提示:此条目页的主题不是
弦函數。
在數學中,正弦(英語:sine、縮寫
)是一種週期函數,是三角函数的一種。它的定义域是整个实数集,值域是
。它是周期函数,其最小正周期为
(
)。在自变量为
(
,其中
为整数)时,该函数有极大值1;在自变量为
(
)时,该函数有极小值-1。正弦函数是奇函数,其图像于原点对称。
正弦 |
![](//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Sin.svg/220px-Sin.svg.png) |
性質 |
奇偶性 | 奇 |
定義域 | (-∞,∞) |
到達域 | [-1,1] |
周期 | ![{\displaystyle 2\pi }](//wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06) ( ) |
特定值 |
當x=0 | 0 |
當x=+∞ | N/A |
當x=-∞ | N/A |
最大值 | ![{\displaystyle \left(\left(2k+{\tfrac {1}{2}}\right)\pi ,1\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/1c491de278446eb917b6391fc8566437df988790)
|
最小值 | ![{\displaystyle \left(\left(2k-{\tfrac {1}{2}}\right)\pi ,-1\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/1413b2096869b074181849a094239f10d47d182d)
|
其他性質 |
渐近线 | N/A |
根 | ![{\displaystyle k\pi }](//wikimedia.org/api/rest_v1/media/math/render/svg/bf859397db5c3d7bddebe20b20a69d8191f2448f) ( ) |
臨界點 | ![{\displaystyle k\pi -{\tfrac {\pi }{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/fffe0581eec379818a0a8f8a0e56da6ee42a8f93) ( ) |
拐點 | ![{\displaystyle k\pi }](//wikimedia.org/api/rest_v1/media/math/render/svg/bf859397db5c3d7bddebe20b20a69d8191f2448f) ( ) |
不動點 | 0 |
k是一個整數。 |
在半个最小正周期内,正弦函数有反函数,称为反正弦函数。
正弦的符号为
,取自拉丁文sinus,词源是梵文的jiva(“弓弦”,如今多写作jya)。这个词在阿拉伯语里转写为jiba(جيب),但该词无意义,阿拉伯语又好省略元音,故只写作jb(جب)。然而在从阿拉伯文翻译到拉丁文时,jb被解释为jayb(جيب),意为“胸部”或“乳房”,而拉丁文sinus便是克雷莫纳的杰拉德由此词翻译而来。该符号最早由法国数学家阿尔贝·热拉尔(Albert Gerard)使用(但他只使用了正弦、余弦和正切;其余三个符号则是被欧拉补足的)。
正弦函数(蓝色)的七阶泰勒公式(粉色)在以原点为中心的一个周期内紧密地逼近原函数
![{\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/20a1d4be32dec2312bf54e5f02a8a5f7c39ffcca)
由于正弦的导数是余弦,余弦的导数是负的正弦,因此正弦函数满足初值問題
![{\displaystyle y''=-y,\,y(0)=0,\,y'(0)=1}](//wikimedia.org/api/rest_v1/media/math/render/svg/7f6329ddb74d494794893217035f530ffd655e8b)
这就是正弦的微分方程定义。
正弦函數的指數定義可由歐拉公式導出:
![{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/67b0d0fcc1c0e6c76ffe716740ff5ee5cc7f2aaa)
![{\displaystyle \sin \left(x+y\right)=\sin x\cos y+\cos x\sin y}](//wikimedia.org/api/rest_v1/media/math/render/svg/e863739245ff9022e0766723a53d52f05fc06017)
![{\displaystyle \sin \left(x-y\right)=\sin x\cos y-\cos x\sin y}](//wikimedia.org/api/rest_v1/media/math/render/svg/e881a8119150497d49cff66cd28e3bf03aa592d6)
![{\displaystyle \sin 2\theta =2\sin \theta \cos \theta \,}](//wikimedia.org/api/rest_v1/media/math/render/svg/c1a9a9901f4cbefa2a7d67a2b97504270668f0da)
![{\displaystyle \sin 3\theta =3\sin \theta -4\sin ^{3}\theta \,}](//wikimedia.org/api/rest_v1/media/math/render/svg/fe2b8785f6f7d9ad1c0c0d3efe15a26f1700ad80)
![{\displaystyle \sin {\frac {\theta }{2}}=\pm \,{\sqrt {\frac {1-\cos \theta }{2}}}.\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/17f51406879bd10580d8ef4aef356a2ed2c905fc)
![{\displaystyle \sin \theta +\sin \phi =2\sin \left({\frac {\theta +\phi }{2}}\right)\cos \left({\frac {\theta -\phi }{2}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/81fbad7cfdaa0650b1a778599b23df0931386fed)
