正弦

正弦的符号为${\displaystyle \sin }$，取自拉丁文sinus。而拉丁文sinus是来自阿拉伯文jiba的误译。阿拉伯文jiba来自梵文jya-ardha，意思是“半根弓弦”。以单位圆方式定义，如果把圆弧想象成一张弓，那么正弦的就好像是弓弦的一半长。该符号最早由瑞士数学家欧拉所使用。

${\displaystyle \int \sin cx\;dx=-{\frac {1}{c}}\cos cx\,\!}$

${\displaystyle \int |\sin x|\;dx=-\cos x\,\!}$

${\displaystyle \int \sin ^{n}{cx}\;dx=-{\frac {\sin ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \sin ^{n-2}cx\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}$

${\displaystyle \int \sin ^{2}{cx}\;dx={\frac {x}{2}}-{\frac {1}{4c}}\sin 2cx={\frac {x}{2}}-{\frac {1}{2c}}\sin cx\cos cx\!}$

${\displaystyle \int {\sqrt {1-\sin {x}}}\;dx=\int {\sqrt {\operatorname {cvs} {x}}}\,dx=2{\frac {\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}{\sqrt {\operatorname {cvs} {x}}}=2{\sqrt {1+\sin {x}}}}$

${\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}}}-{\frac {x\cos cx}{c}}\,\!}$

${\displaystyle \int x^{n}\sin cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!}$

${\displaystyle \int _{\frac {-a}{2}}^{\frac {a}{2}}x^{2}\sin ^{2}{\frac {n\pi x}{a}}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}}}\qquad {\mbox{(for }}n=2,4,6...{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {\sin cx}{x}}\;dx=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}\,\!}$

${\displaystyle \int {\frac {\sin cx}{x^{n}}}\;dx=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx\,\!}$

${\displaystyle \int {\frac {dx}{\sin cx}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|}$

${\displaystyle \int {\frac {dx}{\sin ^{n}cx}}={\frac {\cos cx}{c(1-n)\sin ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\sin ^{n-2}cx}}\qquad {\mbox{(for }}n>1{\mbox{)}}\,\!}$

${\displaystyle \int {\frac {dx}{1\pm \sin cx}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}$

${\displaystyle \int {\frac {x\;dx}{1+\sin cx}}={\frac {x}{c}}\tan \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}$

${\displaystyle \int {\frac {x\;dx}{1-\sin cx}}={\frac {x}{c}}\cot \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}$

${\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\tan \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}$

${\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(for }}|c_{1}|\neq |c_{2}|{\mbox{)}}\,\!}$

正弦定理說明对于任意三角形，它的边是${\displaystyle a}$, ${\displaystyle b}$和${\displaystyle c}$而相对这些边的角是${\displaystyle A}$, ${\displaystyle B}$和${\displaystyle C}$，有：

${\displaystyle {\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}}$

也表示为：

${\displaystyle {\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R}$

它可以通过把三角形分为两个直角三角形并使用正弦的上述定义证明。在这个定理中出现的公共数${\displaystyle {\frac {\sin A}{a}}}$是通过${\displaystyle A}$, ${\displaystyle B}$和${\displaystyle C}$三点的圆的直径的倒数。正弦定理用于在一个三角形的两个角和一个边已知时计算未知边的长度。这是三角测量中常见情况。

• 維基共享資源中与正弦相關的分類

• 餘弦
• 正切
• 餘切
• 正割
• 餘割
• 三角学
• 三角函数
• 函數
• 正弦波