数学三角函数公式 数学三角函数公式
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sin30\u00b0=1/2
sin45\u00b0=\u221a2/2
sin60\u00b0=\u221a3/2
cos30\u00b0=\u221a3/2
cos45\u00b0=\u221a2/2
cos60\u00b0=1/2
tan30\u00b0=\u221a3/3
tan45\u00b0=1
tan60\u00b0=\u221a3[1]
cot30\u00b0=\u221a3
cot45\u00b0=1
cot60\u00b0=\u221a3/3
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sin(A+B)=sinAcosB+cosAsinBsin(A-B)=sinAcosB-sinBcosA
cos(A+B)=cosAcosB-sinAsinBcos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/(1-tanAtanB)tan(A-B)=(tanA-tanB)/(1+tanAtanB)
ctg(A+B)=(ctgActgB-1)/(ctgB+ctgA)ctg(A-B)=(ctgActgB+1)/(ctgB-ctgA)
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tan\u03b1 \u00b7cot\u03b1\uff1d1
sin\u03b1 \u00b7csc\u03b1\uff1d1
cos\u03b1 \u00b7sec\u03b1\uff1d1
sin\u03b1/cos\u03b1\uff1dtan\u03b1\uff1dsec\u03b1/csc\u03b1
cos\u03b1/sin\u03b1\uff1dcot\u03b1\uff1dcsc\u03b1/sec\u03b1
sin2\u03b1\uff0bcos2\u03b1\uff1d1
1\uff0btan2\u03b1\uff1dsec2\u03b1
1\uff0bcot2\u03b1\uff1dcsc2\u03b1
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sin\uff08\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
cos\uff08\uff0d\u03b1\uff09\uff1dcos\u03b1
tan\uff08\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
cot\uff08\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
sin\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcos\u03b1
cos\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dsin\u03b1
tan\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
cot\uff08\u03c0/2\uff0d\u03b1\uff09\uff1dtan\u03b1
sin\uff08\u03c0/2\uff0b\u03b1\uff09\uff1dcos\u03b1
cos\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dsin\u03b1
tan\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
cot\uff08\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dtan\u03b1
sin\uff08\u03c0\uff0d\u03b1\uff09\uff1dsin\u03b1
cos\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcos\u03b1
tan\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
cot\uff08\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
sin\uff08\u03c0\uff0b\u03b1\uff09\uff1d\uff0dsin\u03b1
cos\uff08\u03c0\uff0b\u03b1\uff09\uff1d\uff0dcos\u03b1
tan\uff08\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
cot\uff08\u03c0\uff0b\u03b1\uff09\uff1dcot\u03b1
sin\uff083\u03c0/2\uff0d\u03b1\uff09\uff1d\uff0dcos\u03b1
cos\uff083\u03c0/2\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
tan\uff083\u03c0/2\uff0d\u03b1\uff09\uff1dcot\u03b1
cot\uff083\u03c0/2\uff0d\u03b1\uff09\uff1dtan\u03b1
sin\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcos\u03b1
cos\uff083\u03c0/2\uff0b\u03b1\uff09\uff1dsin\u03b1
tan\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dcot\u03b1
cot\uff083\u03c0/2\uff0b\u03b1\uff09\uff1d\uff0dtan\u03b1
sin\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dsin\u03b1
cos\uff082\u03c0\uff0d\u03b1\uff09\uff1dcos\u03b1
tan\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dtan\u03b1
cot\uff082\u03c0\uff0d\u03b1\uff09\uff1d\uff0dcot\u03b1
sin\uff082k\u03c0\uff0b\u03b1\uff09\uff1dsin\u03b1
cos\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcos\u03b1
tan\uff082k\u03c0\uff0b\u03b1\uff09\uff1dtan\u03b1
cot\uff082k\u03c0\uff0b\u03b1\uff09\uff1dcot\u03b1
(\u5176\u4e2dk\u2208Z)
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sin\uff08\u03b1\uff0b\u03b2\uff09\uff1dsin\u03b1cos\u03b2\uff0bcos\u03b1sin\u03b2
sin\uff08\u03b1\uff0d\u03b2\uff09\uff1dsin\u03b1cos\u03b2\uff0dcos\u03b1sin\u03b2
