# 三角函數公式一覽

（注意：喺呢篇文章裏面，角度嘅單位係弧度。）

## 單位圓定義

${\displaystyle \sin \theta ={\frac {y}{r}}}$

${\displaystyle \cos \theta ={\frac {x}{r}}}$

${\displaystyle \tan \theta ={\frac {y}{x}}}$

${\displaystyle \sec \theta ={\frac {r}{x}}}$

${\displaystyle \csc \theta ={\frac {r}{y}}}$

${\displaystyle \cot \theta ={\frac {x}{y}}}$

## 相互關係

${\displaystyle \csc \theta ={\frac {1}{\sin \theta }}}$

${\displaystyle \sec \theta ={\frac {1}{\cos \theta }}}$

${\displaystyle \cot \theta ={\frac {1}{\tan \theta }}}$

${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {\sec \theta }{\csc \theta }}}$

${\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}={\frac {\csc \theta }{\sec \theta }}}$

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}$

${\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }$

${\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }$

## 移相公式

${\displaystyle \sin(-x)=-\sin(x)}$

${\displaystyle \cos(-x)=\cos(x)}$

${\displaystyle \tan(-x)=-\tan(x)}$

${\displaystyle \sec(-x)=\sec(x)}$

${\displaystyle \csc(-x)=-\csc(x)}$

${\displaystyle \cot(-x)=-\cot(x)}$

${\displaystyle \sin(x+2n\pi )=\sin(x)\quad \forall n\in \mathbb {Z} }$

${\displaystyle \cos(x+2n\pi )=\cos(x)\quad \forall n\in \mathbb {Z} }$

${\displaystyle \tan(x+n\pi )=\tan(x)\quad \forall n\in \mathbb {Z} }$

${\displaystyle \sec(x+2n\pi )=\sec(x)\quad \forall n\in \mathbb {Z} }$

${\displaystyle \csc(x+2n\pi )=\csc(x)\quad \forall n\in \mathbb {Z} }$

${\displaystyle \cot(x+n\pi )=\cot(x)\quad \forall n\in \mathbb {Z} }$

${\displaystyle \sin(x+\pi )=-\sin(x)}$

${\displaystyle \cos(x+\pi )=-\cos(x)}$

${\displaystyle \sec(x+\pi )=-\sec(x)}$

${\displaystyle \csc(x+\pi )=-\csc(x)}$

${\displaystyle \sin(\pi -x)=\sin(x)}$

${\displaystyle \cos(\pi -x)=-\cos(x)}$

${\displaystyle \tan(\pi -x)=-\tan(x)}$

${\displaystyle \sec(\pi -x)=-\sec(x)}$

${\displaystyle \csc(\pi -x)=\csc(x)}$

${\displaystyle \cot(\pi -x)=-\cot(x)}$

${\displaystyle \sin({\frac {\pi }{2}}+x)=\cos(x)}$

${\displaystyle \cos({\frac {\pi }{2}}+x)=-\sin(x)}$

${\displaystyle \tan({\frac {\pi }{2}}+x)=-\cot(x)}$

${\displaystyle \sec({\frac {\pi }{2}}+x)=-\csc(x)}$

${\displaystyle \csc({\frac {\pi }{2}}+x)=\sec(x)}$

${\displaystyle \cot({\frac {\pi }{2}}+x)=-\tan(x)}$

${\displaystyle \sin({\frac {\pi }{2}}-x)=\cos(x)}$

${\displaystyle \cos({\frac {\pi }{2}}-x)=\sin(x)}$

${\displaystyle \tan({\frac {\pi }{2}}-x)=\cot(x)}$

${\displaystyle \sec({\frac {\pi }{2}}-x)=\csc(x)}$

${\displaystyle \csc({\frac {\pi }{2}}-x)=\sec(x)}$

${\displaystyle \cot({\frac {\pi }{2}}-x)=\tan(x)}$

${\displaystyle \sin({\frac {\pi }{4}}+x)={\frac {\sqrt {2}}{2}}(\sin(x)+\cos(x))}$

${\displaystyle \cos({\frac {\pi }{4}}+x)={\frac {\sqrt {2}}{2}}(\cos(x)-\sin(x))}$

${\displaystyle \tan({\frac {\pi }{4}}+x)={\frac {1+\tan(x)}{1-\tan(x)}}}$

${\displaystyle \sec({\frac {\pi }{4}}+x)={\frac {{\sqrt {2}}\sec(x)\csc(x)}{\csc(x)-\sec(x)}}}$

