {\pi }{8}}=\cos 22.5^{\circ }=}$
${\displaystyle {\sqrt {2}}-1\,}$	${\displaystyle \tan {\frac {\pi }{8}}=\tan 22.5^{\circ }=}$
${\displaystyle {\sqrt {2}}+1\,}$	${\displaystyle \cot {\frac {\pi }{8}}=\cot 22.5^{\circ }=}$

סכום זוויות: 12° + 12° = 24°

${\displaystyle {\frac {{\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}}{8}}\,}$	${\displaystyle \sin {\frac {2\pi }{15}}=\sin 24^{\circ }=}$
${\displaystyle {\frac {{\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1}{8}}\,}$	${\displaystyle \cos {\frac {2\pi }{15}}=\cos 24^{\circ }=}$
${\displaystyle {\frac {{\sqrt {50+22{\sqrt {5}}}}-3{\sqrt {3}}-{\sqrt {15}}}{2}}\,}$	${\displaystyle \tan {\frac {2\pi }{15}}=\tan 24^{\circ }=}$
${\displaystyle {\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {15}}-{\sqrt {3}}}{2}}\,}$	${\displaystyle \cot {\frac {2\pi }{15}}=\cot 24^{\circ }=}$

סכום זוויות: 15° + 12° = 27°

${\displaystyle {\frac {({\sqrt {5}}+1){\sqrt {5+{\sqrt {5}}}}-{\sqrt {10}}+{\sqrt {2}}}{8}}\,}$	${\displaystyle \sin {\frac {3\pi }{20}}=\sin 27^{\circ }=}$
${\displaystyle {\frac {({\sqrt {5}}+1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {10}}-{\sqrt {2}})}{8}}\,}$	${\displaystyle \cos {\frac {3\pi }{20}}=\cos 27^{\circ }=}$
${\displaystyle {\sqrt {5}}-1-{\sqrt {5-2{\sqrt {5}}}}\,}$	${\displaystyle \tan {\frac {3\pi }{20}}=\tan 27^{\circ }=}$
${\displaystyle {\sqrt {5}}-1+{\sqrt {5-2{\sqrt {5}}}}\,}$	${\displaystyle \cot {\frac {3\pi }{20}}=\cot 27^{\circ }=}$

משושה, מצולע בעל שש צלעות - 30°

${\displaystyle {\frac {1}{2}}\,}$	${\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }=}$
${\displaystyle {\frac {\sqrt {3}}{2}}\,}$	${\displaystyle \cos {\frac {\pi }{6}}=\cos 30^{\circ }=}$
${\displaystyle {\frac {\sqrt {3}}{3}}\,}$	${\displaystyle \tan {\frac {\pi }{6}}=\tan 30^{\circ }=}$
${\displaystyle {\sqrt {3}}\,}$	${\displaystyle \cot {\frac {\pi }{6}}=\cot 30^{\circ }=}$

סכום זוויות: 18° + 15° = 33°

${\displaystyle {\frac {(2{\sqrt {3}}-2){\sqrt {5+{\sqrt {5}}}}+(1+{\sqrt {3}})({\sqrt {10}}-{\sqrt {2}})}{16}}\,}$	${\displaystyle \sin {\frac {11\pi }{60}}=\sin 33^{\circ }=}$
${\displaystyle {\frac {(2{\sqrt {3}}+2){\sqrt {5+{\sqrt {5}}}}+(1-{\sqrt {3}})({\sqrt {10}}-{\sqrt {2}})}{16}}\,}$	${\displaystyle \cos {\frac {11\pi }{60}}=\cos 33^{\circ }=}$
${\displaystyle {\frac {\left(2-(2-{\sqrt {3}})(3+{\sqrt {5}})\right)\left(2+{\sqrt {2(5-{\sqrt {5}})}}\right)}{4}}\,}$	${\displaystyle \tan {\frac {11\pi }{60}}=\tan 33^{\circ }=}$
${\displaystyle {\frac {\left(2-(2+{\sqrt {3}})(3+{\sqrt {5}})\right)\left(2-{\sqrt {2(5-{\sqrt {5}})}}\right)}{4}}\,}$	${\displaystyle \cot {\frac {11\pi }{60}}=\cot 33^{\circ }=}$

