SIN,COS倍角式

SIN(X+Y)=SIN(X)*COS(Y)+COS(X)*SIN(Y)
COS(X+Y)=COS(X)*COS(Y)-SIN(X)*SIN(Y)
を使っています

PUBLIC NUMERIC MAXLEVEL
OPTION BASE 0
LET MAXLEVEL=20
DIM COSA(MAXLEVEL),COSB(MAXLEVEL),SINA(MAXLEVEL,MAXLEVEL),SINB(MAXLEVEL,MAXLEVEL)
DIM COSINE(MAXLEVEL),SINE(MAXLEVEL,MAXLEVEL)
LET COSA(1)=1
LET SINA(1,0)=1
LET COSB(1)=1
LET SINB(1,0)=1
FOR F=2 TO MAXLEVEL
CALL COSINEX(MIN(F*2,MAXLEVEL),COSA,SINA,COSB,SINB,COSINE)
PRINT "COS(";STR$(F);"X)=";
CALL DISPLAYCOS(F,COSINE)
PRINT
CALL SINEX(MIN(F*2,MAXLEVEL),COSA,SINA,COSB,SINB,SINE)
PRINT "SIN(";STR$(F);"X)=";
CALL DISPLAYSIN(F,SINE)
PRINT
FOR I=0 TO F
LET COSB(I)=COSINE(I)
LET COSINE(I)=0
FOR J=0 TO F
LET SINB(I,J)=SINE(I,J)
LET SINE(I,J)=0
NEXT J
NEXT I
NEXT F
END

EXTERNAL SUB DISPLAYSIN(F,SINE(,))
FOR I=F TO 0 STEP -1
FOR J=F TO 0 STEP -1
IF SINE(I,J)<>0 THEN
IF FL=1 THEN
IF SINE(I,J)<0 THEN PRINT "-"; ELSE PRINT "+";
END IF
IF FL=0 THEN
IF SINE(I,J)<0 THEN PRINT "-";
LET FL=1
END IF
IF I>1 THEN
PRINT STR$(ABS(SINE(I,J)));"*SIN(X)^";STR$(I);
ELSEIF I=1 THEN
PRINT STR$(ABS(SINE(I,J)));"*SIN(X)";
END IF
IF J>1 THEN
PRINT "*COS(X)^";STR$(J);
ELSEIF J=1 THEN
PRINT "*COS(X)";
END IF
END IF
NEXT J
NEXT I
END SUB

EXTERNAL SUB DISPLAYCOS(K,COSINE())
PRINT STR$(COSINE(K));"*COS(X)^";STR$(K);
FOR I=K-1 TO 1 STEP -1
IF COSINE(I)<>0 THEN
IF COSINE(I)<0 THEN PRINT "-"; ELSE PRINT "+";
IF I>1 THEN
PRINT STR$(ABS(COSINE(I)));"*COS(X)^";STR$(I);
ELSE
PRINT STR$(ABS(COSINE(I)));"*COS(X)";
END IF
END IF
NEXT I
IF COSINE(0)<>0 THEN
IF COSINE(0)<0 THEN PRINT "-"; ELSE PRINT "+";
PRINT STR$(ABS(COSINE(0)));
END IF
END SUB

EXTERNAL SUB COSINEX(F,COSA(),SINA(,),COSB(),SINB(,),COSINE())
OPTION BASE 0
DIM SINE(MAXLEVEL,MAXLEVEL),C(2)
LET C(2)=-1
LET C(0)=1
FOR I=0 TO F
FOR J=0 TO F-I
LET COSINE(I+J)=COSINE(I+J)+COSA(I)*COSB(J)
NEXT J
NEXT I
FOR M=0 TO F
FOR N=0 TO F-M
FOR I=0 TO F
FOR J=0 TO F-I
LET SINE(M+N,I+J)=SINE(M+N,I+J)+SINA(M,I)*SINB(N,J)
NEXT J
NEXT I
NEXT N
NEXT M
FOR L=F TO 2 STEP -2
FOR I=0 TO F-2
FOR J=0 TO 2
LET SINE(L-2,I+J)=SINE(L-2,I+J)+SINE(L,I)*C(J)
NEXT J
NEXT I
NEXT L
FOR K=0 TO F
LET COSINE(K)=COSINE(K)-SINE(0,K)
NEXT K
END SUB

