This section will have useful constants (which will be on a separate page), and information concerning maths. Maths is a very broad topic, so some parts of this article might connect to articles outside this page. If you want maths symbols, then check this page out: Tweeting Symbols and Texting: Gharr’s Style Manual [article]:

Area of a hex, and area & volume of hexagonal prism

hex area

Okay… some of the the stuff above is wrong; and the formulas are corrected below.

Base Area of a right-angled triangle = 1/2 * base * height = 1/2 * (1/2*b)* a = 1/4*a*b

Area of triangle consisting of two equal joined right-angled triangles (as shown in hex above) = 2 * 1/4 * a * b = 1/2*a*b (a single triangle or two right-angled triangles that are joined)

area of hex (has six triangles) = 6 * (1/2 * a * b) = 3ab

The hexagonal prism has rectangular sides, area = b * h

There are six rectangular sides, and two hex areas, so the total surface area of a hexagonal prism is = 2*(3ab)+6*(bh) = 6*(ab+bh)

The Volume of a hexagonal prism = area of hex * height = (3ab)*h = 3abh

Maths Notation

Maths notation is needlessly complex. It can and should be better [article]:


The Triangle of Power


                  =       8           = 2^3=8

2            x


                  =       3           = log28=3

2            8


                  =       2           = 3√8=2

x            8


                  =       operation

√            ^

A triangle within a triangle example


                  =       2^(log28) = 2^3 = 8 = x^(logxy) = y

2            x

Miku Hatsune Equation

(◕ω◕) ❤❤❤ [Hatsune Miku equation]:


