~v200 200 ~w158 4 407 640 256 247 0 0 ~f? 14 12 10
? 2 1 1 0 ? ? ? "Arial" ? ? ? 0 ? 0 1
"Times" 12 ? ? 5 0 c n 100 0 0 0 k 288
i"CHEBYCHEV"
-2 1 26177 26178 26115 26178 1 1 1 1 0 0
0 0 -1 0 1 -1 -1 -1 -1 -1 1 1 ? ? 
~Q
]|Expr|[#b @`bb#_b#_b#_})## b$@" *|: ;bP8&c0!)Chebyshev Polynomials|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~V?f0 (T)~p0
1 ~V?v0 (k)~p0
1 ~V?v0 (n)~p0
1 ~V?v0 (y)~p0
1 ~V?v0 (x)~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Trigonometry Rules|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})+# b$L" *|: ;bP8&c0!*Simplify ,M negation|
  | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Hs(sin(-?x)):(-sin(?x))~p0
3 ~Hs(cos(-?x)):(cos(?x))~p0
3 ~Hs(tan(-?x)):(-tan(?x))~p0
3 ~Hs(sin('p)):(0)~p0
3 ~Hs(sin(?n*'p)):(0)~p0
3 ~Hs(cos(1/2*'p)):(0)~p0
3 ~Hs(cos(?n/2*'p)):(0)~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})'# b$L" *|: ;bP8&c0!*Transform to basic|
  | types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Ht(tan(?x)):((sin(?x))/(cos(?x)))~p0
3 ~Ht(csc(?x)):(1/(sin(?x)))~p0
3 ~Ht(sin(?x)):(1/(csc(?x)))~p0
3 ~Ht(sec(?x)):(1/(cos(?x)))~p0
3 ~Ht(cos(?x)):(1/(sec(?x)))~p0
3 ~Ht(cot(?x)):((cos(?x))/(sin(?x)))~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Trig Addition|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Ht(cos(?x+?y)):(cos(?x)*cos(?y)-sin(?x)*sin(?y))~p0
3 ~Ht(sin(?x+?y)):(cos(?x)*sin(?y)+sin(?x)*cos(?y))~p0
3 ~Ht(cos(2*?x)):(2*(cos(?x))^2-1)~p0
3 ~Ht(sin(2*?x)):(2*cos(?x)*sin(?x))~p0
3 ~Ht(sin(?n*?x)):(cos((?n-1)*?x)*sin(?x)+cos(?x)*sin((?n-1)*?x))~p0
3 ~Ht(cos(?n*?x)):(cos(?x)*cos((?n-1)*?x)-sin(?x)*sin((?n-1)*?x))~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})/# b$L" *|: ;bP8&c0!*Transform ,M into|
  | another flavor of trig function}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Ht((sin(?x))^2):(1-(cos(?x))^2)~p0
3 ~Ht((cos(?x))^2):(1-(sin(?x))^2)~p0
3 ~Ht((tan(?x))^2):((sec(?x))^2-1)~p0
3 ~Ht((sec(?x))^2):((tan(?x))^2+1)~p0
3 ~Ht((csc(?x))^2):((cot(?x))^2+1)~p0
3 ~Ht((cot(?x))^2):((csc(?x))^2-1)~p0
3 ~Hs((sin(?x))^2+(cos(?x))^2):(1)~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})b @# b%4" *|: ;bP8&c0!*substituting |
  |z,]tan,Hx,O2,I into a rational function in sin,Hx,I and cos,H|
  |x,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Hs(cos(2*arctan(?z))):((1-?z^2)/(1+?z^2))~p0
3 ~Hs(sin(2*arctan(?z))):(2*?z/(1+?z^2))~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Other rules|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Ht((cos(?x))^2):(1/2*(cos(2*?x)+1))~p0
3 ~Ht((sin(?x))^2):(1/2*(-cos(2*?x)+1))~p0
3 ~Ht(cos(?x)*sin(?x)):(1/2*sin(2*?x))~p0
3 ~Hs(sin(arccos(?x))):(sqrt(-?x^2+1))~p0