EXTERNAL SUB SINEX(F,COSA(),SINA(,),COSB(),SINB(,),SINE(,))
OPTION BASE 0
DIM S(2)
LET S(2)=-1
LET S(0)=1
FOR M=0 TO F
FOR I=0 TO F
FOR J=0 TO F-I
LET SINE(M,I+J)=SINE(M,I+J)+SINA(M,I)*COSB(J)+COSA(J)*SINB(M,I)
NEXT J
NEXT I
NEXT M
FOR L=F TO 2 STEP -1
FOR I=0 TO F-2
FOR J=0 TO 2
LET SINE(I+J,L-2)=SINE(I+J,L-2)+SINE(I,L)*S(J)
NEXT J
LET SINE(I,L)=0
NEXT I
NEXT L
END SUB

!チェビシェフの多項式Tn(x)からｎ倍角の公式を求める

!ｎ倍角の公式
!http://www10.plala.or.jp/rascalhp/math.htm#5

!●cosの場合

!Tk(x)=cos(k*θ)、k=0,1,2,…
!x=cosθ

!漸化式 Tn+2 - 2*x*Tn+1 + Tn = 0
!　第１項　T0=1
!　第２項　T1=x
!が成立する

!漸化式 An+2 - 2*cosθ*An+1 + An = 0
!　第１項　A0=1
!　第２項　A1=cosθ
!となる
!
!A2=2*cosθ*A1-A0
! =2*cosθ*cosθ-1
!A3=2*cosθ*A2-A1
! =2*cosθ*(2*cosθ^2-1)-cosθ
! =4*cosθ^3-3*cosθ
!A4=2*cosθ*A3-A2
! =2*cosθ*(4*cosθ^3-3*cosθ)-(2*cosθ^2-1)
! =8cosθ^4-8cosθ^2+1
! :
! :

OPTION BASE 0

LET N=20 !次数
DIM c0(N),c1(N) !１変数多項式の次数nの係数

MAT c0=ZER !第１項
LET c0(0)=1 !1*cosθ^0

MAT c1=ZER !第２項
LET c1(1)=1 !1*cosθ

LET k=10 !第k項　※1～N

DIM c2(N)
FOR i=1 TO k-1 !An+2 = (2*cosθ)*An+1 - An
MAT c2=ZER
FOR J=N TO 1 STEP -1 !2*cosθ*A1
LET c2(J)=2*c1(J-1)
NEXT J
MAT c2=c2-c0 !-A0

MAT c0=c1 !次へ
MAT c1=c2
NEXT i

PRINT "cos k=";k
FOR i=N TO 1 STEP -1
IF c1(i)<>0 THEN PRINT c1(i);"* cosθ ^";i
NEXT i
IF c1(0)<>0 THEN PRINT c1(i)
PRINT

!●sinの場合

!sin(n*θ)=Sn*sinθ
!(Tn(x))'=n*Sn(x)　※チェビシェフの多項式Tn(x)

!T1(x)=x、S1(x)=1
!T2(x)=2*x^2-1、S2(x)=2*x
!T3(x)=4*x^3-3*x、S3(x)=4*x^2-1
!T4(x)=8*x^4-8*x^2+1、S4(x)=8*x^3-4*x

!sin2θ=S2*sinθ
! =(2cosθ)sinθ
!sin3θ=S3*sinθ
! =(4cosθ^2-1)sinθ　※cos優先
! =(4*(1-sinθ^2)-1)sinθ　※sin優先へ
! =-4sinθ^3+3sinθ
!sin4θ=S4*sinθ
! =(8cosθ^3-4cosθ)sinθ　※cos優先
! =(8(1-sinθ^2)cosθ-4cosθ)sinθ　※sin優先へ
! =-8sinθ^3cosθ+4sinθcosθ
! :
! :

DIM s(N)
MAT s=ZER

FOR i=1 TO N !Sn(x)=Tn(x)'/n
LET s(i-1)=c1(i)*i/k
NEXT i

PRINT "sin k=";k
FOR i=N TO 1 STEP -1
IF s(i)<>0 THEN PRINT s(i);"* cosθ ^";i;" * sinθ"
NEXT i
IF s(0)<>0 THEN PRINT s(i);" * sinθ"