x(t) = ((-7/2 sin(11/7-9 t)-7/8 sin(11/7-3 t)-241/24 sin(11/7-t)+10/3 sin(2 t+11/7)+11/3 sin(4 t+33/7)+61/5 sin(5 t+8/5)+69/5 sin(6 t+8/5)+85/28 sin(7 t+33/7)+19/5 sin(8 t+11/7)+2 sin(10 t+11/7)+1/16 sin(11 t+11/6)+3/7 sin(12 t+33/7)-617/3) theta(131 pi-t) theta(t-127 pi)+(-2/5 sin(11/7-5 t)+27/2 sin(t+11/7)+61/6 sin(2 t+11/7)+1/9 sin(3 t+14/3)+4/3 sin(4 t+11/7)+16/17 sin(6 t+11/7)+1/10 sin(7 t+33/7)+1/5 sin(8 t+8/5)+1/5 sin(9 t+8/5)+2/7 sin(10 t+8/5)+1/27 sin(11 t+14/3)+1/4 sin(12 t+8/5)-572/5) theta(127 pi-t) theta(t-123 pi)+(-1/12 sin(11/7-5 t)-7/5 sin(11/7-3 t)-443/8 sin(11/7-t)+1/3 sin(2 t+11/7)+5/2 sin(4 t+8/5)+4/3 sin(6 t+8/5)+1/13 sin(7 t+32/7)+1/4 sin(8 t+11/7)+1/19 sin(10 t+10/7)+1/9 sin(11 t+14/3)+586/15) theta(123 pi-t) theta(t-119 pi)+(-13/14 sin(3/2-40 t)-4/3 sin(3/2-30 t)-16/11 sin(3/2-29 t)-13/2 sin(11/7-26 t)-25/4 sin(11/7-24 t)-11/3 sin(3/2-17 t)-71/11 sin(11/7-14 t)-275/8 sin(11/7-4 t)-30/7 sin(10/7-2 t)+229/7 sin(t+8/5)+239/34 sin(3 t+32/7)+579/4 sin(5 t+33/7)+128 sin(6 t+11/7)+76/9 sin(7 t+11/7)+175/6 sin(8 t+17/11)+263/8 sin(9 t+11/7)+194/7 sin(10 t+11/7)+149/5 sin(11 t+33/7)+23 sin(12 t+33/7)+5/7 sin(13 t+4)+71/6 sin(15 t+11/7)+69/5 sin(16 t+11/7)+58/5 sin(18 t+11/7)+5/4 sin(19 t+8/5)+16/3 sin(20 t+11/7)+134/15 sin(21 t+33/7)+13/7 sin(22 t+7/5)+8/5 sin(23 t+17/9)+241/30 sin(25 t+14/3)+1/2 sin(27 t+23/11)+29/8 sin(28 t+8/5)+26/7 sin(31 t+17/11)+7 sin(32 t+14/9)+18/5 sin(33 t+3/2)+11/7 sin(34 t+3/2)+6/5 sin(35 t+14/3)+3/2 sin(36 t+33/7)+1/6 sin(37 t+39/11)+18/7 sin(38 t+11/7)+2 sin(39 t+33/7)+1/5 sin(41 t+5/2)+2/7 sin(42 t+11/6)-2277/5) theta(119 pi-t) theta(t-115 pi)+(-221/20 sin(11/7-17 t)-33/2 sin(11/7-15 t)-186/11 sin(11/7-12 t)-111/5 sin(11/7-3 t)-85/4 sin(11/7-2 t)+968/9 sin(t+11/7)+26/7 sin(4 t+8/5)+11/5 sin(5 t+8/5)+47/2 sin(6 t+33/7)+11/6 sin(7 t+23/5)+29/5 sin(8 t+14/9)+90/7 sin(9 t+11/7)+13/8 sin(10 t+33/7)+164/7 sin(11 t+33/7)+22/3 sin(13 t+33/7)+13/3 sin(14 t+11/7)+8/9 sin(16 t+17/9)+2 sin(18 t+8/5)+1/7 sin(19 t+8/7)+36/5 sin(20 t+11/7)+27/8 sin(21 t+8/5)+19/9 sin(22 t+33/7)+6/5) theta(115 pi-t) theta(t-111 pi)+(-1/4 sin(9/7-39 t)-5/7 sin(7/5-35 t)-16/3 sin(17/11-32 t)-20/7 sin(3/2-30 t)-9/10 sin(7/5-29 t)-13/6 sin(3/2-28 t)-23/3 sin(3/2-27 t)-31/5 sin(14/9-24 t)-37/6 sin(14/9-22 t)-4/5 sin(3/2-20 t)-123/5 sin(14/9-16 t)-5/4 sin(6/5-15 t)-42/5 sin(17/11-14 t)-23 sin(14/9-13 t)-34/7 sin(3/2-12 t)-207/13 sin(11/7-11 t)-230/3 sin(11/7-8 t)-655/6 sin(11/7-4 t)+245/4 sin(t+33/7)+91/3 sin(2 t+11/7)+136/7 sin(3 t+8/5)+1036/17 sin(5 t+8/5)+34/5 sin(6 t+11/7)+311/8 sin(7 t+8/5)+475/14 sin(9 t+8/5)+97/5 sin(10 t+8/5)+7/5 sin(17 t+10/7)+11/7 sin(18 t+8/5)+17/4 sin(19 t+13/8)+97/24 sin(21 t+8/5)+9/4 sin(23 t+3/2)+13/9 sin(25 t+3/2)+10/7 sin(26 t+5/3)+10/7 sin(31 t+14/9)+5/2 sin(33 t+11/7)+5/7 sin(34 t+14/3)+3/2 sin(36 t+7/4)+9/8 sin(37 t+8/5)+4/3 sin(38 t+7/4)+5/8 sin(40 t+14/3)+1/7 sin(41 t+10/11)+5/6 sin(42 t+14/3)+4291/8) theta(111 pi-t) theta(t-107 pi)+(-1/3 sin(1/6-59 t)-sin(9/7-36 t)-5/6 sin(6/5-20 t)-25/3 sin(11/7-6 t)-20/3 sin(3/2-2 t)+2/5 sin(t+7/3)+16/5 sin(3 t+9/5)+161/10 sin(4 t+33/7)+93/4 sin(5 t+3/2)+53/4 sin(7 t+13/9)+10/7 sin(8 t+14/3)+51/7 sin(9 t+32/7)+13/7 sin(10 t+15/8)+79/6 sin(11 t+32/7)+27/13 sin(12 t+13/7)+32/7 sin(13 t+23/5)+5/7 sin(14 t+9/4)+6/7 sin(15 t+5/7)+19/5 sin(16 t+4/3)+1/2 sin(17 t+5/3)+35/9 sin(18 t+9/2)+13/12 sin(19 t+8/5)+25/4 sin(21 t+9/7)+20/7 sin(22 t+22/5)+27/4 sin(23 t+17/4)+46/15 sin(24 t+21/5)+33/5 sin(25 t+1)+126/11 sin(26 t+8/7)+56/5 sin(27 t+17/4)+38/5 sin(28 t+45/11)+33/5 sin(29 t+1)+13/8 sin(30 t+6/5)+29/8 sin(31 t+17/4)+18/5 sin(32 t+1)+5/4 sin(33 t+67/17)+19/4 sin(34 t+29/7)+22/5 sin(35 t+13/12)+1/2 sin(37 t+3/2)+10/7 sin(38 t+4/5)+13/5 sin(39 t+4)+1/2 sin(40 t+12/13)+10/7 sin(41 t+5/7)+1/2 sin(42 t+5/6)+sin(43 t+2/7)+3/5 sin(44 t+7/5)+19/9 sin(45 t+4)+2/5 sin(46 t+55/27)+4/5 sin(47 t+2/3)+6/5 sin(48 t+35/9)+1/2 sin(49 t+6/5)+3/5 sin(50 t+37/9)+1/3 sin(51 t+14/5)+1/5 sin(52 t+2/7)+5/8 sin(53 t+18/5)+3/4 sin(54 t+1/3)+1/6 sin(55 t+1/2)+1/3 sin(56 t+11/3)+1/10 sin(57 t+23/5)+1/3 sin(58 t+4/5)+1/7 sin(60 t+3/4)+4/9 sin(61 t+1/7)+4217/19) theta(107 pi-t) theta(t-103 pi)+(-1/21 sin(7/5-18 t)-6/13 sin(17/11-15 t)-3/4 sin(14/9-14 t)-4/3 sin(3/2-13 t)-1/4 sin(16/11-12 t)-6/7 sin(14/9-11 t)-14/5 sin(14/9-8 t)-43/6 sin(11/7-7 t)-77/5 sin(11/7-2 t)+422/5 sin(t+11/7)+107/7 sin(3 t+8/5)+4/7 sin(4 t+11/7)+41/5 sin(5 t+8/5)+12/7 sin(6 t+8/5)+11/7 sin(9 t+8/5)+5/8 sin(10 t+13/8)+1/15 sin(16 t+14/3)+4/7 sin(17 t+8/5)+3/8 sin(20 t+8/5)+432/7) theta(103 pi-t) theta(t-99 pi)+(-2 