3 ~Hs(cos(arcsin(?x))):(sqrt(-?x^2+1))~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Trigonometry|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
1 ~V?c1 ('p)~p0
2 ~V?f256 (sin)~p0
2 ~V?f257 (cos)~p0
2 ~V?f258 (tan)~p0
2 ~V?f261 (sec)~p0
2 ~V?f260 (csc)~p0
2 ~V?f262 (cot)~p0
2 ~V?f272 (arcsin)~p0
2 ~V?f273 (arccos)~p0
2 ~V?f274 (arctan)~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b!," *|: ;bP8&c0!*Input n|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(n=3)~p0
255 ~d~A(T(?n,?x)=cos(?n*arccos(?x)))~p0
0 ~d~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$@["! ) # b$@|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW})%# b$@" *|: ;bP8&c0!*Chebyshev|
  | Polynomial Plot}& b!( b"0 b#8 b$@ b%H b&P!WW}}|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~G1 1 161 386
1 1 4 3 10
(-1.1000000000000001...1.1000000000000001):(~
 -1.1000000000000001...1.1000000000000001):(n=0...16):(1):(16)~
 ~Q
]|Expr|[#b @`bb#_b#_b#_}`f#})!# b$@" *|: ;bP8&c0!*polynomial|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~L1 255 ? (x,T(n,x)):(x=-1...1)~p0
1 ~gc0 60 ? 0 -3724 -3724 0
3120 -3369 -925 0
4030 -3266 -250 0
6241 -3014 1141 0
7411 -2881 1743 0
9362 -2660 2551 0
10272 -2556 2850 0
11866 -2375 3260 0
12823 -2266 3441 0
13262 -2217 3508 0
13994 -2133 3599 0
14563 -2069 3652 0
15213 -1995 3694 0
15456 -1967 3705 0
15603 -1951 3711 0
16383 -1862 3724 0
17684 -1714 3689 0
18074 -1670 3666 0
19894 -1463 3486 0
20235 -1424 3439 0
21845 -1241 3172 0
22413 -1177 3060 0
22541 -1162 3034 0
22755 -1138 2988 0
23193 -1088 2893 0
27046 -650 1871 0
28996 -429 1263 0
30166 -296 879 0
33287 59 -177 0
34327 177 -530 0
35238 281 -836 0
36408 414 -1221 0
38618 665 -1910 0
40569 887 -2459 0
41479 990 -2690 0
42649 1123 -2960 0
43690 1241 -3172 0
44990 1389 -3394 0
45770 1478 -3502 0
46373 1546 -3572 0
47378 1660 -3661 0
47720 1699 -3682 0
49101 1856 -3724 0
49931 1950 -3711 0
50205 1982 -3700 0
50662 2033 -3675 0
50826 2052 -3663 0
51101 2083 -3641 0
51442 2122 -3609 0
52011 2187 -3544 0
53052 2305 -3382 0
53962 2409 -3195 0
56336 2678 -2492 0
56611 2710 -2390 0
57343 2793 -2095 0
58253 2896 -1681 0
59293 3014 -1142 0
62414 3369 924 0
63584 3502 1884 0
65535 3724 3724 0
~
 ~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#@})!# b$@" *|: ;bP8&c0!*roots|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~S17 ? 16711680 ? ? (cos((2*k-1)/(2*n)*'p),0):(k=1...n):(8)~p0
1 ~gc-1 3 ? 0 3225 0 0
36862 0 0 0
65535 -3225 0 0
~Q
]|Expr|[#b @`bb#_b#_b#_})%# b$@" *|: ;bP8&c0!*axes and grids|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~R5592405 ? (x,y):(y=bottom...top):(x=left...right):(0)~p0