![{\displaystyle \sin \theta -\sin \phi =2\cos \left({\theta +\phi \over 2}\right)\sin \left({\theta -\phi \over 2}\right)\;}](//wikimedia.org/api/rest_v1/media/math/render/svg/32808e40e083a7fb62dc44bb24e04be473124f6b)
![{\displaystyle \sin \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f4331d15e171566984437fa575f9b984b17f4507)
![{\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/90927a9524ca2ebb7532b3b642442416f997074c)
![{\displaystyle \int |\sin x|\;dx=-\cos x\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/846431e6e44ad8e51f41719575ce87c1d3c855ce)
![{\displaystyle \int \sin ^{n}{cx}\;dx=-{\frac {\sin ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \sin ^{n-2}cx\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/9b8b9b811f6e7f9d5ab90a2bcb4a52ead77bd18b)
![{\displaystyle \int \sin ^{2}{cx}\;dx={\frac {x}{2}}-{\frac {1}{4c}}\sin 2cx={\frac {x}{2}}-{\frac {1}{2c}}\sin cx\cos cx\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/8e299cc3a3d71b32d9a94abc0a2d8bdddfc7702c)
![{\displaystyle \int {\sqrt {1-\sin {x}}}\;dx=\int {\sqrt {\operatorname {cvs} {x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}{\sqrt {\operatorname {cvs} {x}}}=2{\sqrt {1+\sin {x}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/aec57de36d44af1514dd20cdd00aca2d628c88be)
![{\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/1ba196bb420de05f3a536e722b9fa9b633ae1edb)
![{\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/9504df52028a00ecd0140cefe037d9c33d2dfb86)
![{\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=2,4,6...{\mbox{)}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/7d006ad538fdc542282e62804c82784991ce6474)
![{\displaystyle \int {\frac {\sin cx}{x}}\;dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/1826a014302387415dd1d360bc4a9168ede65017)
![{\displaystyle \int {\frac {\sin cx}{x^{n}}}\;dx=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/26c1849317a483d658b4c1089dff98b610c78758)
![{\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|}](//wikimedia.org/api/rest_v1/media/math/render/svg/33e9b1d708e4a982d400c2bf4ce60212c69b1273)
![{\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/d96874c7eb9e64a763a34b84fc8713b5e47ae156)
![{\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/ce341569f2cd1bf50b4d093c9a4a5c6f236505ac)
![{\displaystyle \int {\frac {x\;dx}{1+\sin cx}}={\frac {x}{c}}\tan \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}](//wikimedia.org/api/rest_v1/media/math/render/svg/b2ad92be2fe75f28b3a47ea3948950df8e6efcb4)
![{\displaystyle \int {\frac {x\;dx}{1-\sin cx}}={\frac {x}{c}}\cot \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}](//wikimedia.org/api/rest_v1/media/math/render/svg/57b478db3bb323ac13e34688e726f9c5b72fb44d)
![{\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\tan \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/b0e1d0c0992f04eb68ac1d4b84f8c8fd96e43064)
![{\displaystyle \int \sin c_{1}x\sin 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{(for }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/7a22f66f9d1b2febaa3694002e5e277de20e5d96)