幾個反三角函數的圖形,其中,反餘切以複變分析定義,因此在原點處出現不連續斷點
在数学中,反三角函数是三角函数的反函数。
數學符號[编辑]
符号
等常用于
等。但是这种符号有时在
和
之间易造成混淆。
在编程中,函数
,
,
通常叫做
,
,
。很多编程语言提供两自变量atan2函数,它计算给定
和
的
的反正切,但是值域为
。
-
在笛卡尔平面上
![{\displaystyle f(x)=\arcsin x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzAdBntK1zNJEotC4aDvBa2e4ytJFoqeOoAhAnjmPzjKPztzDo2aN)
(紅)和
![{\displaystyle f(x)=\arccos x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaNa3zDG2aNhDnAoPatCQa2hAzDvAzgnEzgw4zDlAytzDoNm1ytBE)
(綠)函数的常用主值的图像。
-
在笛卡尔平面上
![{\displaystyle f(x)=\arctan x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oqdEytzBo2aQa2s0aqiOaAw3aDFBzjzDoqo4ajm0ngo1zgnEagdD)
(紅)和
![{\displaystyle f(x)=\operatorname {arccot} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PatoNzgoOa2w0oDi1yqdAaqrEnjnEntlAnjdDz2iQaAs3njJEnqe5)
(綠)函数的常用主值的图像。
-
在笛卡尔平面上
![{\displaystyle f(x)=\operatorname {arcsec} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dagi2ajvFotrAytBAnAs2zNBFntm1ytGQoNe4oNi5aAa1ytrFoAs1)
(紅)和
![{\displaystyle f(x)=\operatorname {arccsc} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaAa0zNC2aNmPoNlDzDG1nAs0nge4oNsPoDa1zNJEnqo0atJCatFF)
(綠)函数的常用主值的图像。
下表列出基本的反三角函数。
名称
|
常用符号
|
定义
|
定义域
|
值域
|
反正弦 |
![{\displaystyle y=\arcsin x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83nqe4a2nDzqsNaDFFzjm3oDBEngeNotnFzqi5oqhBnjGQzjvEytrD) |
![{\displaystyle x=\sin y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oDvDzArEzgeQnDhBngi5a2s4oAvAatvEoDC1zDi4aNhAoteQoAdF) |
![{\displaystyle [-1,1]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aqsNoDJAatvFzAo3ags2atw3aDFDztGNzjm5otlEyqi4oDBCntmP) |
|
反余弦 |
![{\displaystyle y=\arccos x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80aqdAagoPa2zCztrCnjlAnDzDzqa3o2a2aDs4oAnAotrFzDs0nDlD) |
![{\displaystyle x=\cos y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oAi3zNrCzje3atdFzqeOaAePntrBzts3ztK5aNFDnghDoAvEzgs1) |
![{\displaystyle [-1,1]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aqsNoDJAatvFzAo3ags2atw3aDFDztGNzjm5otlEyqi4oDBCntmP) |
|
反正切 |
![{\displaystyle y=\arctan x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81a2eOzjrAzNBBygzDoNmQzthBzNm4oNoOntK5zNs1yqvFaDzCzNrE) |
![{\displaystyle x=\tan y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84yqs4oqo0zqw0ntm1ajrBzNC5aDnFnqhEoNmOnDzDa2iOnDwPngoN) |
![{\displaystyle \mathbb {R} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83yjo4zjBDzNo1ngi3otG0ngdDa2zDntwNntC2oqhCztFFztG2ygvD) |
|
反余切 |
![{\displaystyle y=\operatorname {arccot} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85yqzFagw2zDnFoAe2zqvBaDvCoDmNaNeOagdBoqvFaje1oNFDzNo4) |
![{\displaystyle x=\cot y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnjzBotBEnDoQntK5aDe0aNnBzDlEa2o4zNG3nAs4zAoPzgw3ytC3) |
![{\displaystyle \mathbb {R} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83yjo4zjBDzNo1ngi3otG0ngdDa2zDntwNntC2oqhCztFFztG2ygvD) |
|
反正割 |
![{\displaystyle y=\operatorname {arcsec} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ezje3zArFntBFzDGNoDlFyqe4zNhEajaNzqhAzAs4nqvFaAoNaAa1) |
