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MP3 ? ? ? ? ? 1 1 gbk

8

Trigonometric 60 n

y=sinx 60 n

y=sinx B AC A C AB BC ACB AB AC BAC

RtABC

adjacentb=AC oppositea=BC hypotenuseh=AB adjacentb=AC sin a/h Sine A cos b/h Cosine A tan a/b Tangent A cot b/a Cotangent A sec h/b Secant A csc h/a Cosecant A tancot tgctg

versin

1 0

versin=1-cos vercosin=1+cos coversin=1-sin covercosin=1+sin haversin=(1-cos)/2 havercosin=(1+cos)/2 hacoversin=(1-sin)/2 hacovercosin=(1+sin)/2 exsec=sec-1 excsc=csc-1

/2

x^2+y^2=1 x x y cos sin 1 sin = y/1 cos = x/1 1 2 2 2 k 2 360 180

k (k + 1/2) = (k + 1/2) (k + 1/2) (k + 1/2)

O AB sin AC cos OCversin =1-cos CDtan A AE cot AF sec =OE csc =OF OA A DE exsec = sec-1 /2 x

Un n / Bn n

O P M P x S(1,0) O x S O l MP OM OP l T ST 1 A sincostancot sec csc A sin cos tan cot sec) (csc) 2 sin(90-)=cos cos(90-)=sin, tan(90-)=cot cot(90-)=tan 3 sinA/cosA=tanA sin^2(A)+cos^2(A)=1 sinA=tanAcosA cosA=cotAsinA cotA=cosAcscA tanAcotA=1 ABC , A A A ,

4. 1 2090 3 i ii 090 iii 0A90 0sin1, 1cosA0, 00. A 0 30 45 60 90 sinA 0 1/2 2/2 3/2 1 cosA 1 3/2 2/2 1/2 0 tanA 0 3/3 1 3 None cotA None 3 1 3/3 0 Trigonometry Trigonometrie Trigonometrie Trigonometria Trigonometry ( Bartholomeo Pitiscus,1516-1613) 1595 () ()

()(ABC)(AC) ? AB ()AC ABC

1464 1748 P OM MP()( PM OP )sin=MP/OPcos=OM/OPtan= MP/OM

(AC)(AD) AC AOC ( ) (AB)(AB) AB (AC) dschaib sinus 4 (1631 ) sinus ACACAC (Hipparchus 180~ 125) AB AB ( ) 60 360 60 60 60

partes minutae primaepartes minutae secundaeminutesecond 60o (1/6 ) 60o 60 ( 1/60 ) 120o 90o 72o () 60 60 360 1 1 60 1 60 ,1 60 1 60 60 1854 (Edward Hincks1792-1866) , 2300-1600 Edward Hincks 60 (1) M.(Moritz Benedikt Cantor,1829-1920) 360 360 6 , 60 60 , 365 (12 ) 354 355 , (2)60 234561012 1/21/31/4 1/5, , (3)(G.Kewitsch) 1904 ,1 10 10 6 60 10 , 6 5 1 5 66 , 60 , ( 16-11 ),60 , ,

* 45 60 30

sin^2()+cos^2()=1 cos(2)=cos^2()-sin^2()=1- 2sin^2()=2cos^2()-1 sin(2)=2sin()cos() tan^()+1=1/cos^() 2sin^()=1-cos(2) cot^()+1=1/sin^() sin=tancos cos=cotsin tan=sinsec cot=coscsc sec=tancsc csc=seccot tan cot=1 sin csc=1 cos sec=1 sin/cos=tan=sec/csc cos/sin=cot=csc/sec 180 - y - x 180 +

90 - y=x sin2k+=sin cos2k+=cos tan2k+=tan k cot2k+=cot sec2k+=sec csc2k+=csc sin+=sin + cos+=cos tan+=tan cot+=cot sec(+)=-sec csc(+)=-csc sin=sin - cos=cos tan=tan cot=cot sec(-)=sec csc(-)=-csc sin=sin - cos=cos tan=tan cot=cot sec(-)=-sec csc(-)=csc sin2=sin 2- cos2=cos tan2=tan cot2=cot sec(2-)=sec csc(2-)=-csc sin/2+=cos /2 3/2 cos/2+=sin tan/2+=cot cot/2+=tan sec(/2+)=-csc csc(/2+)=sec sin/2=cos cos/2=sin tan/2=cot cot/2=tan sec(/2-)=csc csc(/2-)=sec sin3/2+=cos cos3/2+=sin