sin(11/7-7 t)+26/5 sin(t+11/7)+122/7 sin(2 t+33/7)+91/4 sin(3 t+11/7)+191/12 sin(4 t+11/7)+9/8 sin(5 t+8/5)+19/7 sin(6 t+11/7)+5/6 sin(8 t+8/5)+5/7 sin(9 t+33/7)+4/7 sin(10 t+8/5)+28/27 sin(11 t+11/7)+13/8 sin(12 t+11/7)+645/23) theta(99 pi-t) theta(t-95 pi)+(-11/7 sin(14/9-12 t)-23/5 sin(14/9-11 t)-13/3 sin(11/7-8 t)-28/9 sin(11/7-7 t)-77/5 sin(11/7-2 t)+1/2 sin(t+14/3)+5/3 sin(3 t+11/7)+31/4 sin(4 t+8/5)+121/30 sin(5 t+8/5)+sin(6 t+33/7)+27/5 sin(9 t+8/5)+3/2 sin(10 t+8/5)-659/6) theta(95 pi-t) theta(t-91 pi)+(-13/5 sin(11/7-10 t)-85/8 sin(14/9-5 t)-343/19 sin(14/9-4 t)-53/4 sin(11/7-3 t)-34/7 sin(11/7-2 t)+45/4 sin(t+11/7)+2/3 sin(6 t+5/3)+44/5 sin(7 t+8/5)+27/4 sin(8 t+13/8)+6/7 sin(9 t+33/7)+49/4 sin(11 t+13/8)+83/8 sin(12 t+8/5)+749/4) theta(91 pi-t) theta(t-87 pi)+(62/3 sin(t+11/7)+1/6 sin(2 t+33/7)+19/9 sin(3 t+11/7)+1/3 sin(4 t+33/7)+5/7 sin(5 t+11/7)+176/9) theta(87 pi-t) theta(t-83 pi)+(58/5 sin(t+11/7)+19/3 sin(2 t+11/7)+11/5 sin(3 t+11/7)+5/9 sin(4 t+8/5)+1/5 sin(5 t+8/5)+1/5 sin(6 t+8/5)+2/5 sin(7 t+8/5)-7/2) theta(83 pi-t) theta(t-79 pi)+(-2/5 sin(3/2-29 t)-4/5 sin(3/2-28 t)-12/13 sin(3/2-27 t)-2/7 sin(17/11-25 t)-19/7 sin(14/9-24 t)-2/7 sin(10/7-22 t)-16/17 sin(11/7-20 t)-23/12 sin(11/7-19 t)-3/7 sin(13/9-17 t)-14/3 sin(11/7-14 t)-4/5 sin(17/11-13 t)-46/5 sin(11/7-11 t)-13/4 sin(14/9-9 t)-75/8 sin(11/7-7 t)-59/7 sin(11/7-6 t)-19/2 sin(14/9-4 t)-130/3 sin(11/7-3 t)-1003/5 sin(11/7-2 t)+625/9 sin(t+11/7)+116/11 sin(5 t+11/7)+11/7 sin(8 t+14/3)+37/12 sin(10 t+11/7)+49/11 sin(12 t+8/5)+9/7 sin(15 t+33/7)+12/5 sin(16 t+11/7)+4/5 sin(18 t+8/5)+2/5 sin(21 t+11/7)+1/4 sin(23 t+32/7)+2/3 sin(26 t+11/7)+1/5 sin(30 t+14/3)-281/2) theta(79 pi-t) theta(t-75 pi)+(-2/5 sin(3/2-21 t)-11/4 sin(11/7-16 t)-9/5 sin(11/7-10 t)-131/6 sin(11/7-5 t)+233/6 sin(t+11/7)+83/6 sin(2 t+11/7)+49/4 sin(3 t+33/7)+29/3 sin(4 t+11/7)+9/10 sin(6 t+14/3)+7/3 sin(7 t+33/7)+37/5 sin(8 t+11/7)+55/7 sin(9 t+11/7)+13/3 sin(11 t+33/7)+13/8 sin(12 t+8/5)+24/7 sin(13 t+11/7)+25/4 sin(14 t+8/5)+4/3 sin(15 t+14/3)+5/3 sin(17 t+11/7)+7/5 sin(18 t+8/5)+5/9 sin(19 t+14/3)+8/5 sin(20 t+8/5)+1/3 sin(22 t+3/2)-3361/8) theta(75 pi-t) theta(t-71 pi)+(-3/4 sin(14/9-38 t)-1/3 sin(11/7-35 t)-10/7 sin(10/7-27 t)-5/2 sin(14/9-26 t)-41/7 sin(14/9-23 t)-4/7 sin(11/7-22 t)-45/7 sin(11/7-21 t)-81/10 sin(11/7-19 t)-41/7 sin(11/7-15 t)+2781/20 sin(t+11/7)+369/5 sin(2 t+33/7)+530/7 sin(3 t+33/7)+731/10 sin(4 t+33/7)+208/7 sin(5 t+33/7)+5/4 sin(6 t+37/8)+13/9 sin(7 t+23/5)+24/5 sin(8 t+11/7)+1/5 sin(9 t+2)+38/37 sin(10 t+11/7)+29/5 sin(11 t+33/7)+33/4 sin(12 t+14/3)+2/3 sin(13 t+9/2)+52/5 sin(14 t+33/7)+41/7 sin(16 t+8/5)+63/16 sin(17 t+33/7)+105/13 sin(18 t+8/5)+11/7 sin(20 t+8/5)+62/7 sin(24 t+8/5)+23/5 sin(25 t+11/7)+29/10 sin(28 t+33/7)+9/4 sin(29 t+8/5)+10/7 sin(30 t+11/7)+11/12 sin(31 t+33/7)+1/9 sin(32 t+1/8)+1/6 sin(33 t+13/3)+1/3 sin(34 t+4/3)+8/9 sin(36 t+11/7)+1/33 sin(37 t+25/7)+2/3 sin(39 t+14/9)+1/7 sin(40 t+9/7)-741/5) theta(71 pi-t) theta(t-67 pi)+(-6/13 sin(3/2-11 t)-2/5 sin(6/5-10 t)-1/4 sin(4/3-9 t)-7/6 sin(3/2-8 t)-17/9 sin(11/8-4 t)-89/10 sin(3/2-3 t)+72/5 sin(t+13/8)+92/13 sin(2 t+8/5)+5/4 sin(5 t+9/5)+6/5 sin(6 t+37/8)+1/5 sin(7 t+5/7)+1/7 sin(12 t+12/5)+1305/7) theta(67 pi-t) theta(t-63 pi)+(-9/7 sin(2/7-11 t)-4/3 sin(1/13-9 t)-6/5 sin(1/16-7 t)-12/11 sin(4/3-6 t)-265/33 sin(1/3-3 t)+81/10 sin(t+28/27)+62/7 sin(2 t+19/5)+39/7 sin(4 t+9/5)+116/23 sin(5 t+1/4)+9/7 sin(8 t+13/4)+3/7 sin(10 t+26/7)+4/5 sin(12 t+16/5)+319/4) theta(63 pi-t) theta(t-59 pi)+(-2/5 sin(1/4-11 t)-1/4 sin(1/3-9 t)-11/10 sin(1-7 t)-26/5 sin(1-3 t)+113/14 sin(t+5/6)+57/7 sin(2 t+24/7)+12/5 sin(4 t+35/8)+8/7 sin(5 t+29/10)+3/8 sin(6 t+4/3)+16/15 sin(8 t+11/6)+2/3 sin(10 t+11/7)+1/4 sin(12 t+2)-371/8) theta(59 pi-t) theta(t-55 pi)+(-3/5 sin(1/4-12 t)-3/4 sin(1/6-10 t)-5/6 sin(1/5-8 t)+299/7 sin(t+11/5)+73/8 sin(2 t+3)+7/2 sin(3 t+35/12)+15/8 sin(4 t+1/6)+9/5 sin(5 t+16/5)+6/7 sin(6 t+1/4)+8/5 sin(7 t+16/5)+sin(9 t+23/7)+5/7 sin(11 t+16/5)-479/5) theta(55 pi-t) theta(t-51 pi)+(-28/5 sin(2/5-9 t)-267/8 sin(6/5-2 t)+5001/20 sin(t+35/8)+97/3 sin(3 t+1/3)+226/9 sin(4 t+11/5)+50/3 sin(5 t+3/8)+61/4 sin(6 t+4/3)+49/5 sin(7 t+1/3)+44/5 sin(8 t+4/5)+141/35 sin(10 t+9/8)+77/38 sin(11 t+1)+18/7 sin(12 t+5/8)-537/5) theta(51 pi-t) theta(t-47 pi)+(-28/27 sin(2/7-11 t)-6/5 sin(1/8-7 t)-41/6 sin(9/7-5 t)-46/5 sin(3/4-3 t)-161/6 sin(1/3-2 t)+1/4 sin(9 t)+385/6 sin(t+21/5)+37/6 sin(4 t+17/7)+13/5 sin(6 t+9/4)+10/7 sin(8 t+9/4)+3/4 sin(10 t+2/3)+1/2 sin(12 t+5/11)+846/5) theta(47 pi-t) theta(t-43 pi)+(-17/11 sin(5/6-22 