![{\displaystyle x=\sec y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83o2hDnqePajvDnjm5nja5aDe1yjm0oDi5otwQnDhEa2dEyqnEnte4) |
![{\displaystyle (-\infty ,-1]\cup [1,+\infty )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ngrCnDFCngoOngzDytzEaAoQzNC5ngnCaDm2nDmQzDvFzNvBytvA) |
|
反余割 |
![{\displaystyle y=\operatorname {arccsc} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83yte2zNm4ytdFyqhAoDmPzgnBzDJDaNm4zjmOnga0atmOygo4ygi0) |
![{\displaystyle x=\csc y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FajFFzjvCaqsOothAytCOote4oDnEzjaNyjw3zNK4ato5ntGPnga4) |
![{\displaystyle (-\infty ,-1]\cup [1,+\infty )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ngrCnDFCngoOngzDytzEaAoQzNC5ngnCaDm2nDmQzDvFzNvBytvA) |
|
(注意:某些數學教科書的作者將
的值域定為
因為當
的定義域落在此區間時,
的值域
,如果
的值域仍定為
,將會造成
,如果希望
,那就必須將
的值域定為
,基於類似的理由
的值域定為
)
如果
允许是复数,则
的值域只适用它的实部。
反三角函数之间的关系[编辑]
余角:
![{\displaystyle \arccos x={\frac {\pi }{2}}-\arcsin x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Co2sQnDnFnAdCnDm4ytw0zDi2nDK3o2vFa2o3zqiQzta0oqzCnte3)
![{\displaystyle \operatorname {arccot} x={\frac {\pi }{2}}-\arctan x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eyqo2oNGPngo1ytGNyjFEytmNzNw2oDsOygdDnDi2ztnFyqi5aNsQ)
![{\displaystyle \operatorname {arccsc} x={\frac {\pi }{2}}-\operatorname {arcsec} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BajrCoqdFagw0oqs1nja2aDC2ytiQaNnFotmPo2nFz2w2aNnCyti1)
负数参数:
![{\displaystyle \arcsin(-x)=-\arcsin x\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AytsPnje0zNvFnDJDo2i5zNmQaghFyqhBnqe4otJBotBCyjrFaqhB)
![{\displaystyle \arccos(-x)=\pi -\arccos x\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nArBotGPnDeQnja3aji4nDm4zDhEnjmPoDrDoArFagvEnDs1nqoQ)
![{\displaystyle \arctan(-x)=-\arctan x\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aNBDnjeOoqvCzNzDngeOaNrAaAhAnDBAotzAyjGQagnAnjs1a2i5)
![{\displaystyle \operatorname {arccot}(-x)=\pi -\operatorname {arccot} x\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81z2dDajrCyghBntm1oNsNyqo4aqeQatJBage4o2nByqhDoNG2zji4)
![{\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ante2oNiOnqa3otG5yqvAa2w5zjnFzDGOo2aPotFDoDG2z2i3oAwO)
![{\displaystyle \operatorname {arccsc}(-x)=-\operatorname {arccsc} x\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oagw4zNhFa2oPoqsNnqw5zNi0o2zDnjwQatvFajBCnto5z2s4nte5)
倒数参数:
![{\displaystyle \arccos {\frac {1}{x}}\,=\operatorname {arcsec} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QagrDoDBBnjK0oqa3a2a2zNBDzjsQotG3z2dCytKOzjBFoAwOyjvE)
![{\displaystyle \arcsin {\frac {1}{x}}\,=\operatorname {arccsc} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dzqa0nAa3oNw4oqdDzDKPzja3zNw1zNhFytlCnDw5ygvFaqaNzDBE)
![{\displaystyle \ x>0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaqzDzjCOntmOzAo3a2zFnDnDyqw4nDhAnqiPzgrFytw5ntJDz2iN)
![{\displaystyle \ x<0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zNw5ygiOajKQoAdAyqzAate2nDzEats4nDaNnDC0aNnDnjaOnjK4)