1 ~gc1 2 ? 0 -3724 -4096 0
65535 -3724 4096 0
~gc1 2 ? 0 -1862 -4096 0
65535 -1862 4096 0
~gc1 2 ? 0 0 -4096 0
65535 0 4096 0
~gc1 2 ? 0 1862 -4096 0
65535 1862 4096 0
~gc1 2 ? 0 3724 -4096 0
65535 3724 4096 0
~
 ~R5592405 ? (x,y):(x=left...right):(y=bottom...top):(0)~p0
1 ~gc1 2 ? 0 -4096 -3724 0
65535 4096 -3724 0
~gc1 2 ? 0 -4096 -1862 0
65535 4096 -1862 0
~gc1 2 ? 0 -4096 0 0
65535 4096 0 0
~gc1 2 ? 0 -4096 1862 0
65535 4096 1862 0
~gc1 2 ? 0 -4096 3724 0
65535 4096 3724 0
~
 ~X1 0 (x,0):(x=left...right):(x)~p0
1 ~gc1 2 ? 0 -4096 0 0
65535 4096 0 0
~X2 0 (0,y):(y=bottom...top):(~
 y)~p0
1 ~gc1 2 ? 0 0 -4096 0
65535 0 4096 0
~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$@" *|: ;bP8&c0!*declarations|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~V?c64 (left)~p0
1 ~V?c65 (right)~p0
1 ~V?c66 (bottom)~p0
1 ~V?c67 (top)~p0
1 ~t~p0
1 ~T~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$R" *|: ;bP8&c0!*Examples|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~Ht(T_?n(?x)):(cos(?n*arccos(?x)))~p0
1 ~A(T_4(x))~p1
0 ~A(T_4(x)=8*x^4-8*x^2+1)~p0
255 ~sb/_!!   }
"&!  c#T"!c"L$_c/__c/__}
^ _~A(T_4(x)=cos(4*arccos(~
 x)))~p0
1 ~sb/^!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(T_4(x)=x*cos(3*arccos(~
 x))-sqrt(-x^2+1)*sin(3*arccos(x)))~p0
2 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(T_4(x)=x*(x*cos(2*~
 arccos(x))-sqrt(-x^2+1)*sin(2*arccos(x)))-sqrt(-x^2+1)*(sqrt(~
 -x^2+1)*cos(2*arccos(x))+x*sin(2*arccos(x))))~p0
3 ~sb/_!!   }
#+!  c#T"!c"L" c"H"!c#L"_c/__c/__!}
#-!  c#T"!c"L"!c"T! c"H"!c#L"_c/__c/__#}
^ _~
 ~A(T_4(x)=x*(x*(2*x^2-1)-sqrt(-x^2+1)*(2*sqrt(-x^2+1)*x))-sqrt(~
 -x^2+1)*(sqrt(-x^2+1)*(2*x^2-1)+x*(2*sqrt(-x^2+1)*x)))~p0
4 ~sb/_!!   }
#/!  c#T"!c"L" c"H"!c"L" c"H"!c#L"_c/__c/__!}
#1!  c#T"!c"L" c"H"!c"L"!c"T! c"H"!c#L"_c/__c/__#}
#1!  c#T"!c"L"!c"T! c"H"!c"L" c"H"!c#L"_c/__c/__!}
#1!  c#T"!c"L"!c"T! c"H"!c"L"!c"H"!c#L"_c/__c/__#}
^ _~
 ~A(T_4(x)=-sqrt(-x^2+1)*(sqrt(-x^2+1)*(2*x^2-1)+2*sqrt(-x^2+1)*~
 x^2)+(-2*(-x^2+1)*x+(2*x^2-1)*x)*x)~p0
5 ~sb/_!!   }
&&!  c#T"!c"L"_c/__c/__}
^ _~A(T_4(x)=4*x^4-4*(-x^~
 2+1)*x^2-4*x^2+1)~p0
6 ~sb/_!!   }
"&!  c#T"!c"L"_c/__c/__}
^ _~A(T_6(x))~p1
0 ~A(T_6(x)=32*x^6-48*x^4+18*x^2-1)~p0
255 ~sb/_!!   }
"&!  c#T"!c"L%_c/__c/__}
^ _~A(T_6(x)=cos(6*arccos(~
 x)))~p0
1 ~sb/^!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(T_6(x)=x*cos(5*arccos(~
 x))-sqrt(-x^2+1)*sin(5*arccos(x)))~p0
2 ~sb/_!!   }
#'!  c#T"!c#L"_c/__c/__!}
^ _~A(T_6(x)=x*(x*cos(4*~
 arccos(x))-sqrt(-x^2+1)*sin(4*arccos(x)))-sqrt(-x^2+1)*(sqrt(~
 -x^2+1)*cos(4*arccos(x))+x*sin(4*arccos(x))))~p0
3 ~sb/_!!   }
#+!  c#T"!c"L" c"H"!c#L"_c/__c/__!}
#-!  c#T"!c"L"!c"T! c"H"!c#L"_c/__c/__#}