END

山中和義さんご返事ありがとうございます。
大変参考になります。

TAN倍角を求めるプログラムをのせておきます。

OPTION BASE 0
PUBLIC NUMERIC MAXLEVEL
LET MAXLEVEL=20
DIM A(MAXLEVEL),B(MAXLEVEL),AA(MAXLEVEL),BB(MAXLEVEL),C(1),D(1)
!' B D
!' ---+---
!' A C C*B+A*D
!'---------- = ---------
!' B D A*C-B*D
!'1- ---*---
!' A C
LET C(0)=1
LET D(1)=1
LET B(1)=1
LET A(0)=1
FOR K=2 TO MAXLEVEL
FOR J=0 TO 1
FOR I=0 TO MAXLEVEL-J
LET AA(I+J)=AA(I+J)+A(I)*C(J)-B(I)*D(J)
LET BB(I+J)=BB(I+J)+B(I)*C(J)+A(I)*D(J)
NEXT I
NEXT J
FOR I=0 TO MAXLEVEL
LET A(I)=AA(I)
LET B(I)=BB(I)
LET AA(I)=0
LET BB(I)=0
NEXT I
PRINT "TAN(";STR$(K);"X)=";
CALL DISPLAY(B)
PRINT "/";
CALL DISPLAY(A)
PRINT
NEXT K
END

EXTERNAL SUB DISPLAY(B())
FOR JJ=MAXLEVEL TO 0 STEP -1
IF B(JJ)<>0 THEN EXIT FOR
NEXT JJ
PRINT "(";
IF ABS(B(JJ))<>1 AND JJ>1 THEN
PRINT STR$(B(JJ));"*TAN(X)^";STR$(JJ);
ELSEIF ABS(B(JJ))=1 AND JJ>1 THEN
IF B(JJ)<0 THEN PRINT "-";
PRINT "TAN(X)^";STR$(JJ);
END IF
FOR J=JJ-1 TO 2 STEP -1
IF B(J)<>0 THEN
IF B(J)<0 THEN PRINT "-"; ELSE PRINT "+";
IF B(J)<>1 THEN
PRINT STR$(ABS(B(J)));"*TAN(X)^";STR$(J);
ELSEIF B(J)=1 THEN
PRINT "TAN(X)^";STR$(J);
END IF
END IF
NEXT J
IF B(1)<>0 THEN
IF JJ>1 THEN
IF B(1)<0 THEN PRINT "-"; ELSE PRINT "+";
END IF
IF B(1)<>1 THEN
PRINT STR$(ABS(B(1)));"*TAN(X)";
ELSEIF B(1)=1 THEN
PRINT "TAN(X)";
END IF
END IF
IF B(0)<>0 THEN
IF B(0)<0 THEN PRINT "-"; ELSE PRINT "+";
PRINT STR$(ABS(B(0)));
END IF
PRINT ")";
END SUB

!tanのｎ倍角の公式

!１変数Xの多項式（N次）のCi*X^iの係数を、配列C(i)=Ciで表す。

OPTION BASE 0

LET N=20 !次数

!演算関連
SUB poly_add(n,a(),b(), x()) !加算x=a+b
MAT x=a+b
END SUB
SUB poly_sub(n,a(),b(), x()) !減算x=a-b
MAT x=a-b
END SUB
SUB poly_mul(n,a(),b(), x()) !乗算x=a*b
DIM xx(100)
MAT xx=ZER(2*n)
FOR i=n TO 0 STEP -1
FOR j=n TO 0 STEP -1
LET xx(i+j)=xx(i+j)+a(i)*b(j) !分配する
NEXT j
NEXT i
MAT x=xx !copy it
END SUB

!表示関連
SUB disp_poly(n,a(),x$) !多項式を表示する　a(X)=ΣAkX^k=AnX^n+An-1X^n-1+…+A1X+A0
CALL disp_mono(n,a(n),x$)
FOR i=n-1 TO 0 STEP -1 !次数の降順
IF a(i)>0 THEN PRINT "+";
CALL disp_mono(i,a(i),x$)
NEXT i
PRINT
END SUB
SUB disp_mono(k,ak,x$) !単項式を表示する　AkX^k
IF ak=0 THEN !係数が０なら
ELSE
IF k<>0 THEN !x^nで
IF ak=1 THEN !係数が１なら
ELSEIF ak=-1 THEN !係数が－１なら
PRINT "-";
ELSE
PRINT STR$(ak);
END IF
END IF
IF k=0 THEN !次数が０なら
PRINT STR$(ak);
ELSEIF k=1 THEN !次数が１なら
PRINT x$;
ELSE
PRINT x$;"^";STR$(k);
END IF
END IF
END SUB