![{\displaystyle \ x>0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaqzDzjCOntmOzAo3a2zFnDnDyqw4nDhAnqiPzgrFytw5ntJDz2iN)
![{\displaystyle \ x<0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zNw5ygiOajKQoAdAyqzAate2nDzEats4nDaNnDC0aNnDnjaOnjK4)
![{\displaystyle \operatorname {arcsec} {\frac {1}{x}}=\arccos x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ota2aDsQzjBBzDlAnqdDatdDaAzFzAw5nta2zgwPoNG1zqvBatGN)
![{\displaystyle \operatorname {arccsc} {\frac {1}{x}}=\arcsin x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CytmQoNG4a2i5yje3otdFoNnDoNG2agrEaAw2aNa3ajs4oDGOajrC)
如果有一段正弦表:
![{\displaystyle \ 0\leq x\leq 1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oDa5oAeNzNi0oNK4atC0zNnFnAnDntG5oNnDaAzAztFBaNzEz2dF)
![{\displaystyle \arctan x=\arcsin {\frac {x}{\sqrt {x^{2}+1}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oDC2aAnDzjm4atC2aAvAzNo2aDGQzAnCntG2aNGNzjJCotePaqw1)
注意只要在使用了复数的平方根的时候,我们选择正实部的平方根(或者正虚部,如果是负实数的平方根的话)。
从半角公式
,可得到:
![{\displaystyle \arcsin x=2\arctan {\frac {x}{1+{\sqrt {1-x^{2}}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AatKNo2e5ajvBoti1ytK2ygzBaNK0otdBoNoOaNsNztdFaDwOaqaO)
![{\displaystyle -1<x\leq +1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ntGQnAa3ntw0aNo5oNs0zAi5oqi3aDlDaNzDo2zDngvBngdAataO)
![{\displaystyle \arctan x=2\arctan {\frac {x}{1+{\sqrt {1+x^{2}}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Paqa2ygs3yji5oDw2oDGQztvDyjeNzqs0oqa2nDiPzDs4yjsPaja5)
三角函數與反三角函數的關係[编辑]
通過定義可知:
|
|
|
|
圖示
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
一般解[编辑]
每个三角函数都周期于它的参数的实部上,在每个
区间内通过它的所有值两次。正弦和余割的周期开始于
结束于
(这里的
是一个整数),在
到
上倒过来。余弦和正割的周期开始于
结束于
,在
到
上倒过来。正切的周期开始于
结束于
,接着(向前)在
到
上重复。余切的周期开始于
结束于
,接着(向前)在
到
上重复。
这个周期性反应在一般反函数上:
![{\displaystyle \sin y=x\ \Leftrightarrow \ (\ y=\arcsin x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=\pi -\arcsin x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BnqrDotsPota5ajoOo2w3zjo5oqa1oAe3a2a0zjBAz2hBzNeQyjKN)
![{\displaystyle \cos y=x\ \Leftrightarrow \ (\ y=\arccos x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=2\pi -\arccos x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BotsOagzDzte5aAw2otJEagvBatGOa2s0ats1aDdDyqa5yge1nDs3)
![{\displaystyle \tan y=x\ \Leftrightarrow \ \ y=\arctan x+k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoAoOzjw0zNsNats2aAzFzqs1ajdAoqo5aghAzgnBa2rBaDCNzjo4)
![{\displaystyle \cot y=x\ \Leftrightarrow \ \ y=\operatorname {arccot} x+k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ytePzjlCzqrAzNoOytoNoNw4agw5ytw5a2eOz2nEnDBFytFDyjsO)
![{\displaystyle \sec y=x\ \Leftrightarrow \ (\ y=\operatorname {arcsec} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=2\pi -\operatorname {arcsec} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EngvFo2zAzjrBzDdCaqwQngi0zgvEnqaQnts0zAi4oDeOaDiPoDwP)