t)-7/4 sin(1/5-19 t)-6/5 sin(7/5-15 t)-74/25 sin(2/3-8 t)-31/4 sin(9/8-6 t)-33/2 sin(11/10-4 t)-166/5 sin(9/10-2 t)+293/4 sin(t+23/6)+61/4 sin(3 t+31/10)+22/5 sin(5 t+23/9)+83/11 sin(7 t+5/4)+40/7 sin(9 t+4/7)+3/5 sin(10 t+13/4)+17/8 sin(11 t+9/8)+21/5 sin(12 t+22/7)+4/3 sin(13 t+6/5)+16/7 sin(14 t+49/16)+23/12 sin(16 t+15/4)+3/7 sin(17 t+9/5)+1/3 sin(18 t+22/5)+3 sin(20 t+30/7)+3 sin(21 t+10/7)+8/5 sin(23 t+3/5)+13/4 sin(24 t+25/6)-2410/7) theta(43 pi-t) theta(t-39 pi)+(-11/4 sin(5/4-22 t)-11/4 sin(2/5-17 t)-9/4 sin(3/5-10 t)-61/7 sin(5/4-8 t)+1250/9 sin(t+8/5)+203/4 sin(2 t+35/12)+139/9 sin(3 t+14/9)+103/6 sin(4 t+15/4)+79/10 sin(5 t+3/2)+208/19 sin(6 t+30/7)+71/12 sin(7 t+5/11)+8/3 sin(9 t+2/3)+5 sin(11 t+12/7)+21/11 sin(12 t+1/7)+19/8 sin(13 t+5/2)+37/36 sin(14 t+22/7)+5/2 sin(15 t+22/7)+5/4 sin(16 t+19/10)+11/5 sin(18 t+13/3)+1/3 sin(19 t+14/15)+13/9 sin(20 t+13/4)+10/3 sin(21 t+5/6)+15/7 sin(23 t+3)+13/12 sin(24 t+14/9)+806/5) theta(39 pi-t) theta(t-35 pi)+(-4/3 sin(4/9-8 t)+881/20 sin(t+1/7)+409/8 sin(2 t+24/7)+313/12 sin(3 t+11/3)+319/20 sin(4 t+21/5)+23/7 sin(5 t+11/4)+11/5 sin(6 t+4)+13/4 sin(7 t+13/12)+26/9 sin(9 t+2)+8/5 sin(10 t+71/35)+7/5 sin(11 t+13/7)+1/5 sin(12 t+11/8)+2619/8) theta(35 pi-t) theta(t-31 pi)+(-11/10 sin(3/7-6 t)-28/5 sin(3/4-4 t)-89/4 sin(2/3-3 t)-740/13 sin(5/8-2 t)+174/7 sin(t+15/7)+15/4 sin(5 t+19/9)+9/4 sin(7 t+3/2)+7/5 sin(8 t+1/5)+11/7 sin(9 t+9/5)+sin(10 t+23/5)+5/8 sin(11 t+13/4)+2/5 sin(12 t+24/7)-1777/5) theta(31 pi-t) theta(t-27 pi)+(-5/7 sin(3/4-15 t)-4/5 sin(1/3-13 t)-10/7 sin(1/17-11 t)-26/25 sin(1/7-7 t)-5/3 sin(3/4-6 t)+1345/14 sin(t+21/5)+55/3 sin(2 t+25/12)+27/4 sin(3 t+11/4)+17/4 sin(4 t+7/4)+13/12 sin(5 t+37/12)+8/5 sin(8 t+145/36)+7/8 sin(9 t+2/5)+6/5 sin(10 t+32/9)+sin(12 t+17/6)+7/8 sin(14 t+13/5)+5/9 sin(16 t+12/7)+333/5) theta(27 pi-t) theta(t-23 pi)+(-5/4 sin(3/7-8 t)+379/3 sin(t+29/7)+93/5 sin(2 t+7/3)+89/6 sin(3 t+10/3)+10/3 sin(4 t+14/9)+16/3 sin(5 t+13/5)+16/7 sin(6 t+31/16)+21/11 sin(7 t+17/7)+7/4 sin(9 t+11/5)+5/7 sin(10 t+18/5)+4/5 sin(11 t+17/7)+2/3 sin(12 t+17/7)-860/7) theta(23 pi-t) theta(t-19 pi)+(-2/7 sin(3/2-19 t)-16/7 sin(5/6-10 t)-12/5 sin(1/6-6 t)+486/7 sin(t+4/5)+230/11 sin(2 t+14/3)+7/3 sin(3 t+7/4)+3 sin(4 t+7/3)+74/15 sin(5 t+5/7)+63/31 sin(7 t+1/22)+8/9 sin(8 t+5/2)+7/4 sin(9 t+25/6)+3/7 sin(11 t+3/5)+1/2 sin(12 t+18/5)+1/2 sin(13 t+5/4)+1/5 sin(14 t+4/5)+5/9 sin(15 t+3/5)+1/16 sin(16 t+13/6)+5/11 sin(17 t+3/4)+1/3 sin(18 t+9/5)+1/4 sin(20 t+23/5)+515/2) theta(19 pi-t) theta(t-15 pi)+(-5/8 sin(1/26-12 t)-37/5 sin(2/3-9 t)-21/11 sin(5/4-5 t)-85/4 sin(3/4-4 t)-139/6 sin(7/5-3 t)+1614/19 sin(t+17/5)+181/6 sin(2 t+21/5)+49/3 sin(6 t+3/5)+67/7 sin(7 t+23/5)+51/5 sin(8 t+11/6)+26/7 sin(10 t+11/3)+5 sin(11 t+1/2)+1031/5) theta(15 pi-t) theta(t-11 pi)+(-7/8 sin(2/5-11 t)-13/7 sin(7/5-5 t)+152/7 sin(t+25/6)+82/7 sin(2 t+3/5)+40/3 sin(3 t+7/5)+133/22 sin(4 t+63/16)+3 sin(6 t+34/11)+19/3 sin(7 t+11/10)+17/3 sin(8 t+5/2)+10/3 sin(9 t+4/3)+19/5 sin(10 t+12/5)+13/6 sin(12 t+18/5)+1467/8) theta(11 pi-t) theta(t-7 pi)+(-12/13 sin(4/3-10 t)-6/5 sin(1/5-9 t)-13/8 sin(6/5-7 t)-7/5 sin(1/4-5 t)-79/5 sin(1/4-3 t)-811/18 sin(1/6-t)+211/7 sin(2 t+5/11)+29/6 sin(4 t+17/7)+65/8 sin(6 t+23/7)+13/9 sin(8 t+27/7)+12/7 sin(11 t+5/11)+10/9 sin(12 t+17/5)-637/3) theta(7 pi-t) theta(t-3 pi)+(-1/3 sin(5/6-60 t)-4/9 sin(3/4-56 t)-5/8 sin(4/7-53 t)-10/7 sin(5/4-51 t)-11/7 sin(1/26-47 t)-10/7 sin(1/24-39 t)-23/9 sin(22/21-32 t)-9/7 sin(3/4-30 t)-11/4 sin(4/5-26 t)-23/6 sin(33/32-17 t)-20/3 sin(1/4-16 t)-45/7 sin(2/7-14 t)-67/5 sin(3/5-12 t)-8/3 sin(9/10-11 t)-82/27 sin(6/5-10 t)+7679/12 sin(t+17/11)+715/4 sin(2 t+10/3)+1728/19 sin(3 t+22/5)+335/7 sin(4 t+15/4)+345/7 sin(5 t+32/7)+143/5 sin(6 t+7/3)+223/5 sin(7 t+14/3)+45/7 sin(8 t+10/7)+74/5 sin(9 t+3/8)+7/2 sin(13 t+1/3)+45/8 sin(15 t+23/6)+38/7 sin(18 t+17/8)+16/7 sin(19 t+41/9)+1/5 sin(20 t+2)+25/13 sin(21 t+2/5)+13/4 sin(22 t+22/9)+2 sin(23 t+12/5)+5/6 sin(24 t+1/4)+16/5 sin(25 t+21/20)+19/4 sin(27 t+18/7)+13/3 sin(28 t+1/5)+3/5 sin(29 t+13/7)+13/5 sin(31 t+8/5)+7/4 sin(33 t+19/6)+10/7 sin(34 t+19/8)+22/7 sin(35 t+1/16)+19/5 sin(36 t+17/4)+8/3 sin(37 t+9/4)+7/5 sin(38 t+19/7)+11/5 sin(40 t+49/12)+19/7 sin(41 t+8/3)+9/4 sin(42 t+12/7)+15/8 sin(43 t+1/17)+sin(44 t+19/6)+13/8 sin(45 t+16/7)+11/6 sin(46 t+5/3)+1/10 sin(48 t+25/7)+2/5 sin(49 t+13/5)+3/2 sin(50 t+1)+3/5 sin(52 t+5/2)+11/12 sin(54 t+61/15)+1/3 sin(55 t+21/11)+8/7 sin(57 t+14/3)+11/7 sin(58 t+13/4)+sin(59 t+8/7)+467/7) theta(3 pi-t) theta(t+pi)) theta(sqrt(sgn(sin(t/2))))