!その他
DIM c1(2*N)
LET c1(0)=1 !定数１
!------------------------------ ここまでがサブルーチン

つづき

!２倍角の公式
!●tan2θ=(tanθ+tanθ)/(1-tanθtanθ)

!●tanθ=(tan０+tanθ)/(1-tan０tanθ)=t
!●tan2θ=(tanθ+tanθ)/(1-tanθtanθ)=2*t/(1-t^2)
!●tan3θ=(tan2θ+tanθ)/(1-tan2θtanθ)=(3*t-t^3)/(1-3*t^2)
!●tan4θ=(tan3θ+tanθ)/(1-tan3θtanθ)=(4*t-4*t^3)/(1-6*t^2+t^4)
!　　:
!　　:

DIM t(2*N)
LET t(1)=1 !tanθの定義

DIM P00(2*N),Q00(2*N) !１つ前の分子、分母　tan(w-1)θ=P/Q
LET P00(0)=0 !tan０
LET Q00(0)=1

DIM P99(2*N),Q99(2*N) !分子、分母
FOR w=1 TO N
PRINT "tan";w;"θ="

!分子の部
DIM T1(2*N)
CALL poly_mul(N,Q00,t, T1) !tan(w-1)θ+tanθ=P/Q+t=(P+Qt)/Q
CALL poly_add(N,T1,P00, P99) !通分して分子のみ　※分母は同じため約せる
CALL disp_poly(n,P99,"tanθ")

PRINT REPEAT$("-",w*7)

!分母の部
CALL poly_mul(N,P00,t, T1) !1-tan(w-1)θtanθ=1-P/Q*t=(Q-P*t)/Q
CALL poly_sub(N,Q00,T1, Q99)
CALL disp_poly(n,Q99,"tanθ")
PRINT

MAT P00=P99 !次へ
MAT Q00=Q99
NEXT w

END

なお、sin、cosのｎ倍角の公式は加法定理を使って、
２変数（sinθ、cosθ）の多項式で導出できます。

SIN,COS倍角式	返事を書くノートメニュー
しばっち <dihjvcfsyu> 2008/02/08 20:45:02
SIN(X+Y)=SIN(X)COS(Y)+COS(X)SIN(Y) COS(X+Y)=COS(X)COS(Y)-SIN(X)SIN(Y) を使っています PUBLIC NUMERIC MAXLEVEL OPTION BASE 0 LET MAXLEVEL=20 DIM COSA(MAXLEVEL),COSB(MAXLEVEL),SINA(MAXLEVEL,MAXLEVEL),SINB(MAXLEVEL,MAXLEVEL) DIM COSINE(MAXLEVEL),SINE(MAXLEVEL,MAXLEVEL) LET COSA(1)=1 LET SINA(1,0)=1 LET COSB(1)=1 LET SINB(1,0)=1 FOR F=2 TO MAXLEVEL CALL COSINEX(MIN(F2,MAXLEVEL),COSA,SINA,COSB,SINB,COSINE) PRINT "COS(";STR$(F);"X)="; CALL DISPLAYCOS(F,COSINE) PRINT CALL SINEX(MIN(F2,MAXLEVEL),COSA,SINA,COSB,SINB,SINE) PRINT "SIN(";STR$(F);"X)="; CALL DISPLAYSIN(F,SINE) PRINT FOR I=0 TO F LET COSB(I)=COSINE(I) LET COSINE(I)=0 FOR J=0 TO F LET SINB(I,J)=SINE(I,J) LET SINE(I,J)=0 NEXT J NEXT I NEXT F END EXTERNAL SUB DISPLAYSIN(F,SINE(,)) FOR I=F TO 0 STEP -1 FOR J=F TO 0 STEP -1 IF SINE(I,J)<>0 THEN IF FL=1 THEN IF SINE(I,J)<0 THEN PRINT "-"; ELSE PRINT "+"; END IF IF FL=0 THEN IF SINE(I,J)<0 THEN PRINT "-"; LET FL=1 END IF IF I>1 THEN PRINT STR$(ABS(SINE(I,J)));"SIN(X)^";STR$(I); ELSEIF I=1 THEN PRINT STR$(ABS(SINE(I,J)));"SIN(X)"; END IF IF J>1 THEN PRINT "COS(X)^";STR$(J); ELSEIF J=1 THEN PRINT "COS(X)"; END IF END IF NEXT J NEXT I END SUB EXTERNAL SUB DISPLAYCOS(K,COSINE()) PRINT STR$(COSINE(K));"COS(X)^";STR$(K); FOR I=K-1 TO 1 STEP -1 IF COSINE(I)<>0 THEN IF COSINE(I)<0 THEN PRINT "-"; ELSE PRINT "+"; IF I>1 THEN PRINT STR$(ABS(COSINE(I)));"COS(X)^";STR$(I); ELSE PRINT STR$(ABS(COSINE(I)));"COS(X)"; END IF END IF NEXT I IF COSINE(0)<>0 THEN IF COSINE(0)<0 THEN PRINT "-"; ELSE PRINT "+"; PRINT STR$(ABS(COSINE(0))); END IF END SUB EXTERNAL SUB COSINEX(F,COSA(),SINA(,),COSB(),SINB(,),COSINE()) OPTION BASE 0 DIM SINE(MAXLEVEL,MAXLEVEL),C(2) LET C(2)=-1 LET C(0)=1 FOR I=0 TO F FOR J=0 TO F-I LET COSINE(I+J)=COSINE(I+J)+COSA(I)COSB(J) NEXT J NEXT I FOR M=0 TO F FOR N=0 TO F-M FOR I=0 TO F FOR J=0 TO F-I LET SINE(M+N,I+J)=SINE(M+N,I+J)+SINA(M,I)SINB(N,J) NEXT J NEXT I NEXT N NEXT M FOR L=F TO 2 STEP -2 FOR I=0 TO F-2 FOR J=0 TO 2 LET SINE(L-2,I+J)=SINE(L-2,I+J)+SINE(L,I)C(J) NEXT J NEXT I NEXT L FOR K=0 TO F LET COSINE(K)=COSINE(K)-SINE(0,K) NEXT K END SUB