cos\uff08\u03b1\uff0b\u03b2\uff09\uff1dcos\u03b1cos\u03b2\uff0dsin\u03b1sin\u03b2
cos\uff08\u03b1\uff0d\u03b2\uff09\uff1dcos\u03b1cos\u03b2\uff0bsin\u03b1sin\u03b2
tan\u03b1\uff0btan\u03b2 tan\u03b1\uff0dtan\u03b2
tan\uff08\u03b1\uff0b\u03b2\uff09\uff1d\u2014\u2014\u2014\u2014\u2014\u2014 tan\uff08\u03b1\uff0d\u03b2\uff09\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0dtan\u03b1
\u00b7tan\u03b2 1\uff0btan\u03b1 \u00b7tan\u03b2
2tan(\u03b1/2)
sin\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0btan2(\u03b1/2)
1\uff0dtan2(\u03b1/2)
cos\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0btan2(\u03b1/2)
2tan(\u03b1/2)
tan\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0dtan2(\u03b1/2)
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sin2\u03b1\uff1d2sin\u03b1cos\u03b1
cos2\u03b1\uff1dcos2\u03b1\uff0dsin2\u03b1\uff1d2cos2\u03b1\uff0d1\uff1d1\uff0d2sin2\u03b1
2tan\u03b1
tan2\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014
1\uff0dtan2\u03b1
sin3\u03b1\uff1d3sin\u03b1\uff0d4sin3\u03b1
cos3\u03b1\uff1d4cos3\u03b1\uff0d3cos\u03b1
3tan\u03b1\uff0dtan3\u03b1
tan3\u03b1\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0d3tan2\u03b1
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;
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\u3001
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sin(\u03b8+
)\uff0c\u8fd9\u91cc\u8f85\u52a9\u89d2
\u6240\u5728\u8c61\u9650\u7531a\u3001b\u7684\u7b26\u53f7\u786e\u5b9a\uff0c
\u89d2\u7684\u503c\u7531tan
=
\u786e\u5b9a\u3002
sin(-a)=-sin(a)
cos(-a)=cos(a)
sin(pi/2-a)=cos(a)
cos(pi/2-a)=sin(a)
sin(pi/2+a)=cos(a)
cos(pi/2+a)=-sin(a)
sin(pi-a)=sin(a)
cos(pi-a)=-cos(a)
sin(pi+a)=-sin(a)
cos(pi+a)=-cos(a)
tga=tana=sina/cosa
两角和与差的三角函数
sin(a+b)=sin(a)cos(b)+cos(α)sin(b)
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
sin(a-b)=sin(a)cos(b)-cos(a)sin(b)
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))
tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b))
三角函数和差化积公式
sin(a)+sin(b)=2sin((a+b)/2)cos((a-b)/2)
sin(a)?sin(b)=2cos((a+b)/2)sin((a-b)/2)
cos(a)+cos(b)=2cos((a+b)/2)cos((a-b)/2)
cos(a)-cos(b)=-2sin((a+b)/2)sin((a-b)/2)
积化和差公式
sin(a)sin(b)=-1/2*[cos(a+b)-cos(a-b)]
cos(a)cos(b)=1/2*[cos(a+b)+cos(a-b)]
sin(a)cos(b)=1/2*[sin(a+b)+sin(a-b)]
二倍角公式
sin(2a)=2sin(a)cos(a)
cos(2a)=cos^2(a)-sin^2(a)=2cos^2(a)-1=1-2sin^2(a)
半角公式
sin^2(a/2)=(1-cos(a))/2
cos^2(a/2)=(1+cos(a))/2
tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))
万能公式
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))
其它公式
a*sin(a)+b*cos(a)=sqrt(a^2+b^2)sin(a+c)
[其中，tan(c)=b/a]
a*sin(a)-b*cos(a)=sqrt(a^2+b^2)cos(a-c)
[其中，tan(c)=a/b]
1+sin(a)=(sin(a/2)+cos(a/2))^2
1-sin(a)=(sin(a/2)-cos(a/2))^2
其他非重点三角函数
csc(a)=1/sin(a)
sec(a)=1/cos(a)
双曲函数
sinh(a)=(e^a-e^(-a))/2
cosh(a)=(e^a+e^(-a))/2
tgh(a)=sinh(a)/cosh(a)
绛旓細涓夎鍑芥暟鍏紡鏈夌Н鍖栧拰宸叕寮忋佸拰宸寲绉叕寮忋佷笁鍊嶈鍏紡銆佹寮︿簩鍊嶈鍏紡銆佷綑寮︿簩鍊嶈鍏紡銆佷綑寮﹀畾鐞嗙瓑銆1绉寲鍜屽樊鍏紡銆俿in伪路cos尾=(1/2)*[sin(伪+尾)+sin(伪-尾)]锛沜os伪路sin尾=(1/2)*[sin(伪+尾)-sin(伪-尾)];cos伪路cos尾=(1/2)*[cos(伪+尾)+cos(伪-尾)];sin伪路...
绛旓細cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB 浜屽嶈涓夎鍑芥暟鍏紡 涓囪兘涓夎鍑芥暟鍏紡 鍏朵粬涓夎鍑芥暟鍏紡 01 涓夎鍑芥暟鍏紡涓锛氳伪涓轰换鎰忚锛岀粓杈圭浉鍚岀殑瑙掔殑鍚屼竴涓夎鍑芥暟鐨勫肩浉绛夛細sin(2k蟺+伪)= sin伪 cos(2k蟺+伪)= cos伪 tan(2k蟺+伪)= tan伪 cot(2k蟺+伪)= cot伪 ...
绛旓細sin鐨勫钩鏂广乧os鐨勫钩鏂广 tan鐨勫钩鏂 鐨鍏紡鏄細1銆乻in²伪+cos²伪=1 2銆1+tan²伪=sec²伪 3銆1+cot²伪=csc²伪 4銆乻in²伪=(1-cos2a)/2 5銆乧os²a=(1+cos2a)/2 6銆乼an²a=(2tana-1)/(tan2a)...