y(t) = ((-4/5 sin(11/7-12 t)-1/6 sin(14/9-11 t)-13/5 sin(11/7-8 t)-1/5 sin(3/2-7 t)-18/7 sin(11/7-5 t)-161/9 sin(11/7-t)+1/2 sin(2 t+11/7)+54/7 sin(3 t+11/7)+13/5 sin(4 t+33/7)+6/7 sin(6 t+11/7)+3/4 sin(9 t+8/5)+5/7 sin(10 t+8/5)-1582/5) theta(131 pi-t) theta(t-127 pi)+(-87/4 sin(11/7-t)+37/5 sin(2 t+11/7)+5/7 sin(3 t+8/5)+17/11 sin(4 t+11/7)+2/7 sin(5 t+8/5)+7/8 sin(6 t+11/7)+1/7 sin(7 t+8/5)+3/7 sin(8 t+8/5)+1/36 sin(9 t+14/3)+1/6 sin(10 t+8/5)+1/6 sin(11 t+33/7)-139/2) theta(127 pi-t) theta(t-123 pi)+(-113/9 sin(11/7-t)+75/4 sin(2 t+8/5)+16/7 sin(3 t+8/5)+19/4 sin(4 t+8/5)+5/4 sin(5 t+8/5)+7/4 sin(6 t+8/5)+1/5 sin(7 t+7/4)+11/12 sin(8 t+8/5)+1/3 sin(9 t+14/3)+3/4 sin(10 t+8/5)+1/5 sin(11 t+13/8)+4/7 sin(12 t+8/5)-911/8) theta(123 pi-t) theta(t-119 pi)+(-37/7 sin(11/7-37 t)-2/7 sin(11/12-30 t)-27/7 sin(3/2-20 t)-4/5 sin(4/5-17 t)-375/8 sin(14/9-2 t)+927/8 sin(t+33/7)+395/6 sin(3 t+14/3)+966/5 sin(4 t+33/7)+2992/5 sin(5 t+33/7)+7825/24 sin(6 t+11/7)+754/7 sin(7 t+11/7)+227/3 sin(8 t+17/11)+109/2 sin(9 t+8/5)+318/11 sin(10 t+3/2)+390/11 sin(11 t+8/5)+401/11 sin(12 t+14/9)+86/5 sin(13 t+11/7)+107/7 sin(14 t+17/11)+143/6 sin(15 t+3/2)+642/7 sin(16 t+11/7)+95/9 sin(18 t+3/2)+4 sin(19 t+7/4)+17/3 sin(21 t+9/5)+19/5 sin(22 t+4/3)+3/2 sin(23 t+31/7)+14/3 sin(24 t+7/5)+85/9 sin(25 t+8/5)+34/5 sin(26 t+11/7)+568/21 sin(27 t+33/7)+3/8 sin(28 t+19/6)+23/4 sin(29 t+23/5)+25/8 sin(31 t+5/3)+11/8 sin(32 t+13/12)+1/5 sin(33 t+5/2)+13/3 sin(34 t+11/7)+2 sin(35 t+7/4)+9/4 sin(36 t+33/7)+34/3 sin(38 t+11/7)+12/5 sin(39 t+10/7)+20/3 sin(40 t+3/2)+17/4 sin(41 t+14/9)+7/6 sin(42 t+4/3)-5377/8) theta(119 pi-t) theta(t-115 pi)+(-39/8 sin(11/7-21 t)-171/10 sin(11/7-17 t)-199/10 sin(11/7-13 t)-131/7 sin(11/7-10 t)-247/5 sin(11/7-8 t)-109/8 sin(14/9-5 t)+125/3 sin(t+11/7)+213/5 sin(2 t+11/7)+17/4 sin(3 t+3/2)+185/4 sin(4 t+8/5)+98/5 sin(6 t+8/5)+47/4 sin(7 t+8/5)+sin(9 t+19/5)+167/9 sin(11 t+11/7)+165/4 sin(12 t+8/5)+14/3 sin(14 t+14/3)+11/5 sin(15 t+5/3)+11/12 sin(16 t+14/9)+41/9 sin(18 t+8/5)+2/5 sin(19 t+7/4)+10/3 sin(20 t+11/7)+27/5 sin(22 t+8/5)+3815/9) theta(115 pi-t) theta(t-111 pi)+(-18/5 sin(10/7-41 t)-8/9 sin(7/6-40 t)-47/8 sin(11/7-37 t)-53/7 sin(16/11-35 t)-47/7 sin(13/9-34 t)-52/7 sin(3/2-33 t)-3/4 sin(5/4-32 t)-47/7 sin(11/7-29 t)-63/16 sin(11/7-27 t)-19/18 sin(4/3-26 t)-8/9 sin(3/2-25 t)-43/3 sin(13/9-22 t)-33 sin(14/9-21 t)-5/2 sin(7/6-20 t)-389/26 sin(3/2-19 t)-15/4 sin(3/2-18 t)-73/5 sin(14/9-14 t)-173/4 sin(14/9-13 t)-13/4 sin(3/2-10 t)-141/4 sin(14/9-9 t)-443/5 sin(11/7-8 t)-756/5 sin(14/9-7 t)-788/3 sin(11/7-6 t)-337 sin(11/7-5 t)-168 sin(11/7-3 t)-592/5 sin(11/7-t)+927/8 sin(2 t+33/7)+1742/3 sin(4 t+8/5)+46/7 sin(11 t+14/3)+241/4 sin(12 t+8/5)+38/3 sin(15 t+11/7)+94/7 sin(16 t+8/5)+19/5 sin(17 t+5/3)+23/7 sin(23 t+8/5)+18/5 sin(24 t+13/8)+78/11 sin(28 t+14/9)+23/7 sin(30 t+8/5)+7 sin(31 t+5/3)+4/7 sin(36 t+6/7)+9/8 sin(38 t+11/8)+7/2 sin(39 t+12/7)+8/7 sin(42 t+3/2)-7295/12) theta(111 pi-t) theta(t-107 pi)+(-6/11 sin(3/8-61 t)-4/7 sin(5/6-43 t)-2/5 sin(16/17-15 t)-19 sin(3/2-7 t)-132/7 sin(11/7-t)+43/11 sin(2 t+22/5)+64 sin(3 t+3/2)+195/7 sin(4 t+32/7)+272/5 sin(5 t+14/3)+212/5 sin(6 t+3/2)+81 sin(8 t+10/7)+111/5 sin(9 t+6/5)+356/9 sin(10 t+7/5)+46/5 sin(11 t+8/7)+11/4 sin(12 t+41/10)+23/7 sin(13 t+33/7)+11/6 sin(14 t+57/14)+25/8 sin(16 t+4/3)+1/7 sin(17 t+1)+4/3 sin(18 t+30/7)+1/3 sin(19 t+5/6)+2/5 sin(20 t+19/10)+8/5 sin(21 t+14/3)+25/7 sin(22 t+4/3)+26/5 sin(23 t+31/7)+5/3 sin(24 t+11/7)+13/8 