反三角函数的公式有哪些 反三角函数公式
\u53cd\u4e09\u89d2\u51fd\u6570\u516c\u5f0f\u90fd\u6709\u4ec0\u4e48\u03c0\u662f\u4e09\u89d2\u51fd\u6570\u7684\u4e00\u4e2a\u7279\u6b8a\u503c arcsin arccos \u662f\u8868\u793a\u53cd\u4e09\u89d2\u51fd\u6570\u7684\u7279\u6b8a\u503c\u5c31\u6bd4\u5982 sin\u03c0/6=1/2 \u7528\u53cd\u4e09\u89d2\u51fd\u6570\u8868\u793a\u5c31\u662f \u03b1=arcsin1/2 \u03b1\u5c31\u662f\u03c0/6 \u7279\u6b8a\u503c\u5c31\u4e0d\u7528\u8868\u793a\u540e\u9762\u7684 arcsin \u8868\u793a\u89d2\u5ea6\u5c31\u76f4\u63a5\u8868\u793a\u51fa\u6765\u5c31\u53ef\u4ee5\u4e86\uff0c\u50cf\u4f60\u51fa\u7684\u5c31\u4e0d\u662f\u7279\u6b8a\u503c\uff0c\u6240\u4ee5\u5c31\u8868\u793a\u4e3a \u03b1=-arcsin5/6\u56e0\u4e3a\u4f60\u51fa\u7684\u662f-5/6 \u4e3a\u7b2c\u56db\u8c61\u9650\u89d2\uff0c\u6240\u4ee5\u4e3a\u8d1f\u7684\u03c0\u662f\u7279\u6b8a\u89d2\uff0c\u76f4\u63a5\u8868\u793a\u5c31\u53ef\u4ee5\u4e86\uff0c\u4e0d\u7528\u52a0arccossin\u03c0=0 cos\u03c0=-1 \u5982\u679c\u4f60\u8981\u8868\u793a\u5c31\u662f \u03c0=arcsin0 \u03c0=-arccos1
\u53cd\u4e09\u89d2\u51fd\u6570\u516c\u5f0f:
arcsin(-x)=-arcsinx
arccos(-x)=\u220f\uff0darccosx
arctan(-x)=-arctanx
arccot(-x)=\u220f\uff0darccotx
arcsinx+arccosx=\u220f/2=arctanx+arccotx
sin(arcsinx)=x=cos(arccosx)=tan(arctanx)=cot(arccotx)
\u5f53x\u2208\u3014\u2014\u220f/2\uff0c\u220f/2\u3015\u65f6\uff0c\u6709arcsin(sinx)=x
\u5f53x\u2208\u30140,\u220f\u3015,arccos(cosx)=x
x\u2208(\u2014\u220f/2\uff0c\u220f/2),arctan(tanx)=x
x\u2208(0\uff0c\u220f),arccot(cotx)=x
x\u30090,arctanx=arctan1/x,arccotx\u7c7b\u4f3c
\u82e5(arctanx+arctany)\u2208(\u2014\u220f/2\uff0c\u220f/2),\u5219arctanx+arctany=arctan(x+y/1-xy)
\u540c\u89d2\u4e09\u89d2\u51fd\u6570\u7684\u57fa\u672c\u5173\u7cfb\u5f0f
\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
\u8bf1\u5bfc\u516c\u5f0f
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
tan\u03b1\uff0btan\u03b2
tan\uff08\u03b1\uff0b\u03b2\uff09\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
1\uff0dtan\u03b1
\u00b7tan\u03b2
tan\u03b1\uff0dtan\u03b2
tan\uff08\u03b1\uff0d\u03b2\uff09\uff1d\u2014\u2014\u2014\u2014\u2014\u2014
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)
\u534a\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f
\u4e09\u89d2\u51fd\u6570\u7684\u964d\u5e42\u516c\u5f0f
\u4e8c\u500d\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f
\u4e09\u500d\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u548c\u6b63\u5207\u516c\u5f0f
