FORMULAS FOR CALCULATING THE SINES

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

```