求三角函数的公式
\u4e09\u89d2\u51fd\u6570\u5168\u90e8\u516c\u5f0f\u4e00\u4e2a\u5b9a\u4e49\uff0c\u4e09\u89d2\u51fd\u6570\uff0c
\u4e24\u79cd\u5236\u5ea6\uff0c\u89d2\u5ea6\u5f27\u5ea6\u3002
\u4e09\u5957\u516c\u5f0f\uff0c\u7262\u56fa\u8bb0\u5fc6\uff0c
\u540c\u89d2\u8bf1\u5bfc\uff0c\u52a0\u6cd5\u5b9a\u7406\u3002
\u540c\u89d2\u516c\u5f0f\uff0c\u516b\u4e2a\u4e09\u7ec4\uff0c
\u5e73\u65b9\u5173\u7cfb\uff0c\u5bfc\u6570\u5546\u6570\u3002
\u8bf1\u5bfc\u516c\u5f0f\uff0c\u4e24\u7c7b\u4e5d\u7ec4\uff0c
\u8c61\u9650\u5b9a\u53f7\uff0c\u5076\u540c\u5947\u4f59\u3002
\u4e09\u89d2\u7684\u6240\u6709\u516c\u5f0f\u5982\u4e0b\uff1a
\u5012\u6570\u5173\u7cfb: \u5546\u7684\u5173\u7cfb\uff1a \u5e73\u65b9\u5173\u7cfb\uff1a
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
\uff08\u516d\u8fb9\u5f62\u8bb0\u5fc6\u6cd5\uff1a\u56fe\u5f62\u7ed3\u6784\u201c\u4e0a\u5f26\u4e2d\u5207\u4e0b\u5272\uff0c\u5de6\u6b63\u53f3\u4f59\u4e2d\u95f41\u201d\uff1b\u8bb0\u5fc6\u65b9\u6cd5\u201c\u5bf9\u89d2\u7ebf\u4e0a\u4e24\u4e2a\u51fd\u6570\u7684\u79ef\u4e3a1\uff1b\u9634\u5f71\u4e09\u89d2\u5f62\u4e0a\u4e24\u9876\u70b9\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u5e73\u65b9\u548c\u7b49\u4e8e\u4e0b\u9876\u70b9\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u5e73\u65b9\uff1b\u4efb\u610f\u4e00\u9876\u70b9\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7b49\u4e8e\u76f8\u90bb\u4e24\u4e2a\u9876\u70b9\u7684\u4e09\u89d2\u51fd\u6570\u503c\u7684\u4e58\u79ef\u3002\u201d\uff09
\u8bf1\u5bfc\u516c\u5f0f\uff08\u53e3\u8bc0:\u5947\u53d8\u5076\u4e0d\u53d8\uff0c\u7b26\u53f7\u770b\u8c61\u9650\u3002\uff09
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)
\u4e24\u89d2\u548c\u4e0e\u5dee\u7684\u4e09\u89d2\u51fd\u6570\u516c\u5f0f \u4e07\u80fd\u516c\u5f0f
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
csc²A-cot²A=1
sec²A-tan²A=1
sin(α±β)=sinαcosβ±cosαsinβ
cos(α±β)=cosαcosβ-±sinαsinβ
tan(α±β)=(tanα±tanβ)÷(1-±tanαtanβ)
cot(α±β)=(cotαcotβ-±1)÷(cotβ±cotα)
sinα+sinβ=2sin[(α+β)÷2]cos[(α-β)÷2]
sinα-sinβ=2cos[(α+β)÷2]sin[(α-β)÷2]
cosα+cosβ=2cos[(α+β)÷2]cos[(α-β)÷2]
cosα-cosβ=-2sin[(α+β)÷2]sin[(α-β)÷2]
sinαsinβ=1÷2×[cos(α-β)-cos(α+β)]
cosαcosβ=1÷2×[cos(α+β)+cos(α-β)]
sinαcosβ=1÷2×[sin(α+β)+sin(α-β)]
sin2α=2sinαcosα
sin3α=sinα(3-4sin²α)
cos2α=2cos²α-1
cos3α=cosα(4cos²α-3)
tan2α=2tanα÷(1-tan²α)
tan3α=(3tanα-tan³α)÷(1-3tan³α)
sin(α÷2)=±√[(1-cosα)÷2]
=±1÷2×√(1+sinα)±1÷2×√(1-sinα)
cos(α÷2)=±√[(1＋cosα)÷2]
=±1÷2×√(1+sinα)-±1÷2×√(1-sinα)
tan(α÷2)=sinα÷(1+cosα)
=(1-cosα)÷sinα
=±√[(1-cosα)÷(1+cosα)]
万能公式: sinα=2tan(α÷2)÷[1+tan²(α÷2)]
cosα=[1-tan²(α÷2)]÷[1+tan²(α÷2)]
tanα=2tan(α÷2)÷[1-tan²(α÷2)]
平面三角形内的公式:
正弦定理 :
a÷sinA=b÷sinB=c÷sinC=2R (R为外接圆半径=abc/4S)
余弦定理:
a²=b²+c²-2bccosA
b²=a²+c²-2bccosB
c²=a²+b²-2abcosC
正切定理:
(a-b)÷(a+b)=tan[(A-B)÷2]÷tan[(A+B)÷2]
半角公式:
sin(A÷2)=√[(s-b)(s-c)÷bc]
cos(A÷2)=√[s(s-a)÷bc]
tan(A÷2)=√[(s-b)(s-c)÷s(s-a)]
三角形面积=√[s(s-a)(s-b)(s-c)]=0.5c²sinAsinB/sin(A+B)
内切圆半径：r=S/s=stan(α/2)tan(β/2)tan(γ/2)
=√[(s-a)(s-b)(s-c)/c]
其中s=(a+b+c)÷2
射影定理:
acosB+bcosA=c
模尔外得公式:
(a+b)÷c=cos[(A-B)÷2]÷sin(C÷2)
(a-b)÷c=sin[(A-B)÷2]÷cos(C÷2)
绛旓細1銆佸叕寮忎竴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 sin(2k蟺+伪)=sin伪(k鈭圸)cos(2k蟺+伪)=cos伪(k鈭圸)tan(2k蟺+伪)=tan伪(k鈭圸)cot(2k蟺+伪)=cot伪(k鈭圸)2銆佸叕寮忎簩锛氳伪涓轰换鎰忚锛屜+伪鐨勪笁瑙掑嚱鏁板间笌伪鐨勪笁瑙掑嚱鏁板间箣闂寸殑鍏崇郴 sin(蟺+伪)=锛峴in伪 cos(蟺+伪...