Re: SIN,COS倍角式	返事を書くノートメニュー
しばっち <dihjvcfsyu> 2008/02/08 20:45:37
EXTERNAL SUB SINEX(F,COSA(),SINA(,),COSB(),SINB(,),SINE(,)) OPTION BASE 0 DIM S(2) LET S(2)=-1 LET S(0)=1 FOR M=0 TO F FOR I=0 TO F FOR J=0 TO F-I LET SINE(M,I+J)=SINE(M,I+J)+SINA(M,I)COSB(J)+COSA(J)SINB(M,I) NEXT J NEXT I NEXT M FOR L=F TO 2 STEP -1 FOR I=0 TO F-2 FOR J=0 TO 2 LET SINE(I+J,L-2)=SINE(I+J,L-2)+SINE(I,L)*S(J) NEXT J LET SINE(I,L)=0 NEXT I NEXT L END SUB

Re: EXTERNALSUBSINEX(F,COSA(),SINA(,),COSB()...	返事を書くノートメニュー
山中和義 <drdlxujciw> 2008/02/09 15:09:21
!チェビシェフの多項式Tn(x)からｎ倍角の公式を求める !ｎ倍角の公式 !http://www10.plala.or.jp/rascalhp/math.htm#5 !●cosの場合 !Tk(x)=cos(kθ)、k=0,1,2,… !x=cosθ !漸化式 Tn+2 - 2xTn+1 + Tn = 0 !　第１項　T0=1 !　第２項　T1=x !が成立する !漸化式 An+2 - 2cosθAn+1 + An = 0 !　第１項　A0=1 !　第２項　A1=cosθ !となる ! !A2=2cosθA1-A0 ! =2cosθcosθ-1 !A3=2cosθA2-A1 ! =2cosθ(2cosθ^2-1)-cosθ ! =4cosθ^3-3cosθ !A4=2cosθA3-A2 ! =2cosθ(4cosθ^3-3cosθ)-(2cosθ^2-1) ! =8cosθ^4-8cosθ^2+1 ! : ! : OPTION BASE 0 LET N=20 !次数 DIM c0(N),c1(N) !１変数多項式の次数nの係数 MAT c0=ZER !第１項 LET c0(0)=1 !1cosθ^0 MAT c1=ZER !第２項 LET c1(1)=1 !1cosθ LET k=10 !第k項　※1～N DIM c2(N) FOR i=1 TO k-1 !An+2 = (2cosθ)An+1 - An MAT c2=ZER FOR J=N TO 1 STEP -1 !2cosθA1 LET c2(J)=2c1(J-1) NEXT J MAT c2=c2-c0 !-A0 MAT c0=c1 !次へ MAT c1=c2 NEXT i PRINT "cos k=";k FOR i=N TO 1 STEP -1 IF c1(i)<>0 THEN PRINT c1(i);"* cosθ ^";i NEXT i IF c1(0)<>0 THEN PRINT c1(i) PRINT !●sinの場合 !sin(nθ)=Snsinθ !(Tn(x))'=nSn(x)　※チェビシェフの多項式Tn(x) !T1(x)=x、S1(x)=1 !T2(x)=2x^2-1、S2(x)=2x !T3(x)=4x^3-3x、S3(x)=4x^2-1 !T4(x)=8x^4-8x^2+1、S4(x)=8x^3-4x !sin2θ=S2sinθ ! =(2cosθ)sinθ !sin3θ=S3sinθ ! =(4cosθ^2-1)sinθ　※cos優先 ! =(4(1-sinθ^2)-1)sinθ　※sin優先へ ! =-4sinθ^3+3sinθ !sin4θ=S4sinθ ! =(8cosθ^3-4cosθ)sinθ　※cos優先 ! =(8(1-sinθ^2)cosθ-4cosθ)sinθ　