绛旓細鍏紡涓簊inA=a/c锛宑osA=b/c锛宼anA=a/b銆傚湪鐩磋涓夎褰腑锛屽綋骞抽潰涓婄殑涓夌偣A銆丅銆丆鐨勮繛绾匡紝AB銆丄C銆丅C锛屾瀯鎴愪竴涓洿瑙掍笁瑙掑舰锛屽叾涓垹ACB涓虹洿瑙掋傚鈭燘AC鑰岃█锛屽杈筧=BC銆佹枩杈筩=AB銆侀偦杈筨=AC銆
绛旓細涓夎鍑芥暟楂樹腑鎵鏈夊叕寮忓涓嬶細1銆佷袱瑙掑拰鍏紡锛sin(A+B)=sinAcosB+cosAsinB sin(A-B)=sinAcosB-sinBcosA cos(A+B)=cosAcosB-sinAsinB cos(A-B)=cosAcosB+sinAsinB tan(A+B)=(tanA+tanB)/(1-tanAtanB) tan(A-B)=(tanA-tanB)/(1+tanAtanB)ctg(A+B)=(ctgActgB-1)/(ctgB+ctgA) ctg...
绛旓細鍏紡瑙佷笅闈細涓夎鍑芥暟鐨勫繀鑳屽叕寮忓寘鎷崐瑙掑叕寮忥紝鍊嶈鍏紡锛屼袱瑙掑拰涓庡樊鍏紡锛绉寲鍜屽樊鍏紡锛屽拰宸寲绉叕寮忋俿in(A/2)=卤鈭((1-cosA)/2)锛宑os(A/2)=卤鈭((1+cosA)/2)锛宼an(A/2)=卤鈭((1-cosA)/((1+cosA))銆備笁瑙掑嚱鏁版槸鏁板涓睘浜庡垵绛夊嚱鏁颁腑鐨勮秴瓒婂嚱鏁扮殑鍑芥暟銆傞氬父鏄湪骞抽潰鐩磋鍧愭爣绯讳腑...
绛旓細1銆乻inx+cosx=鈭2(sinx*鈭2/2+cosx*鈭2)鍥犱负cosx=鈭2/2,sinx=鈭2/2 鎵浠inx+cosx=鈭2(sinxcos蟺/4+cosxsin蟺/4)=鈭2sin(x+蟺/4)2銆乻inx锛媍osx =鈭2(鈭2/2 * sinx+鈭2/2 * cosx)=鈭2(sinxcos45搴+cosxsin45搴)=鈭2sin(x+45搴)涓夎鍑芥暟鍏紡鐪嬩技寰堝銆佸緢澶嶆潅锛屼絾鍙鎺屾彙...
绛旓細鏁板涓夎鍑芥暟鍏紡 姝ｅ鸡(sin)锛氬杈规瘮鏂滆竟锛涘嵆sinA=a/c锛屼笁瑙掑嚱鏁板叕寮 閿愯涓夎鍑芥暟瀹氫箟 閿愯瑙扐鐨勬寮(sin),浣欏鸡(cos)鍜屾鍒(tan),浣欏垏(cot)浠ュ強姝ｅ壊(sec)锛屼綑鍓(csc)閮藉彨鍋氳A鐨勯攼瑙掍笁瑙掑嚱鏁般傛寮(sin)锛氬杈规瘮鏂滆竟锛屽嵆sinA=a/c锛涗綑寮(cos)锛氶偦杈规瘮鏂滆竟锛屽嵆cosA=b/c锛涙鍒(tan)锛...
绛旓細鏁板涓夎鍑芥暟鍏紡鏄涓嬶細1銆乻in2伪=2sin伪cos伪銆2銆乼an2伪=2tan伪/(1-tan^2(伪))銆3銆乧os2伪=cos^2(伪)-sin^2(伪)=2cos^2(伪)-1=1-2sin^2(伪) 銆4銆乻in^2(伪/2)=(1-cos伪)/2銆5銆乧os^2(伪/2)=(1+cos伪)/2銆6銆乼an^2(伪/2)=(1-cos伪)/(1+cos伪)銆7銆...
绛旓細涓夎鍑芥暟鍏紡鏄敤浜庢弿杩颁笁瑙掑嚱鏁颁箣闂寸殑鍏崇郴鐨鏁板鍏紡銆備互涓嬫槸涓浜涘父瑙佺殑涓夎鍑芥暟鍏紡鍙婂叾鍚箟锛1. 姝ｅ鸡鍑芥暟鍏紡锛歴in^2(x) + cos^2(x) = 1銆傝繖涓叕寮忕О涓烘寮﹀嚱鏁扮殑骞虫柟鍔犱綑寮﹀嚱鏁扮殑骞虫柟绛変簬1鐨勬亽绛夊紡銆傚畠琛ㄧず鍦ㄤ换浣曡搴 x 涓婏紝姝ｅ鸡鍑芥暟鐨勫钩鏂逛笌浣欏鸡鍑芥暟鐨勫钩鏂逛箣鍜屾绘槸绛変簬1銆2. 浣欏鸡鍑芥暟鍏紡锛...