sin(25 t+1/3)+27/5 sin(26 t+10/9)+19/4 sin(27 t+4/5)+8/3 sin(28 t+16/15)+25/13 sin(29 t+5/8)+7/6 sin(30 t+11/8)+15/7 sin(31 t+17/4)+4/7 sin(32 t+3/2)+2/5 sin(33 t+2/3)+1/21 sin(34 t+26/7)+1/3 sin(35 t+4/5)+1/5 sin(36 t+21/10)+4/3 sin(37 t+25/6)+7/8 sin(38 t+6/5)+3/5 sin(39 t+14/3)+2/5 sin(40 t+12/7)+1/2 sin(41 t+1/5)+sin(42 t+11/3)+19/9 sin(44 t+3/4)+5/2 sin(45 t+1/2)+17/9 sin(46 t+2/3)+4/3 sin(47 t+33/8)+5/6 sin(48 t+29/8)+12/11 sin(49 t+1/2)+15/8 sin(50 t+5/7)+4/3 sin(51 t+19/5)+13/7 sin(52 t+11/3)+36/35 sin(53 t+4/9)+13/7 sin(54 t+5/8)+5/4 sin(55 t+26/7)+15/16 sin(56 t+11/3)+11/7 sin(57 t+4/9)+3/4 sin(58 t+10/3)+5/4 sin(59 t+26/7)+5/3 sin(60 t+4/7)-6181/10) theta(107 pi-t) theta(t-103 pi)+(-1/4 sin(3/2-20 t)-2/3 sin(3/2-17 t)-9/10 sin(3/2-16 t)-1/9 sin(16/15-14 t)-24/5 sin(14/9-13 t)-1/9 sin(1/4-12 t)-8/9 sin(3/2-11 t)-1/12 sin(4/3-10 t)-8/3 sin(11/7-9 t)-11/3 sin(3/2-8 t)-279/20 sin(11/7-7 t)-119/24 sin(14/9-6 t)-119/10 sin(11/7-3 t)-6/7 sin(11/7-t)+107/6 sin(2 t+11/7)+104/7 sin(4 t+8/5)+4/7 sin(5 t+8/5)+11/8 sin(15 t+8/5)+15/8 sin(18 t+8/5)+1/3 sin(19 t+14/3)+1531/8) theta(103 pi-t) theta(t-99 pi)+(-1/8 sin(11/7-11 t)-27/11 sin(11/7-9 t)-14/13 sin(11/7-7 t)-9/5 sin(11/7-5 t)-88/7 sin(11/7-3 t)+103/7 sin(t+11/7)+26/5 sin(2 t+33/7)+8 sin(4 t+11/7)+15/16 sin(6 t+8/5)+10/3 sin(8 t+11/7)+1/6 sin(10 t+14/3)+6/11 sin(12 t+11/7)-219/5) theta(99 pi-t) theta(t-95 pi)+(-151/19 sin(11/7-10 t)-17/2 sin(11/7-8 t)-44/5 sin(11/7-2 t)+13/5 sin(t+8/5)+1/3 sin(3 t+5/3)+31/4 sin(4 t+8/5)+55/18 sin(5 t+8/5)+68/9 sin(6 t+8/5)+3/5 sin(7 t+3/2)+19/10 sin(9 t+11/7)+17/3 sin(11 t+8/5)+11/7 sin(12 t+8/5)+2572/5) theta(95 pi-t) theta(t-91 pi)+(-20/3 sin(14/9-11 t)-1/2 sin(4/3-9 t)-207/13 sin(14/9-8 t)-29/14 sin(10/7-7 t)-13/2 sin(14/9-6 t)-3/2 sin(13/9-5 t)-39/2 sin(14/9-4 t)-467/11 sin(11/7-3 t)+215/18 sin(t+8/5)+417/19 sin(2 t+11/7)+23/4 sin(10 t+8/5)+27/8 sin(12 t+11/7)+6349/12) theta(91 pi-t) theta(t-87 pi)+(5 sin(t+11/7)+1/6 sin(2 t+33/7)+sin(3 t+11/7)+1/3 sin(4 t+8/5)+2/3 sin(5 t+11/7)+2209/10) theta(87 pi-t) theta(t-83 pi)+(-121/7 sin(11/7-t)+33/5 sin(2 t+11/7)+2/7 sin(3 t+33/7)+4/5 sin(4 t+11/7)+4/5 sin(5 t+33/7)+3/7 sin(6 t+11/7)+1/13 sin(7 t+8/5)+2672/11) theta(83 pi-t) theta(t-79 pi)+(-6/13 sin(10/7-26 t)-2/3 sin(14/9-25 t)-11/8 sin(14/9-24 t)-7/5 sin(11/7-18 t)-16/17 sin(11/7-15 t)-14/3 sin(11/7-13 t)-129/13 sin(11/7-7 t)-87/5 sin(11/7-3 t)+106/3 sin(t+11/7)+101/5 sin(2 t+11/7)+31/6 sin(4 t+8/5)+14/3 sin(5 t+11/7)+46 sin(6 t+8/5)+19/5 sin(8 t+33/7)+10/7 sin(9 t+8/5)+24/5 sin(10 t+11/7)+9/8 sin(11 t+14/9)+23/4 sin(12 t+8/5)+3 sin(14 t+8/5)+37/7 sin(16 t+8/5)+1/8 sin(17 t+41/9)+7/3 sin(19 t+8/5)+1/5 sin(20 t+14/9)+4/7 sin(21 t+14/3)+19/6 sin(22 t+8/5)+2/3 sin(23 t+33/7)+1/2 sin(27 t+13/8)+1/8 sin(28 t+11/8)+1/4 sin(29 t+32/7)+1/2 sin(30 t+8/5)-5258/7) theta(79 pi-t) theta(t-75 pi)+(-4/3 sin(14/9-19 t)-16/7 sin(11/7-7 t)-71/8 sin(11/7-t)+574/5 sin(2 t+11/7)+16/5 sin(3 t+14/3)+162/5 sin(4 t+11/7)+79/10 sin(5 t+8/5)+40/9 sin(6 t+11/7)+5/4 sin(8 t+5/3)+13/5 sin(9 t+33/7)+100/11 sin(10 t+11/7)+36/5 sin(11 t+11/7)+9/5 sin(12 t+14/3)+19/3 sin(13 t+33/7)+26/5 sin(14 t+11/7)+2 sin(15 t+11/7)+34/11 sin(16 t+14/3)+7/8 sin(17 t+8/5)+29/7 sin(18 t+11/7)+1/2 sin(20 t+23/5)+sin(21 t+8/5)+13/14 sin(22 t+11/7)-1976/3) theta(75 pi-t) theta(t-71 pi)+(-2/3 sin(14/9-39 t)-1/7 sin(24/23-38 t)-1/4 sin(1/39-32 t)-13/4 sin(3/2-28 t)-1/4 sin(9/7-24 t)-23/7 sin(11/7-23 t)-8/5 sin(3/2-19 t)-85/3 sin(11/7-15 t)-107/4 sin(11/7-12 t)-7/4 