מחומש, מצולע בעל חמש צלעות - 36°

${\displaystyle {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\,}$	${\displaystyle \sin {\frac {\pi }{5}}=\sin 36^{\circ }=}$
${\displaystyle {\frac {1+{\sqrt {5}}}{4}}={\frac {\varphi }{2}}\,}$	${\displaystyle \cos {\frac {\pi }{5}}=\cos 36^{\circ }=}$
${\displaystyle {\sqrt {5-2{\sqrt {5}}}}\,}$	${\displaystyle \tan {\frac {\pi }{5}}=\tan 36^{\circ }=}$
${\displaystyle {\frac {\sqrt {25+10{\sqrt {5}}}}{5}}\,}$	${\displaystyle \cot {\frac {\pi }{5}}=\cot 36^{\circ }=}$

סכום זוויות: 21° + 18° = 39°

${\displaystyle {\frac {(2-2{\sqrt {3}}){\sqrt {5-{\sqrt {5}}}}+({\sqrt {6}}+{\sqrt {2}})({\sqrt {5}}+1)}{16}}\,}$	${\displaystyle \sin {\frac {13\pi }{60}}=\sin 39^{\circ }=}$
${\displaystyle {\frac {(2+2{\sqrt {3}}){\sqrt {5-{\sqrt {5}}}}+({\sqrt {6}}-{\sqrt {2}})({\sqrt {5}}+1)}{16}}\,}$	${\displaystyle \cos {\frac {13\pi }{60}}=\cos 39^{\circ }=}$
${\displaystyle {\frac {\left((2-{\sqrt {3}})(3-{\sqrt {5}})-2\right)\left(2-{\sqrt {2(5+{\sqrt {5}})}}\right)}{4}}\,}$	${\displaystyle \tan {\frac {13\pi }{60}}=\tan 39^{\circ }=}$
${\displaystyle {\frac {\left((2+{\sqrt {3}})(3-{\sqrt {5}})-2\right)\left(2+{\sqrt {2(5+{\sqrt {5}})}}\right)}{4}}\,}$	${\displaystyle \cot {\frac {13\pi }{60}}=\cot 39^{\circ }=}$

סכום זוויות: 21° + 21° = 42°

${\displaystyle {\frac {{\sqrt {30+6{\sqrt {5}}}}-{\sqrt {5}}+1}{8}}\,}$	${\displaystyle \sin {\frac {7\pi }{30}}=\sin 42^{\circ }=}$
${\displaystyle {\frac {{\sqrt {10+2{\sqrt {5}}}}+{\sqrt {15}}-{\sqrt {3}}}{8}}\,}$	${\displaystyle \cos {\frac {7\pi }{30}}=\cos 42^{\circ }=}$
${\displaystyle {\frac {{\sqrt {15}}+{\sqrt {3}}-{\sqrt {10+2{\sqrt {5}}}}}{2}}\,}$	${\displaystyle \tan {\frac {7\pi }{30}}=\tan 42^{\circ }=}$
${\displaystyle {\frac {{\sqrt {50-22{\sqrt {5}}}}+3{\sqrt {3}}-{\sqrt {15}}}{2}}\,}$	${\displaystyle \cot {\frac {7\pi }{30}}=\cot 42^{\circ }=}$

ריבוע, מצולע בעל ארבע צלעות - 45°

${\displaystyle {\frac {\sqrt {2}}{2}}\,}$	${\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }=}$
${\displaystyle {\frac {\sqrt {2}}{2}}\,}$	${\displaystyle \cos {\frac {\pi }{4}}=\cos 45^{\circ }=}$
${\displaystyle 1\,}$	${\displaystyle \tan {\frac {\pi }{4}}=\tan 45^{\circ }=}$
${\displaystyle 1\,}$	${\displaystyle \cot {\frac {\pi }{4}}=\cot 45^{\circ }=}$

משולש, מצולע בעל שלוש צלעות - 60°

${\displaystyle {\frac {\sqrt {3}}{2}}\,}$	${\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }=}$
${\displaystyle {\frac {1}{2}}\,}$	${\displaystyle \cos {\frac {\pi }{3}}=\cos 60^{\circ }=}$
${\displaystyle {\sqrt {3}}\,}$	${\displaystyle \tan {\frac {\pi }{3}}=\tan 60^{\circ }=}$
${\displaystyle {\frac {\sqrt {3}}{3}}\,}$	${\displaystyle \cot {\frac {\pi }{3}}=\cot 60^{\circ }=}$

קישורים חיצוניים

• Constructible Regular Polygons
• Naming polygons
• קבועים טריגונומטריים מדויקים, באתר MathWorld (באנגלית)