${\displaystyle \csc({\frac {\pi }{4}}+x)={\frac {{\sqrt {2}}\sec(x)\csc(x)}{\csc(x)+\sec(x)}}}$

${\displaystyle \cot({\frac {\pi }{4}}+x)={\frac {\cot(x)-1}{\cot(x)+1}}}$

## 複角公式

${\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)}$

${\displaystyle \sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)}$

${\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)}$

${\displaystyle \cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)}$

${\displaystyle \tan(x+y)={\frac {\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}}}$

${\displaystyle \tan(x-y)={\frac {\tan(x)-\tan(y)}{1+\tan(x)\tan(y)}}}$

${\displaystyle \sec(x+y)={\frac {\sec(x)\sec(y)\csc(x)\csc(y)}{\csc(x)\csc(y)-\sec(x)\sec(y)}}}$

${\displaystyle \sec(x-y)={\frac {\sec(x)\sec(y)\csc(x)\csc(y)}{\csc(x)\csc(y)+\sec(x)\sec(y)}}}$

${\displaystyle \csc(x+y)={\frac {\sec(x)\sec(y)\csc(x)\csc(y)}{\sec(x)\csc(y)+\csc(x)\sec(y)}}}$

${\displaystyle \csc(x-y)={\frac {\sec(x)\sec(y)\csc(x)\csc(y)}{\sec(x)\csc(y)-\csc(x)\sec(y)}}}$

${\displaystyle \cot(x+y)={\frac {\cot(x)\cot(y)-1}{\cot(x)+\cot(y)}}}$

${\displaystyle \cot(x-y)={\frac {\cot(x)\cot(y)+1}{\cot(y)-\cot(x)}}}$

## 倍角公式

${\displaystyle \sin(2x)=2\sin(x)\cos(x)}$

${\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)}$

${\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}}$

${\displaystyle \sec(2x)={\frac {\sec ^{2}(x)\csc ^{2}(x)}{\csc ^{2}(x)-\sec ^{2}(x)}}={\frac {\sec ^{2}(x)}{2-\sec ^{2}(x)}}={\frac {\csc ^{2}(x)}{\csc ^{2}(x)-2}}}$

${\displaystyle \csc(2x)={\frac {\sec(x)\csc(x)}{2}}}$

${\displaystyle \cot(2x)={\frac {\cot ^{2}(x)-1}{2\cot(x)}}}$

${\displaystyle \sin(3x)=3\sin(x)-4\sin ^{3}(x)=4\sin(x)\sin({\frac {\pi }{3}}-x)\sin({\frac {\pi }{3}}+x)}$

${\displaystyle \cos(3x)=4\cos ^{3}(x)-3\cos(x)=4\cos(x)\cos({\frac {\pi }{3}}-x)\cos({\frac {\pi }{3}}+x)}$

${\displaystyle \tan(3x)={\frac {\tan ^{3}(x)-3\tan(x)}{3\tan ^{2}(x)-1}}=\tan(x)\tan({\frac {\pi }{3}}-x)\tan({\frac {\pi }{3}}+x)}$