sin(3/2-9 t)-53/4 sin(11/7-5 t)+53/8 sin(t+33/7)+221/4 sin(2 t+11/7)+33/4 sin(3 t+11/7)+64/7 sin(4 t+11/7)+69/4 sin(6 t+11/7)+91/5 sin(7 t+11/7)+45/4 sin(8 t+11/7)+88/5 sin(10 t+8/5)+5/2 sin(11 t+3/2)+35/3 sin(13 t+8/5)+4/5 sin(14 t+2)+23/3 sin(16 t+8/5)+59/12 sin(17 t+14/3)+43/6 sin(18 t+11/7)+29/5 sin(20 t+8/5)+30/7 sin(21 t+11/7)+6/5 sin(22 t+8/5)+23/9 sin(25 t+8/5)+53/6 sin(26 t+11/7)+15/4 sin(27 t+8/5)+1/13 sin(29 t+5/4)+1/4 sin(30 t+9/5)+30/7 sin(31 t+8/5)+9/4 sin(33 t+11/7)+24/7 sin(34 t+8/5)+9/4 sin(35 t+11/7)+2/3 sin(36 t+10/7)+5/8 sin(37 t+32/7)+18/5 sin(40 t+8/5)-4188/7) theta(71 pi-t) theta(t-67 pi)+(-6/7 sin(4/3-10 t)-3/4 sin(11/7-8 t)-11/6 sin(4/3-7 t)-7/2 sin(11/7-6 t)-64/9 sin(3/2-4 t)+1/6 sin(11 t)+7/5 sin(t+13/9)+9 sin(2 t+5/3)+7 sin(3 t+8/5)+sin(5 t+11/8)+1/5 sin(9 t+23/6)+1/3 sin(12 t+23/5)+116/23) theta(67 pi-t) theta(t-63 pi)+(-9/5 sin(9/10-7 t)-41/5 sin(7/5-5 t)-22/3 sin(1/5-4 t)+53/7 sin(t+2/7)+37/3 sin(2 t+7/3)+59/7 sin(3 t+21/5)+9/5 sin(6 t+17/5)+3/7 sin(8 t+23/6)+1/4 sin(9 t+8/5)+2/5 sin(10 t+12/7)+2/5 sin(11 t+105/26)+3/7 sin(12 t+5/3)+1265/4) theta(63 pi-t) theta(t-59 pi)+(-1/2 sin(3/4-12 t)-6/5 sin(1/2-10 t)-6/7 sin(3/7-8 t)-11/5 sin(1/2-6 t)-28/3 sin(1/2-t)+57/5 sin(2 t+19/9)+9 sin(3 t+17/5)+17/4 sin(4 t+43/14)+55/14 sin(5 t+7/4)+21/11 sin(7 t+19/6)+4/9 sin(9 t+17/6)+4/5 sin(11 t+17/5)+1448/5) theta(59 pi-t) theta(t-55 pi)+(-6/13 sin(3/5-12 t)-6/11 sin(9/10-10 t)-4/7 sin(15/16-8 t)-20/19 sin(8/7-6 t)-53/7 sin(3/8-2 t)+827/14 sin(t+11/7)+8 sin(3 t+19/10)+7/4 sin(4 t+35/8)+12/5 sin(5 t+12/5)+9/7 sin(7 t+7/3)+8/9 sin(9 t+21/8)+3/5 sin(11 t+11/4)+126/5) theta(55 pi-t) theta(t-51 pi)+(-11/5 sin(1/3-12 t)-1/2 sin(8/9-6 t)-1827/5 sin(2/3-t)+311/5 sin(2 t+12/5)+1065/38 sin(3 t+10/7)+59/4 sin(4 t+16/5)+343/19 sin(5 t+11/6)+132/19 sin(7 t+8/3)+55/6 sin(8 t+11/8)+26/5 sin(9 t+1/2)+9/4 sin(10 t+1/5)+14/3 sin(11 t+1/5)-1115/3) theta(51 pi-t) theta(t-47 pi)+(-11/5 sin(7/8-10 t)-sin(10/9-6 t)+285/4 sin(t+14/5)+61 sin(2 t+9/2)+199/10 sin(3 t+19/6)+430/39 sin(4 t+7/4)+21/5 sin(5 t+5/4)+11/8 sin(7 t+3/2)+3/7 sin(8 t+6/5)+12/5 sin(9 t+1/5)+9/8 sin(11 t+16/5)+3/5 sin(12 t+25/8)+51/5) theta(47 pi-t) theta(t-43 pi)+(-17/8 sin(1/4-20 t)-24/23 sin(1/6-16 t)-178/7 sin(5/4-3 t)-242/3 sin(1/3-2 t)-1228/3 sin(6/5-t)+43/2 sin(4 t+1/2)+49/5 sin(5 t+7/8)+9 sin(6 t+1/2)+13/4 sin(7 t+8/5)+18/5 sin(8 t+13/5)+27/5 sin(9 t+25/13)+4/7 sin(10 t+25/7)+2 sin(11 t+32/9)+6/5 sin(12 t+13/4)+7/5 sin(13 t+19/5)+3/5 sin(14 t+2/7)+9/7 sin(15 t+9/2)+6/5 sin(17 t+31/8)+7/8 sin(18 t+1/6)+15/14 sin(19 t+16/7)+11/6 sin(21 t+10/3)+17/18 sin(22 t+5/11)+4/5 sin(23 t+18/7)+5/3 sin(24 t+1/30)-5897/5) theta(43 pi-t) theta(t-39 pi)+(-4/7 sin(1/4-24 t)-8/5 sin(2/5-21 t)-1/2 sin(10/9-19 t)-2 sin(17/11-17 t)-13/7 sin(3/7-12 t)-27/13 sin(2/5-11 t)-173/6 sin(3/5-3 t)-3245/8 sin(10/11-t)+237/4 sin(2 t+1/5)+45/2 sin(4 t+5/3)+61/6 sin(5 t+7/5)+37/5 sin(6 t+2)+14/5 sin(7 t+13/3)+17/4 sin(8 t+31/8)+17/5 sin(9 t+4)+19/18 sin(10 t+7/8)+14/13 sin(13 t+23/7)+11/6 sin(14 t+4/3)+5/6 sin(15 t+28/9)+9/5 sin(16 t+16/17)+10/7 sin(18 t+12/5)+6/5 sin(20 t+15/7)+14/9 sin(22 t+19/5)+4/5 sin(23 t+21/11)-4829/4) theta(39 pi-t) theta(t-35 pi)+(-6/5 sin(3/2-11 t)-2/5 sin(2/5-6 t)-27/11 sin(1/4-5 t)-534/5 sin(4/3-t)+511/10 sin(2 t+69/34)+37/5 sin(3 t+15/7)+45/8 sin(4 t+7/3)+6/5 sin(7 t+35/8)+23/11 sin(8 t+1/4)+14/15 sin(9 t+1/5)+9/4 sin(10 t+1/2)+3/4 sin(12 t+5/4)-5623/3) theta(35 pi-t) theta(t-31 pi)+(-3 sin(1/8-9 t)-11/5 sin(1-8 t)-17/5 sin(1/13-7 t)-3/4 