![{\displaystyle \csc y=x\ \Leftrightarrow \ (\ y=\operatorname {arccsc} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ \lor \ y=\pi -\operatorname {arccsc} x+2k\pi {\text{ }}\forall {\text{ }}k\in \mathbb {Z} \ )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzDeNzNhBoDnEzta4aNw5aDmOaqsNzqeOoNa3aDsNoDaQntCQoAvA)
反三角函数的导数[编辑]
对于实数
的反三角函數的导函数如下:
![{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin x&{}={\frac {1}{\sqrt {1-x^{2}}}};\qquad |x|<1\\{\frac {\mathrm {d} }{\mathrm {d} x}}\arccos x&{}={\frac {-1}{\sqrt {1-x^{2}}}};\qquad |x|<1\\{\frac {\mathrm {d} }{\mathrm {d} x}}\arctan x&{}={\frac {1}{1+x^{2}}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccot} x&{}={\frac {-1}{1+x^{2}}}\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsec} x&{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\{\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccsc} x&{}={\frac {-1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9ByjJBnAo0ngrAzjrAzqzCnDK4aNi2nAo5zjCPzNi3oNrCzDi1nDrE)
舉例說明,设
,得到:
![{\displaystyle {\frac {d\arcsin x}{dx}}={\frac {d\theta }{d\sin \theta }}={\frac {1}{\cos \theta }}={\frac {1}{\sqrt {1-\sin ^{2}\theta }}}={\frac {1}{\sqrt {1-x^{2}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ajaNoDwNaDa4ztvAo2dDnDGNoqoNoqhCngo0zDnCagw0yta0aNrC)
因為要使根號內部恆為正,所以在條件加上
,其他導數公式同理可證[1]。
表达为定积分[编辑]
积分其导数并固定在一点上的值给出反三角函数作为定积分的表达式:
![{\displaystyle {\begin{aligned}\arcsin x&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arccos x&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arctan x&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arccot} x&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcsec} x&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arccsc} x&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PztBFz2i5zDmOzjzAzNC0oAoOygvFaqoNatG0oAwOatw5oteQyjlD)
当
等于1时,在有极限的域上的积分是瑕积分,但仍是良好定义的。
无穷级数[编辑]
如同正弦和余弦函数,反三角函数可以使用无穷级数计算如下:
![{\displaystyle {\begin{aligned}\arcsin z&{}=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \\&{}=\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{2n+1}}{(2n+1)}};\qquad |z|\leq 1\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DyjBCytw1ajK2oNhAo2s5njCOoqw1ztBBoNePaNlAotG4a2a1oAzB)
![{\displaystyle {\begin{aligned}\arccos z&{}={\frac {\pi }{2}}-\arcsin z\\&{}={\frac {\pi }{2}}-\left[z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \right]\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{2n+1}}{(2n+1)}};\qquad |z|\leq 1\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaqsPzNnAnAvDnqzBoDw4zAeOoAvByqe3ytG5ytrByts0zji0a2nB)
![{\displaystyle {\begin{aligned}\arctan z&{}=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \\&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ygaQoNaOaDmNo2vBzDwNytJFajC2oAw0nDo4zDCPnDi3aqe5yjBC)
![{\displaystyle {\begin{aligned}\operatorname {arccot} z&{}={\frac {\pi }{2}}-\arctan z\\&{}={\frac {\pi }{2}}-\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \right)\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoNo2zNdBnjwOzgwQzAdDzqeNatsNo2wQnDBDzNK2o2vDzjo1zjaP)