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
\u4e09\u89d2\u51fd\u6570\u7684\u548c\u5dee\u5316\u79ef\u516c\u5f0f
\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5316\u548c\u5dee\u516c\u5f0f
\u03b1\uff0b\u03b2
\u03b1\uff0d\u03b2
sin\u03b1\uff0bsin\u03b2\uff1d2sin\u2014\uff0d\uff0d\u00b7cos\u2014\uff0d\u2014
2
2
\u03b1\uff0b\u03b2
\u03b1\uff0d\u03b2
sin\u03b1\uff0dsin\u03b2\uff1d2cos\u2014\uff0d\uff0d\u00b7sin\u2014\uff0d\u2014
2
2
\u03b1\uff0b\u03b2
\u03b1\uff0d\u03b2
cos\u03b1\uff0bcos\u03b2\uff1d2cos\u2014\uff0d\uff0d\u00b7cos\u2014\uff0d\u2014
2
2
\u03b1\uff0b\u03b2
\u03b1\uff0d\u03b2
cos\u03b1\uff0dcos\u03b2\uff1d\uff0d2sin\u2014\uff0d\uff0d\u00b7sin\u2014\uff0d\u2014
2
2
1
sin\u03b1
\u00b7cos\u03b2\uff1d-[sin\uff08\u03b1\uff0b\u03b2\uff09\uff0bsin\uff08\u03b1\uff0d\u03b2\uff09]
2
1
cos\u03b1
\u00b7sin\u03b2\uff1d-[sin\uff08\u03b1\uff0b\u03b2\uff09\uff0dsin\uff08\u03b1\uff0d\u03b2\uff09]
2
1
cos\u03b1
\u00b7cos\u03b2\uff1d-[cos\uff08\u03b1\uff0b\u03b2\uff09\uff0bcos\uff08\u03b1\uff0d\u03b2\uff09]
2
1
sin\u03b1
\u00b7sin\u03b2\uff1d\uff0d
-[cos\uff08\u03b1\uff0b\u03b2\uff09\uff0dcos\uff08\u03b1\uff0d\u03b2\uff09]
2
arccos(-x)=∏－arccosx
arctan(-x)=-arctanx
arccot(-x)=∏－arccotx
arcsinx+arccosx=∏/2=arctanx+arccotx
sin(arcsinx)=x=cos(arccosx)=tan(arctanx)=cot(arccotx)
当x∈〔—∏/2，∏/2〕时，有arcsin(sinx)=x
当x∈〔0,∏〕,arccos(cosx)=x
x∈(—∏/2，∏/2),arctan(tanx)=x
x∈(0，∏),arccot(cotx)=x
x〉0,arctanx=arctan1/x,arccotx类似
若(arctanx+arctany)∈(—∏/2，∏/2),则arctanx+arctany=arctan(x+y/1-xy)
这个不在高考范围里，没必要深学
绛旓細1銆佸弽姝ｅ鸡鍑芥暟鐨姹傚锛(arcsinx)'=1/鈭(1-x^2)2銆佸弽浣欏鸡鍑芥暟鐨勬眰瀵硷細(arccosx)'=-1/鈭(1-x^2)3銆佸弽姝ｅ垏鍑芥暟鐨勬眰瀵硷細(arctanx)'=1/(1+x^2)4銆佸弽浣欏垏鍑芥暟鐨勬眰瀵硷細(arccotx)'=-1/(1+x^2)涓洪檺鍒鍙嶄笁瑙掑嚱鏁涓哄崟鍊煎嚱鏁帮紝灏嗗弽姝ｅ鸡鍑芥暟鐨勫紋闄愬湪-蟺/2鈮鈮は/2锛屽皢y浣滀负鍙嶆寮...
绛旓細涓夎鍑芥暟涓鍙嶄笁瑙掑嚱鏁扮殑鍏崇郴鍏紡锛歴in(A+B)=sinAcosB+cosAsinBsin(A-B)銆傚綋涓夎鍑芥暟涓殑鑷彉閲忓拰鍥犲彉閲忚皟鎹㈠悗锛屼笁瑙掑嚱鏁扮殑鍙嶅嚱鏁锛屽氨鏄弽涓夎鍑芥暟銆傚弽涓夎鍑芥暟瀹為檯涓婂苟涓嶈兘鍙仛鍑芥暟锛屽洜涓哄畠骞朵笉婊¤冻涓涓嚜鍙橀噺瀵瑰簲涓涓嚱鏁板肩殑瑕佹眰锛屽叾鍥惧儚涓庡叾鍘熷嚱鏁板叧浜庡嚱鏁皔=x瀵圭О銆傚叾姒傚康棣栧厛鐢辨鎷夋彁鍑猴紝骞朵笖棣栧厛...
绛旓細鍙嶄笁瑙掑嚱鏁板叕寮:1銆乤rcsin(-x)=-arcsinx 2銆乤rccos(-x)=蟺锛峚rccosx 3銆乤rctan(-x)=-arctanx 4銆乤rccot(-x)=蟺锛峚rccotx 5銆乤rcsinx+arccosx=蟺/2=arctanx+arccotx 6銆乻in(arcsinx)=x=cos(arccosx)=tan(arctanx)=cot(arccotx)7銆佸綋x鈭堛斺斚/2,蟺/2銆曟椂,鏈塧rcsin(sinx)=x...
绛旓細鍙嶄笁瑙掑嚱鏁扮殑绉垎鍙互閫氳繃涓浜涘熀鏈殑绉垎鍏紡鏉ユ眰銆備互涓嬫槸涓浜涘熀鏈殑鍙嶄笁瑙掑嚱鏁扮殑绉垎鍏紡锛1. 鍙嶆寮﹀嚱鏁帮細$\int \arcsin(x) \, dx = x \arcsin(x) + \sqrt{1 - x^2} + C 2. 鍙嶄綑寮﹀嚱鏁帮細$\int \arccos(x) \, dx = x \arccos(x) - \sqrt{1 - x^2} + C 3. 鍙嶆鍒囧嚱鏁...
