FORMULAS FOR CALCULATING THE SINES
(На русском: Аналитические формулы для синусов)
Some values of sines can be calculated using radicals. Here you can find formulas for calculating the sines of the angles in increments of three degrees. Not only these formulas are correct, there are other expressions, and they can be derived.
__________
___ ___ ___ ___ ___ ___ / ___
\/ 30 + \/ 10 - \/ 6 - \/ 2 - ( \/ 6 - \/ 2 ) * \/ 10 + \/ 20
sin(03) = -------------------------------------------------------------------
16
____________ ___________
/ ___ / ___
\/ 30 - 6*\/ 5 - \/ 6 + 2*\/ 5
sin(06) = ----------------------------------
8
_________ _________
/ ___ / ___
\/ 3 + \/ 5 - \/ 5 - \/ 5
sin(09) = -----------------------------
4
____________
/ ___ ___ ___
\/ 10 + 2*\/ 5 + \/ 3 - \/ 15
sin(12) = ---------------------------------
8
_________
/ ___
\/ 2 - \/ 3
sin(15) = --------------
2
_________
/ ___
\/ \/ 5 - 1
sin(18) = --------------
4
____________________________________________________________
/ _______________________________________________
/ / ____________
\ / \ / ___ ___ / ___
\/ 2 - 1/2 * \/ 7 - \/ 5 + 1/2 * (\/ 5 - 1) * \/ 30 + 6*\/ 5
sin(21) = --------------------------------------------------------------------
2
___________________________
/ ____________
\ / ___ / ___
\/ 7 + \/ 5 - \/ 30 + 6*\/ 5
sin(24) = ----------------------------------
4
_________________________
/ ____________
\ / / ___
\/ 2 - 1/2 * \/ 10 - 2*\/ 5
sin(27) = --------------------------------
2
sin(30) = 1/2
________________________________________
/ ___________________________
/ / ____________
\ / \ / ___ / ___
\/ 2 - 1/2 * \/ 7 + \/ 5 - \/ 30 + 6*\/ 5
sin(33) = ------------------------------------------------
2
____________
/ ___
\/ 10 - 2*\/ 5
sin(36) = -----------------
4
_____________________________________________
/ ____________
\ / / ___ ___ ___
\/ 2 - 1/4 * ( \/ 10 + 2*\/ 5 + \/ 3 - \/ 15 )
sin(39) = ----------------------------------------------------
2
________________________________________________
/ ____________
\ / ___ ___ / ___
\/ 9 + \/ 5 - 1/2 * (\/ 5 - 1) * \/ 30 + 6*\/ 5
sin(42) = -------------------------------------------------------
4
___
\/ 2
sin(45) = -------
2
________________________________________________
/ ____________
\ / ___ ___ / ___
\/ 7 - \/ 5 + 1/2 * (\/ 5 - 1) * \/ 30 + 6*\/ 5
sin(48) = -------------------------------------------------------
4
_____________________________________________
/ ____________
\ / / ___ ___ ___
\/ 2 + 1/4 * ( \/ 10 + 2*\/ 5 + \/ 3 - \/ 15 )
sin(51) = ----------------------------------------------------
2
___________
/ ___
\/ 6 + 2*\/ 5
sin(54) = ----------------
4
________________________________________
/ ___________________________
/ / ____________
\ / \ / ___ / ___
\/ 2 + 1/2 * \/ 7 + \/ 5 - \/ 30 + 6*\/ 5
sin(57) = ------------------------------------------------
2
___
\/ 3
sin(60) = -------
2
_________________________
/ ____________
\ / / ___
\/ 2 + 1/2 * \/ 10 - 2*\/ 5
sin(63) = --------------------------------
2
___________________________
/ ____________
\ / ___ / ___
\/ 9 - \/ 5 + \/ 30 + 6*\/ 5
sin(66) = ----------------------------------
4
____________________________________________________________
/ _______________________________________________
/ / ____________
\ / \ / ___ ___ / ___
\/ 2 + 1/2 * \/ 7 - \/ 5 + 1/2 * (\/ 5 - 1) * \/ 30 + 6*\/ 5
sin(69) = --------------------------------------------------------------------
2
____________
/ ___
\/ 10 + 2*\/ 5
sin(72) = -----------------
4
_________
/ ___
\/ 2 + \/ 3
sin(75) = --------------
2
________________________________________________
/ ____________
\ / ___ ___ / ___
\/ 9 + \/ 5 + 1/2 * (\/ 5 - 1) * \/ 30 + 6*\/ 5
sin(78) = -------------------------------------------------------
4
_________________________
/ ____________
\ / / ___
\/ 2 + 1/2 * \/ 10 + 2*\/ 5
sin(81) = --------------------------------
2
___________________________
/ ____________
\ / ___ / ___
\/ 7 + \/ 5 + \/ 30 + 6*\/ 5
sin(84) = ----------------------------------
4
________________________________________
/ ___________________________
/ / ____________
\ / \ / ___ / ___
\/ 2 + 1/2 * \/ 7 + \/ 5 + \/ 30 + 6*\/ 5
sin(87) = ------------------------------------------------
2
sin(90) = 1