^ _~
 ~A(T_6(x)=-(-x^2+1)*cos(4*arccos(x))+x^2*cos(4*arccos(x))-2*sqrt(~
 -x^2+1)*x*sin(4*arccos(x)))~p0
4 ~sb/_!!   }
"&!  c#T"!c"L"_c/__c/__}
^ _~A(T_6(x)=-(-x^2+1)*(~
 x*cos(3*arccos(x))-sqrt(-x^2+1)*sin(3*arccos(x)))+x^2*(x*cos(~
 3*arccos(x))-sqrt(-x^2+1)*sin(3*arccos(x)))-2*sqrt(-x^2+1)*x*~
 (sqrt(-x^2+1)*cos(3*arccos(x))+x*sin(3*arccos(x))))~p0
5 ~sb/_!!   }
#-!  c#T"!c"L# c"T! c"H"!c#L"_c/__c/__!}
#+!  c#T"!c"L#!c"H"!c#L"_c/__c/__!}
#-!  c#T"!c"L#"c"T! c"H$#c#L"_c/__c/__#}
^ _~
 ~A(T_6(x)=2*x^3*cos(3*arccos(x))-x*cos(3*arccos(x))-2*(-x^2+1)*~
 x*cos(3*arccos(x))+sqrt(-x^2+1)*sin(3*arccos(x))-4*sqrt(-x^2+~
 1)*x^2*sin(3*arccos(x)))~p0
6 ~sb/_!!   }
"&!  c#T"!c"L#_c/__c/__}
^ _~A(T_6(x)=4*x^3*cos(3*~
 arccos(x))-3*x*cos(3*arccos(x))+sqrt(-x^2+1)*sin(3*arccos(x))-~
 4*sqrt(-x^2+1)*x^2*sin(3*arccos(x)))~p0
7 ~sb/_!!   }
"&!  c#T"!c"L%_c/__c/__}
^ _~A(T_6(x)=4*x^3*(x*cos(~
 2*arccos(x))-sqrt(-x^2+1)*sin(2*arccos(x)))-3*x*(x*cos(2*arccos(~
 x))-sqrt(-x^2+1)*sin(2*arccos(x)))+sqrt(-x^2+1)*(sqrt(-x^2+1)*~
 cos(2*arccos(x))+x*sin(2*arccos(x)))-4*sqrt(-x^2+1)*x^2*(sqrt(~
 -x^2+1)*cos(2*arccos(x))+x*sin(2*arccos(x))))~p0
8 ~sb/_!!   }
#+!  c#T"!c"L$ c"H#"c#L"_c/__c/__!}
#-!  c#T"!c"L$!c"T! c"H#"c#L"_c/__c/__!}
#+!  c#T"!c"L$"c"H"!c#L"_c/__c/__#}
#-!  c#T"!c"L$#c"T! c"H$#c#L"_c/__c/__#}
^ _~
 ~A(T_6(x)=(-x^2+1)*cos(2*arccos(x))+4*x^4*cos(2*arccos(x))-3*~
 x^2*cos(2*arccos(x))-4*(-x^2+1)*x^2*cos(2*arccos(x))-8*sqrt(-~
 x^2+1)*x^3*sin(2*arccos(x))+4*sqrt(-x^2+1)*x*sin(2*arccos(x)))~p0
9 ~sb/_!!   }
"&!  c#T"!c"L$_c/__c/__}
^ _~A(T_6(x)=cos(2*arccos(~
 x))+8*x^4*cos(2*arccos(x))-8*x^2*cos(2*arccos(x))-8*sqrt(-x^2+~
 1)*x^3*sin(2*arccos(x))+4*sqrt(-x^2+1)*x*sin(2*arccos(x)))~p0
10 ~sb/_!!   }
"&!  c#T"!c"L&_c/__c/__}
^ _~A(T_6(x)=(2*x^2-1)+8*~
 x^4*(2*x^2-1)-8*x^2*(2*x^2-1)-8*sqrt(-x^2+1)*x^3*(2*sqrt(-x^2+~
 1)*x)+4*sqrt(-x^2+1)*x*(2*sqrt(-x^2+1)*x))~p0
11 ~sb/_!!   }
#)!  c#T"!c"L% c#L"_c/__c/__!}
#+!  c#T"!c"L%!c"H#"c#L"_c/__c/__!}
#-!  c#T"!c"L%"c"T! c"H#"c#L"_c/__c/__!}
#-!  c#T"!c"L%#c"T! c"H$#c#L"_c/__c/__#}
#+!  c#T"!c"L%$c"H$#c#L"_c/__c/__#}
^ _~t~p0
1 ~c1
3 62 -1 9 62 -1 ~c2
4 62 -1 2 62 -1 1 62 -1 ~c3
5 62 -1 4 62 -1 28 -1 44 -1 ~c4
6 62 -1 5 62 -1 28 -1 44 -1 27 -1 ~c4
7 62 -1 6 62 -1 28 -1 44 -1 27 -1 ~c1
8 62 -1 7 62 -1 ~c1
9 62 -1 8 62 -1 ~c1
11 62 -1 22 62 -1 ~c2
12 62 -1 10 62 -1 1 62 -1 ~c3
13 62 -1 12 62 -1 28 -1 44 -1 ~c4
14 62 -1 13 62 -1 28 -1 44 -1 27 -1 ~c1
15 62 -1 14 62 -1 ~c4
16 62 -1 15 62 -1 28 -1 44 -1 27 -1 ~c1
17 62 -1 16 62 -1 ~c1
18 62 -1 17 62 -1 ~c4
19 62 -1 18 62 -1 28 -1 44 -1 27 -1 ~c1
20 62 -1 19 62 -1 ~c1
21 62 -1 20 62 -1 ~c4
22 62 -1 21 62 -1 28 -1 44 -1 27 -1 ~e