※sin優先へ ! =-8sinθ^3cosθ+4sinθcosθ ! : ! : DIM s(N) MAT s=ZER FOR i=1 TO N !Sn(x)=Tn(x)'/n LET s(i-1)=c1(i)i/k NEXT i PRINT "sin k=";k FOR i=N TO 1 STEP -1 IF s(i)<>0 THEN PRINT s(i);" cosθ ^";i;" * sinθ" NEXT i IF s(0)<>0 THEN PRINT s(i);" * sinθ" END

Re: !チェビシェフの多項式Tn(x)からｎ倍角の公...	返事を書くノートメニュー
しばっち <dihjvcfsyu> 2008/02/10 17:34:35 ** この記事は1回修正されてます
山中和義さんご返事ありがとうございます。大変参考になります。 TAN倍角を求めるプログラムをのせておきます。 OPTION BASE 0 PUBLIC NUMERIC MAXLEVEL LET MAXLEVEL=20 DIM A(MAXLEVEL),B(MAXLEVEL),AA(MAXLEVEL),BB(MAXLEVEL),C(1),D(1) !' B D !' ---+--- !' A C CB+AD !'---------- = --------- !' B D AC-BD !'1- ------ !' A C LET C(0)=1 LET D(1)=1 LET B(1)=1 LET A(0)=1 FOR K=2 TO MAXLEVEL FOR J=0 TO 1 FOR I=0 TO MAXLEVEL-J LET AA(I+J)=AA(I+J)+A(I)C(J)-B(I)D(J) LET BB(I+J)=BB(I+J)+B(I)C(J)+A(I)D(J) NEXT I NEXT J FOR I=0 TO MAXLEVEL LET A(I)=AA(I) LET B(I)=BB(I) LET AA(I)=0 LET BB(I)=0 NEXT I PRINT "TAN(";STR$(K);"X)="; CALL DISPLAY(B) PRINT "/"; CALL DISPLAY(A) PRINT NEXT K END EXTERNAL SUB DISPLAY(B()) FOR JJ=MAXLEVEL TO 0 STEP -1 IF B(JJ)<>0 THEN EXIT FOR NEXT JJ PRINT "("; IF ABS(B(JJ))<>1 AND JJ>1 THEN PRINT STR$(B(JJ));"TAN(X)^";STR$(JJ); ELSEIF ABS(B(JJ))=1 AND JJ>1 THEN IF B(JJ)<0 THEN PRINT "-"; PRINT "TAN(X)^";STR$(JJ); END IF FOR J=JJ-1 TO 2 STEP -1 IF B(J)<>0 THEN IF B(J)<0 THEN PRINT "-"; ELSE PRINT "+"; IF B(J)<>1 THEN PRINT STR$(ABS(B(J)));"TAN(X)^";STR$(J); ELSEIF B(J)=1 THEN PRINT "TAN(X)^";STR$(J); END IF END IF NEXT J IF B(1)<>0 THEN IF JJ>1 THEN IF B(1)<0 THEN PRINT "-"; ELSE PRINT "+"; END IF IF B(1)<>1 THEN PRINT STR$(ABS(B(1)));"TAN(X)"; ELSEIF B(1)=1 THEN PRINT "TAN(X)"; END IF END IF IF B(0)<>0 THEN IF B(0)<0 THEN PRINT "-"; ELSE PRINT "+"; PRINT STR$(ABS(B(0))); END IF PRINT ")"; END SUB