绛旓細(x-y)/(1+xy)] 銆傚叾涓紝瑙勫垯 1 鍜 2 鏄弽姝ｅ垏鍑芥暟鐨鍩烘湰鍊硷紝瑙勫垯 3 鏄弽姝ｅ垏鍑芥暟 鐨瀵圭О鎬ц川锛岃鍒 4 鍜 5 鏄弽姝ｅ垏鍑芥暟鐨勫姞鍑鍏紡锛岃鍒 6 鍜 7 鏄弽 姝ｅ垏鍑芥暟鐨勫拰宸叕寮忋 閫氳繃杩欎簺瑙勫垯锛屾垜浠彲浠ュ揩閫熷噯纭湴璁＄畻鍙嶆鍒囧嚱鏁扮殑鍊硷紝甯姪 鎴戜滑鏇村ソ鍦扮悊瑙ｅ拰澶勭悊涓夎鍑芥暟鐩稿叧鐨勯棶棰樸
绛旓細2銆乼an(a+b)=(tana+tanb)/(1-tana*tanb)娉細鑻ユ槸a-b锛屽垯鎶婂悗闈㈢殑鍔犲噺閮芥崲涓涓嬨3銆1/tanb=cotb锛堣繖涓叕寮忎笉甯哥敤锛屽伓灏旂敤涔熺粡甯稿啓鎴愭鍒囩殑鍊掓暟鐨勫舰寮忥級4銆乼anB=q锛堝父鏁帮級鍒欒B=acttan(q)锛岃繖鏄弽鍑芥暟鐨勫叕寮忋鍙嶄笁瑙掑嚱鏁扮殑鍏紡锛氬弽涓夎鍑芥暟鐨勫拰宸叕寮忎笌瀵瑰簲鐨勪笁瑙掑嚱鏁扮殑鍜屽樊鍏紡娌℃湁鍏崇郴锛...
绛旓細cos(arcsinx)=(1-x^2)^0.5 arcsin(-x)=-arcsinx arccos(-x)=蟺-arccosx arctan(-x)=-arctanx arccot(-x)=蟺-arccotx arcsinx+arccosx=蟺/2=arctanx+arccotx sin(arcsinx)=cos(arccosx)=tan(arctanx)=cot(arccotx)=x arcsin x = x + x^3/(2*3) + (1*3)x^5/(2*...
绛旓細鎺㈢储鏁板涓殑绁炵鍔涢噺锛7涓笉鍙垨缂虹殑鍙嶄笁瑙掑嚱鏁鍏紡</ 鍦ㄦ暟瀛︾殑娴╂负鏄熸捣涓紝鍙嶄笁瑙掑嚱鏁板氨鍍忕拃鐠ㄧ殑鏄熻景锛屽畠浠笉浠呮彮绀轰簡瑙掑害涓庣洿瑙掍笁瑙掑舰杈归暱涔嬮棿鐨勫濡欒仈绯伙紝杩樺湪鍚勭鏁板闂鍜屽伐绋嬪簲鐢ㄤ腑鍙戞尌鐫鑷冲叧閲嶈鐨勪綔鐢ㄣ傝鎴戜滑涓璧锋繁鍏ヤ簡瑙ｄ竴涓嬭繖涓冧釜鍩虹涓斿己澶鐨勫叕寮锛屽畠浠垎鍒槸锛氭寮︾殑閫嗗嚱鏁帮紙arcsin鎴杝in^(...
绛旓細鍙嶄笁瑙掑嚱鏁扮殑鍏崇郴鍏紡 浣欒鍏崇郴鍏紡 arcsin(x)+arccos(x)=蟺/2 arctan(x)+arccot(x)=蟺/2 arcsec(x)+arccsc(x)=蟺/2 璐熸暟鍏崇郴鍏紡 arcsin(-x)=-arcsin(x)arccos(-x)=蟺-arccos(x)arctan(-x)=-arctan(x)arccot(-x)=蟺-arccot(x)arcsec(-x)=蟺-arcsec(x)arcsec(-x)=...
绛旓細arcsinx鐨勫鏁版槸y'=1/cosy=1/鈭歔1-(siny)²]=1/鈭(1-x²)鎺ㄥ杩囩▼璇存槑锛歽=arcsinx y'=1/鈭氾紙1-x²锛夊弽鍑芥暟鐨瀵兼暟锛歽=arcsinx,閭ｄ箞锛宻iny=x,姹傚寰楀埌锛宑osy*y'=1 鍗硑'=1/cosy=1/鈭歔1-(siny)²]=1/鈭(1-x²)鍙嶄笁瑙掑嚱鏁浠嬬粛 鍙嶄笁瑙掑嚱鏁版槸姝ｅ鸡锛...