tan3/2+=cot cot3/2+=tan sec(3/2+)=csc csc(3/2+)=-sec sin3/2=cos cos3/2=sin tan3/2=cot cot3/2=tan sec(3/2-)=-csc csc(3/2-)=-sec sin cos tan cot sec csc 2k+ sin cos tan cot sec csc (1/2)k- cos sin cot tan csc sec (1/2)k+ cos -sin -cot -tan -csc sec k- sin -cos -tan -cot -sec csc k+ -sin -cos tan cot -sec -csc (3/2)k- -cos -sin cot tan -csc -sec (3/2)k+ -cos sin -cot -tan csc -sec 2k- -sin cos -tan -cot sec -csc -sin cos -tan -cot sec -csc 90+ 90+ ) 2 K/ K sin cos tan sin x cos y tan 90+90 90 90+ sin(90+)=cos , cos(90+)=-sin ~ sin(90+)90 cos 90+ sin(90+)=cos

y=sinx x=k+/2kz) (k0kz) y=cosx x=kkz) k+/20(kz) y=tanx k0(kz) cos(+)=coscos-sinsin cos(-)=coscos+sinsin sin()=sincoscossin tan(+)=(tan+tan)/(1-tantan) tan(-)=(tan-tan)/(1+tantan) sin+sin=2sin[(+)/2]cos[(-)/2] sin-sin=2cos[(+)/2]sin[(-)/2] cos+cos=2cos[(+)/2]cos[(-)/2] cos-cos=-2sin[(+)/2]sin[(-)/2] sincos=(1/2)[sin(+)+sin(-)] cossin=(1/2)[sin(+)-sin(-)] coscos=(1/2)[cos(+)+cos(-)] sinsin=-(1/2)[cos(+)-cos(-)] sin(2)=2sincos=2/(tan+cot) cos(2)=cos^2;-sin^2;=2cos^2;-1=1-2sin^2; tan(2)=2tan/(1-tan^2;) cot(2)=(cot^2;-1)/(2cot) sec(2)=sec^2;/(1-tan^2;) csc(2)=1/2*seccsc sin(3) = 3sin-4sin^3; = 4sinsin(60+)sin(60-) cos(3) = 4cos^3;-3cos = 4coscos(60+)cos(60-) tan(3) = (3tan-tan^3;)/(1-3tan^2;) = tantan(/3+)tan(/3-) cot(3)=(cot^3;-3cot)/(3cot-1) n sin(n)=ncos^(n-1)sin-C(n,3)cos^(n-3)sin^3+C(n,5)cos^(n-5)sin^5- cos(n)=cos^n-C(n,2)cos^(n-2)sin^2+C(n,4)cos^(n-4)sin^4- sin(/2)=((1-cos)/2) cos(/2)=((1+cos)/2) tan(/2)=((1-cos)/(1+cos))=sin/(1+cos)=(1-cos)/sin

cot(/2)=((1+cos)/(1-cos))=(1+cos)/sin=sin/(1-cos) sec(/2)=((2sec/(sec+1)) csc(/2)=((2sec/(sec-1)) Asin+Bcos=(A^2;+B^2;)sin(+arctan(B/A)) Asin+Bcos=(A^2;+B^2;)cos(-arctan(A/B)) sin(a)= (2tan(a/2))/(1+tan^2;(a/2)) cos(a)= (1-tan^2;(a/2))/(1+tan^2;(a/2)) tan(a)= (2tan(a/2))/(1-tan^2;(a/2)) sin^2;=(1-cos(2))/2=versin(2)/2 cos^2;=(1+cos(2))/2=covers(2)/2 tan^2;=(1-cos(2))/(1+cos(2)) sin(++)=sincoscos+cossincos+coscossin-sinsinsin cos(++)=coscoscos-cossinsin-sincossin-sinsincos tan(++)=(tan+tan+tan-tantantan)(1-tantan-tantan-tantan) 0 0 1 0 /6 1/2 3/2 3/3 3 /4 2/2 2/2 1 1 /3 3/2 1/2 3 3/3 /2 1 0 0 0 -1 0 c0+c1x+c2x2+...+cnxn+...=cnxn (n=0..) c0+c1(x-a)+c2(x-a)2+...+cn(x-a)n+...=cn(x-a)n (n=0..) , c0,c1,c2,...cn... a f(x)=f(a)+f'(a)/1!*(x-a)+f''(a)/2!*(x-a)2+...+f(n)(a)/n!*(x-a)n+ e^x = 1+x+x^2/2!+x^3/3!++x^n/n!+ ln(1+x)=x-x^2/2+x^3/3-+(-1)^(k-1)*(x^k)/k(|x|