Re: SIN,COS倍角式	返事を書くノートメニュー
山中和義 <drdlxujciw> 2008/02/10 17:34:41 ** この記事は1回修正されてます
!tanのｎ倍角の公式 !１変数Xの多項式（N次）のCiX^iの係数を、配列C(i)=Ciで表す。 OPTION BASE 0 LET N=20 !次数 !演算関連 SUB poly_add(n,a(),b(), x()) !加算x=a+b MAT x=a+b END SUB SUB poly_sub(n,a(),b(), x()) !減算x=a-b MAT x=a-b END SUB SUB poly_mul(n,a(),b(), x()) !乗算x=ab DIM xx(100) MAT xx=ZER(2n) FOR i=n TO 0 STEP -1 FOR j=n TO 0 STEP -1 LET xx(i+j)=xx(i+j)+a(i)b(j) !分配する NEXT j NEXT i MAT x=xx !copy it END SUB !表示関連 SUB disp_poly(n,a(),x$) !多項式を表示する　a(X)=ΣAkX^k=AnX^n+An-1X^n-1+…+A1X+A0 CALL disp_mono(n,a(n),x$) FOR i=n-1 TO 0 STEP -1 !次数の降順 IF a(i)>0 THEN PRINT "+"; CALL disp_mono(i,a(i),x$) NEXT i PRINT END SUB SUB disp_mono(k,ak,x$) !単項式を表示する　AkX^k IF ak=0 THEN !係数が０なら ELSE IF k<>0 THEN !x^nで IF ak=1 THEN !係数が１なら ELSEIF ak=-1 THEN !係数が－１なら PRINT "-"; ELSE PRINT STR$(ak); END IF END IF IF k=0 THEN !次数が０なら PRINT STR$(ak); ELSEIF k=1 THEN !次数が１なら PRINT x$; ELSE PRINT x$;"^";STR$(k); END IF END IF END SUB !その他 DIM c1(2*N) LET c1(0)=1 !定数１ !------------------------------ ここまでがサブルーチン

Re: !tanのｎ倍角の公式	返事を書くノートメニュー
山中和義 <drdlxujciw> 2008/02/10 17:35:22 ** この記事は1回修正されてます
つづき !２倍角の公式 !●tan2θ=(tanθ+tanθ)/(1-tanθtanθ) !●tanθ=(tan０+tanθ)/(1-tan０tanθ)=t !●tan2θ=(tanθ+tanθ)/(1-tanθtanθ)=2t/(1-t^2) !●tan3θ=(tan2θ+tanθ)/(1-tan2θtanθ)=(3t-t^3)/(1-3t^2) !●tan4θ=(tan3θ+tanθ)/(1-tan3θtanθ)=(4t-4t^3)/(1-6t^2+t^4) !　　: !　　: DIM t(2N) LET t(1)=1 !tanθの定義 DIM P00(2N),Q00(2N) !１つ前の分子、分母　tan(w-1)θ=P/Q LET P00(0)=0 !tan０ LET Q00(0)=1 DIM P99(2N),Q99(2N) !分子、分母 FOR w=1 TO N PRINT "tan";w;"θ=" !分子の部 DIM T1(2N) CALL poly_mul(N,Q00,t, T1) !tan(w-1)θ+tanθ=P/Q+t=(P+Qt)/Q CALL poly_add(N,T1,P00, P99) !通分して分子のみ　※分母は同じため約せる CALL disp_poly(n,P99,"tanθ") PRINT REPEAT$("-",w7) !分母の部 CALL poly_mul(N,P00,t, T1) !1-tan(w-1)θtanθ=1-P/Qt=(Q-P*t)/Q CALL poly_sub(N,Q00,T1, Q99) CALL disp_poly(n,Q99,"tanθ") PRINT MAT P00=P99 !次へ MAT Q00=Q99 NEXT w END なお、sin、cosのｎ倍角の公式は加法定理を使って、２変数（sinθ、cosθ）の多項式で導出できます。