sin(2/5-6 t)+601/6 sin(t+9/2)+156/5 sin(2 t+19/20)+134/9 sin(3 t+21/5)+11 sin(4 t+29/8)+20/3 sin(5 t+17/5)+5/8 sin(10 t+23/6)+7/5 sin(11 t+3/7)+2 sin(12 t+17/7)-31740/17) theta(31 pi-t) theta(t-27 pi)+(-6/5 sin(4/3-11 t)-11/10 sin(7/5-10 t)-29/10 sin(6/5-9 t)-3/7 sin(17/16-5 t)-221/5 sin(13/14-t)+287/11 sin(2 t+2)+55/3 sin(3 t+32/11)+58/5 sin(4 t+14/5)+10/9 sin(6 t+19/8)+8/7 sin(7 t+1/3)+2/3 sin(8 t+15/8)+4/9 sin(12 t+7/4)+3/8 sin(13 t+29/7)+3/8 sin(14 t+11/5)+3/8 sin(15 t+3/5)+3/5 sin(16 t+13/12)+110) theta(27 pi-t) theta(t-23 pi)+(-9/7 sin(3/7-11 t)-131/13 sin(33/32-2 t)+1591/6 sin(t+9/2)+183/7 sin(3 t+65/16)+3 sin(4 t+13/3)+27/4 sin(5 t+25/7)+9/4 sin(6 t+79/20)+13/8 sin(7 t+35/12)+44/15 sin(8 t+38/13)+5/6 sin(9 t+1/21)+12/5 sin(10 t+12/5)+11/5 sin(12 t+2)-1478/7) theta(23 pi-t) theta(t-19 pi)+(-43/8 sin(1-6 t)-13 sin(14/9-5 t)-156/7 sin(1/7-t)+61/7 sin(2 t+29/7)+29/8 sin(3 t+11/7)+28/3 sin(4 t+11/4)+38/7 sin(7 t+22/5)+11/8 sin(8 t+14/5)+sin(9 t+17/7)+1/2 sin(10 t+11/10)+5/7 sin(11 t+1/9)+3/5 sin(12 t+7/6)+2/5 sin(13 t+4/7)+1/6 sin(14 t+4)+2/5 sin(15 t+17/4)+12/13 sin(16 t+17/6)+4/9 sin(17 t+30/7)+1/6 sin(18 t+1)+1/5 sin(19 t+32/7)+2/7 sin(20 t+7/2)-4921/6) theta(19 pi-t) theta(t-15 pi)+(-31/16 sin(3/4-11 t)-12/7 sin(6/5-7 t)-32/5 sin(1/5-3 t)-51/2 sin(1/2-2 t)-3899/11 sin(1-t)+121/8 sin(4 t+3/4)+34/5 sin(5 t+23/5)+4 sin(6 t+11/7)+35/9 sin(8 t+5/8)+14/5 sin(9 t+17/4)+19/5 sin(10 t+8/5)+14/15 sin(12 t+14/5)-823/2) theta(15 pi-t) theta(t-11 pi)+(-37/4 sin(2/5-5 t)-291/7 sin(9/7-t)+43/4 sin(2 t+9/7)+23 sin(3 t+3)+62/7 sin(4 t+1/3)+59/9 sin(6 t+23/6)+53/13 sin(7 t+17/8)+79/10 sin(8 t+4)+16/3 sin(9 t+17/5)+13/5 sin(10 t+22/5)+3/5 sin(11 t+13/6)+12/7 sin(12 t+2/7)+1699/5) theta(11 pi-t) theta(t-7 pi)+(-11/7 sin(3/7-11 t)-30/7 sin(5/7-9 t)-1818/7 sin(13/9-t)+247/4 sin(2 t+12/7)+71/5 sin(3 t+3/2)+183/5 sin(4 t+9/5)+58/5 sin(5 t+7/4)+14 sin(6 t+17/9)+17/11 sin(7 t+2/5)+25/13 sin(8 t+11/6)+2/5 sin(10 t+21/11)+8/3 sin(12 t+7/3)-5529/5) theta(7 pi-t) theta(t-3 pi)+(-5/3 sin(28/27-55 t)-15/14 sin(1/39-44 t)-30/29 sin(1/2-42 t)-23/12 sin(1/14-38 t)-143/16 sin(4/3-25 t)-41/8 sin(4/3-24 t)-57/4 sin(9/10-21 t)-73/8 sin(16/17-20 t)-47/5 sin(1/6-19 t)-24/5 sin(2/3-18 t)-104/11 sin(3/4-17 t)-18 sin(3/4-16 t)-175/9 sin(3/7-12 t)-523/7 sin(12/11-t)+2909/3 sin(2 t+9/2)+501/5 sin(3 t+1/15)+336/11 sin(4 t+7/2)+447/14 sin(5 t+9/5)+117/4 sin(6 t+105/26)+292/5 sin(7 t+1/6)+7/4 sin(8 t+4/7)+93/5 sin(9 t+3/2)+159/10 sin(10 t+1)+95/3 sin(11 t+1/5)+9/2 sin(13 t+4/7)+11 sin(14 t+1/2)+125/6 sin(15 t+1/12)+67/9 sin(22 t+19/5)+25/6 sin(23 t+2/5)+5 sin(26 t+29/10)+15/7 sin(27 t+1/7)+13/4 sin(28 t+23/6)+14/5 sin(29 t+29/8)+61/10 sin(30 t+10/11)+27/5 sin(31 t+50/11)+32/5 sin(32 t+5/2)+17/6 sin(33 t+9/7)+21/4 sin(34 t+1/6)+17/4 sin(35 t+23/6)+5 sin(36 t+9/5)+11/5 sin(37 t+9/10)+11/7 sin(39 t+18/7)+3/4 sin(40 t+1/2)+12/5 sin(41 t+11/6)+3/2 sin(43 t+11/5)+9/5 sin(45 t+9/4)+6/5 sin(46 t+23/9)+4/5 sin(47 t+21/11)+sin(48 t+10/11)+2/3 sin(49 t+11/6)+52/17 sin(50 t+17/7)+9/5 sin(51 t+1/2)+3/7 sin(52 t+14/3)+23/11 sin(53 t+75/37)+19/8 sin(54 t+7/5)+5/7 sin(56 t+11/4)+5/2 sin(57 t+5/3)+39/19 sin(58 t+1/5)+13/6 sin(59 t+21/5)+7/8 sin(60 t+14/13)-2692/5) theta(3 pi-t) theta(t+pi)) theta(sqrt(sgn(sin(t/2))))


Trees are useful in list searching, and computer game design (computer programming).

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Shortened link to article: Maths [article]:


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