![{\displaystyle {\begin{aligned}\operatorname {arcsec} z&{}=\arccos \left(z^{-1}\right)\\&{}={\frac {\pi }{2}}-\left[z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots \right]\\&{}={\frac {\pi }{2}}-\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{-(2n+1)}}{(2n+1)}};\qquad \left|z\right|\geq -4\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CajBCaNo4zthDzte4ngoNngs5nDaNnjG1ota3zDaPaqwQytm1z2nD)
![{\displaystyle {\begin{aligned}\operatorname {arccsc} z&{}=\arcsin \left(z^{-1}\right)\\&{}=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{-7}}{7}}+\cdots \\&{}=\sum _{n=0}^{\infty }\left[{\frac {(2n)!}{2^{2n}(n!)^{2}}}\right]{\frac {z^{-(2n+1)}}{2n+1}};\qquad \left|z\right|\geq 1\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oNw3aNdEyts4zjlCaAvBzjBDagaNoNw4aDo3o2zDoqw1zti5ythB)
欧拉发现了反正切的更有效的级数:
。
(注意对
在和中的项是空积1。)
反三角函数的不定积分[编辑]
![{\displaystyle {\begin{aligned}\int \arcsin x\,dx&{}=x\,\arcsin x+{\sqrt {1-x^{2}}}+C,\qquad x\leq 1\\\int \arccos x\,dx&{}=x\,\arccos x-{\sqrt {1-x^{2}}}+C,\qquad x\leq 1\\\int \arctan x\,dx&{}=x\,\arctan x-{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arccot} x\,dx&{}=x\,\operatorname {arccot} x+{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arcsec} x\,dx&{}=x\,\operatorname {arcsec} x-\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C=x\,\operatorname {arcsec} x+\operatorname {sgn}(x)\ln \left|x-{\sqrt {x^{2}-1}}\right|+C\\\int \operatorname {arccsc} x\,dx&{}=x\,\operatorname {arccsc} x+\operatorname {sgn}(x)\ln \left|x+{\sqrt {x^{2}-1}}\right|+C=x\,\operatorname {arccsc} x-\operatorname {sgn}(x)\ln \left|x-{\sqrt {x^{2}-1}}\right|+C\\\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83agdAoDKNoNlBzgo1aja0yjaOaqdAo2s4ngiNoqnAzjeQytrDaNrC)
使用分部积分法和上面的简单导数很容易得出它们。
使用
,設
![{\displaystyle {\begin{aligned}u&{}=&\arcsin x&\quad \quad \mathrm {d} v=\mathrm {d} x\\\mathrm {d} u&{}=&{\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}&\quad \quad {}v=x\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AotiNzNzAzjzFoNBDags3nDm1oNnBathDoDnFaNm1aNrBnjw1aDFA)
則
![{\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin x-\int {\frac {x}{\sqrt {1-x^{2}}}}\,\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Da2zEytzBotFDa2e1yqhDatwOzgdEythCyjw5oAwOnAs2a2i2nAaP)
換元
![{\displaystyle k=1-x^{2}.\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85o2dAyqhCnqo3zjvCzDFAyja2nga4aAe1otGPoqi2nAsNotm1nDiQ)
則
![{\displaystyle \mathrm {d} k=-2x\,\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aNi0njlDaAo3zte5ytK3zDe3nDi3zNe0zDvEzgrCygzDyjwOytm4)
且
![{\displaystyle \int {\frac {x}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=-{\frac {1}{2}}\int {\frac {\mathrm {d} k}{\sqrt {k}}}=-{\sqrt {k}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnAzDoAzEo2zFoqa5nghAotBEzNoQzqe4oNG5aDJCnqe0ztGPyjFB)
換元回x得到
![{\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin x+{\sqrt {1-x^{2}}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaAvEoDFDyjdAzAs2ntiNnts1aAsQoqs4ytJDagiQzNC2zNe4zqwN)