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绛旓細1锛峵an2(伪/2)鍗婅鐨勬寮︺佷綑寮﹀拰姝ｅ垏鍏紡 涓夎鍑芥暟 鐨闄嶅箓鍏紡 浜屽嶈鐨勬寮︺佷綑寮﹀拰姝ｅ垏鍏紡
绛旓細鍗婅鍏紡锛(sina)^=(1-cos2a)/2 (灏哸鐢╝/2浠ｆ浛鍗冲緱鍗婅鎻忚堪)(cosa)^=(1+cos2a)/2 (tga)^=(1-cos2a)/(1+cos2a)涓夊嶈鍏紡锛歴in3a= 3sina-4sin^3 a cos3a=-3cosa+4cos^3 a 绉寲鍜屽樊鍏紡锛歴inacosb= [sin(a+b)+sin(a-b)]/2 (灏嗕笂闈㈠叧浜巗in鐨勫拰宸叕寮忕浉鍔犻櫎浠2鍗...
绛旓細1銆鍏紡涓锛岃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鍊肩浉绛夛細sin锛2k蟺+伪锛=sin伪 锛坘鈭圸锛塩os锛2k蟺+伪锛=cos伪 锛坘鈭圸锛塼an锛2k蟺+伪锛=tan伪 锛坘鈭圸锛塩ot锛2k蟺+伪锛=cot伪锛坘鈭圸锛2銆佸叕寮忎簩锛岃伪涓轰换鎰忚锛屜+伪鐨勪笁瑙掑嚱鏁板间笌伪鐨勪笁瑙掑嚱鏁板间箣闂寸殑鍏崇郴锛歴in锛埾+伪锛=...
绛旓細鍗婅鐨勬寮︺佷綑寮﹀拰姝ｅ垏鍏紡 涓夎鍑芥暟鐨闄嶅箓鍏紡 浜屽嶈鐨勬寮︺佷綑寮﹀拰姝ｅ垏鍏紡 涓夊嶈鐨勬寮︺佷綑寮﹀拰姝ｅ垏鍏紡 sin2伪锛2sin伪cos伪 cos2伪锛漜os2伪锛峴in2伪锛2cos2伪锛1锛1锛2sin2伪 2tan伪 tan2伪锛濃斺斺1锛峵an2伪 sin3伪锛3sin伪锛4sin3伪 cos3伪锛4cos3伪锛3cos伪 3tan伪锛峵a...
绛旓細涓夎鍑芥暟鐨勫拰宸寲绉叕寮忎笁瑙掑嚱鏁扮殑绉寲鍜屽樊鍏紡 伪锛嬑 伪锛嵨 sin伪锛媠in尾锛2sin鈥旓紞锛•cos鈥旓紞鈥2 2 伪锛嬑 伪锛嵨 sin伪锛峴in尾锛2cos鈥旓紞锛•sin鈥旓紞鈥2 2 伪锛嬑 伪锛嵨 cos伪锛媍os尾锛2cos鈥旓紞锛•cos鈥旓紞鈥...
绛旓細sin鐨勫钩鏂广乧os鐨勫钩鏂广 tan鐨勫钩鏂 鐨勫叕寮鏄細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1.璇卞鍏紡 sin(-a)=-sin(a)cos(-a)=cos(a)sin(蟺2-a)=cos(a)cos(蟺2-a)=sin(a)sin(蟺2+a)=cos(a)cos(蟺2+a)=-sin(a)sin(蟺-a)=sin(a)cos(蟺-a)=-cos(a)sin(蟺+a)=-sin(a)cos(蟺+a)=-cos(a)2.涓よ鍜屼笌宸殑涓夎鍑芥暟 sin(a+b)=sin(a)cos(b)+cos(伪)...