加法公式和減法公式[编辑]
arcsin x + arcsin y[编辑]
![{\displaystyle \arcsin x+\arcsin y=\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),xy\leq 0\lor x^{2}+y^{2}\leq 1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oqvEzNwOzqe5aNaQzNdDaAe5ztdEzqs1aNhEntKNaNlEo2iQoDCN)
![{\displaystyle \arcsin x+\arcsin y=\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),x>0,y>0,x^{2}+y^{2}>1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DaNnEoAdEzjGQygs1aNBCygw0yjaNats2otzAaDG2yqw0ztBDnAs5)
![{\displaystyle \arcsin x+\arcsin y=-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),x<0,y<0,x^{2}+y^{2}>1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BoDw0zti1atCPygiQnga1yjw3yjnEzqvDnjCQytFDzgnFygs4njFC)
arcsin x - arcsin y[编辑]
![{\displaystyle \arcsin x-\arcsin y=\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right),xy\geq 0\lor x^{2}+y^{2}\leq 1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80otCQots1yjiNajnCzjG3zDFAoqaQoDe3ztK0atK1oto0a2hDotCO)
![{\displaystyle \arcsin x-\arcsin y=\pi -\arcsin \left(x{\sqrt {1-y^{2}}}-y{\sqrt {1-x^{2}}}\right),x>0,y<0,x^{2}+y^{2}>1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QataQo2rFoNrFaAiOnDrDaNe3z2s3yqa1zjs2ngsQoNlDyjw3oNnC)
![{\displaystyle \arcsin x-\arcsin y=-\pi -\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right),x<0,y>0,x^{2}+y^{2}>1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nDlFztGOztrAnqi5o2i2z2nFntw0zjnEaNw5agdEaAwPaArEz2iN)
arccos x + arccos y[编辑]
![{\displaystyle \arccos x+\arccos y=\arccos \left(xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x+y\geq 0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoDw2ntvEoAzCnjeOztG0oDiPyjKQntlEaqwPaAa1zDwOo2dFzqi2)
![{\displaystyle \arccos x+\arccos y=2\pi -\arccos \left(xy-{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x+y<0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaAw4aNFFaNiNygaNajvAoDo5ythAnDdBngw4oNw0yjKOnqzBatwN)
arccos x - arccos y[编辑]
![{\displaystyle \arccos x-\arccos y=-\arccos \left(xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x\geq y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oo2sPaNGOajs3z2hAntBDoAvCaNC3zta5ajFEo2dFoNdAo2w0ygzF)
![{\displaystyle \arccos x-\arccos y=\arccos \left(xy+{\sqrt {1-x^{2}}}\cdot {\sqrt {1-y^{2}}}\right),x<y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FyqnDyjsNzNmPotK1ajG1njFCyjKQoqaOnjG3nDrEaDa4ztm2nqa5)
arctan x + arctan y[编辑]
![{\displaystyle \arctan \,x+\arctan \,y=\arctan \,{\frac {x+y}{1-xy}},xy<1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zts1nta1oNm5aDe0oqdEotJBageOzjs4zNC5nje0oNo4nDsPzgw5)
![{\displaystyle \arctan \,x+\arctan \,y=\pi +\arctan \,{\frac {x+y}{1-xy}},x>0,xy>1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pz2e1yjo3ajFEatC2oDBAo2a1nqsOajvAztaNagzBzDvFoqi2ajKN)
![{\displaystyle \arctan \,x+\arctan \,y=-\pi +\arctan \,{\frac {x+y}{1-xy}},x<0,xy>1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NytBBnjeQntaPo2zDnDFDoAwPatnAa2aNaDvBota5aDK2atGPnge1)
arctan x - arctan y[编辑]
![{\displaystyle \arctan x-\arctan y=\arctan {\frac {x-y}{1+xy}},xy>-1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zjiQytsOyjmOoNwPyji1aNi0aDvBaDFEajm0zNw3aDaNoArFz2w0)
![{\displaystyle \arctan x-\arctan y=\pi +\arctan {\frac {x-y}{1+xy}},x>0,xy<-1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DajaQa2zAato4yjePnjzEnqiOnqa2atePzjhEoDzFnjvEzAsOaNsN)
![{\displaystyle \arctan x-\arctan y=-\pi +\arctan {\frac {x-y}{1+xy}},x<0,xy<-1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NajlBote0aDKOnDKOygnDoqaQztK3njaQaDJFoNK5yqzBnghCoqhA)
arccot x + arccot y[编辑]
![{\displaystyle \operatorname {arccot} x+\operatorname {arccot} y=\operatorname {arccot} {\frac {xy-1}{x+y}},x>-y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Ozgw1oNGQzts4oqe4oDwPaNzEnAe0a2sQagoPaghAyqw4oNoQzAo3)
![{\displaystyle \operatorname {arccot} x+\operatorname {arccot} y=\operatorname {arccot} {\frac {xy-1}{x+y}}+\pi ,x<-y}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nDzCoAvCo2oOagw2nDCNytm3aqoOzgi0yjo4zAhBz2hDzghAoqi0)
arcsin x + arccos y[编辑]
![{\displaystyle \arcsin x+\arccos x={\frac {\pi }{2}},|x|\leq 1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82aDJDaDJDotBDoNhEzDFCzDKNnji1zDdEz2e3ateNzAo4ngdBoDFD)
arctan x + arccot y[编辑]
![{\displaystyle \arctan x+\operatorname {arccot} x={\frac {\pi }{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Cage5nqa3ygw5njzAaqiQatw4ajnAoDoOyqwOntG2nqePnqoPzjwN)
- ^
设
,得到:
![{\displaystyle {\frac {d\arccos x}{dx}}={\frac {d\theta }{d\cos \theta }}={\frac {-1}{\sin \theta }}={\frac {1}{\sqrt {1-\cos ^{2}\theta }}}={\frac {-1}{\sqrt {1-x^{2}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oNBCoDrDzjK3z2zDoNmQatK2agrAnArByjwPygdBzDrBygzAoNK2)
因為要使根號內部恆為正,所以在條件加上
。
设
,得到:
![{\displaystyle {\frac {d\arctan x}{dx}}={\frac {d\theta }{d\tan \theta }}={\frac {1}{\sec ^{2}\theta }}={\frac {1}{1+\tan ^{2}\theta }}={\frac {1}{1+x^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80agiOzqiQoqwQyjK2oAvFytBAzNFBo2o3oDJAajJEoqs3zNK4otK4)
设
,得到:
![{\displaystyle {\frac {d\operatorname {arccot} x}{dx}}={\frac {d\theta }{d\cot \theta }}={\frac {-1}{\csc ^{2}\theta }}={\frac {1}{1+\cot ^{2}\theta }}={\frac {-1}{1+x^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oDmOytdFaNoOo2vFaqs3zqoNotiQaNJAytGPnqi2otdFnjC1a2oQ)
设
,得到:
![{\displaystyle {\frac {d\operatorname {arcsec} x}{dx}}={\frac {d\theta }{d\sec \theta }}={\frac {1}{\sec \theta \tan \theta }}={\frac {1}{\left|x\right|{\sqrt {x^{2}-1}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AztlAotsNoNdEnqa2zAiOoNdFoNi5nDBEoto4zgePoNvByghAo2rD)
因為要使根號內部恆為正,所以在條件加上
,比較容易被忽略是
產生的絕對值
的定義域是
,所以
,所以
要加绝对值。
设
,得到:
![{\displaystyle {\frac {d\operatorname {arccsc} x}{dx}}={\frac {d\theta }{d\csc \theta }}={\frac {-1}{\csc \theta \cot \theta }}={\frac {-1}{\left|x\right|{\sqrt {x^{2}-1}}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnDo4zqo0ajoPoDG3oDK3zAdEo2i0atrByqs5nji0zDKNotdDyghA)
因為要使根號內部恆為正,所以在條件加上
,比較容易被忽略是
產生的絕對值
的定義域是
。
外部链接[编辑]