${\displaystyle \sec(3x)={\frac {\sec ^{3}(x)}{4-3\sec ^{2}(x)}}}$

${\displaystyle \csc(3x)={\frac {\csc ^{3}(x)}{3\csc ^{2}(x)-4}}}$

${\displaystyle \cot(3x)={\frac {\cot ^{3}(x)-3\cot(x)}{3\cot ^{2}(x)-1}}}$

## 積化和差公式

${\displaystyle \sin(x)\cos(y)={\frac {1}{2}}(\sin(x+y)+\sin(x-y))}$

${\displaystyle \cos(x)\sin(y)={\frac {1}{2}}(\sin(x+y)-\sin(x-y))}$

${\displaystyle \cos(x)\cos(y)={\frac {1}{2}}(\cos(x+y)+\cos(x-y))}$

${\displaystyle \sin(x)\sin(y)={\frac {1}{2}}(\cos(x-y)-\cos(x+y))}$

## 和差化積公式

${\displaystyle \sin(x)+\sin(y)=2\sin({\frac {x+y}{2}})\cos({\frac {x-y}{2}})}$

${\displaystyle \sin(x)-\sin(y)=2\cos({\frac {x+y}{2}})\sin({\frac {x-y}{2}})}$

${\displaystyle \cos(x)+\cos(y)=2\cos({\frac {x+y}{2}})\cos({\frac {x-y}{2}})}$

${\displaystyle \cos(x)-\cos(y)=-2\sin({\frac {x+y}{2}})\sin({\frac {x-y}{2}})}$

${\displaystyle \tan(x)+\tan(y)={\frac {\sin(x+y)}{\cos(x)\cos(y)}}}$

${\displaystyle \tan(x)-\tan(y)={\frac {\sin(x-y)}{\cos(x)\cos(y)}}}$

## 三隻角總和係 ${\displaystyle \pi }$ 嘅條件恆等式

${\displaystyle \tan(x)+\tan(y)+\tan(z)=\tan(x)\tan(y)\tan(z)}$

${\displaystyle \cot({\frac {x}{2}})+\cot({\frac {y}{2}})+\cot({\frac {z}{2}})=\cot({\frac {x}{2}})\cot({\frac {y}{2}})\cot({\frac {z}{2}})}$

${\displaystyle \tan({\frac {x}{2}})\tan({\frac {y}{2}})+\tan({\frac {x}{2}})\tan({\frac {z}{2}})+\tan({\frac {y}{2}})\tan({\frac {z}{2}})=1}$

${\displaystyle \cot(x)\cot(y)+\cot(x)\cot(z)+\cot(y)\cot(z)=1}$

${\displaystyle \sin(x)+\sin(y)+\sin(z)=4\cos({\frac {x}{2}})\cos({\frac {y}{2}})\cos({\frac {z}{2}})}$

${\displaystyle -\sin(x)+\sin(y)+\sin(z)=4\cos({\frac {x}{2}})\sin({\frac {y}{2}})\sin({\frac {z}{2}})}$

${\displaystyle \cos(x)+\cos(y)+\cos(z)=4\sin({\frac {x}{2}})\sin({\frac {y}{2}})\sin({\frac {z}{2}})+1}$

${\displaystyle -\cos(x)+\cos(y)+\cos(z)=4\sin({\frac {x}{2}})\cos({\frac {y}{2}})\cos({\frac {z}{2}})-1}$

${\displaystyle \sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\sin(y)\sin(z)}$

${\displaystyle -\sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\cos(y)\cos(z)}$

${\displaystyle \cos(2x)+\cos(2y)+\cos(2z)=-4\cos(x)\cos(y)\cos(z)-1}$

${\displaystyle -\cos(2x)+\cos(2y)+\cos(2z)=-4\cos(x)\sin(y)\sin(z)+1}$

${\displaystyle \sin ^{2}(x)+\sin ^{2}(y)+\sin ^{2}(z)=2\cos(x)\cos(y)\cos(z)+2}$

${\displaystyle -\sin ^{2}(x)+\sin ^{2}(y)+\sin ^{2}(z)=2\cos(x)\sin(y)\sin(z)}$

${\displaystyle \cos ^{2}(x)+\cos ^{2}(y)+\cos ^{2}(z)=-2\cos(x)\cos(y)\cos(z)+1}$

${\displaystyle -\cos ^{2}(x)+\cos ^{2}(y)+\cos ^{2}(z)=-2\cos(x)\sin(y)\sin(z)+1}$

${\displaystyle -\sin ^{2}(2x)+\sin ^{2}(2y)+\sin ^{2}(2z)=-2\cos(2x)\sin(2y)\sin(2z)}$

${\displaystyle -\cos ^{2}(2x)+\cos ^{2}(2y)+\cos ^{2}(2z)=2\cos(2x)\sin(2y)\sin(2z)+1}$

${\displaystyle \sin ^{2}({\frac {x}{2}})+\sin ^{2}({\frac {y}{2}})+\sin ^{2}({\frac {z}{2}})+2\sin({\frac {x}{2}})\sin({\frac {y}{2}})\sin({\frac {z}{2}})=1}$

## 泰勒級數

${\displaystyle \sin(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$

${\displaystyle \cos(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }$

${\displaystyle \sin ^{-1}(x)=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}=x+{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}+\cdots }$

${\displaystyle \cos ^{-1}(x)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}={\frac {\pi }{2}}-x-{\frac {x^{3}}{6}}-{\frac {3x^{5}}{40}}-\cdots }$

${\displaystyle \tan ^{-1}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots }$

## 複數公式

${\displaystyle (\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)}$狄默夫公式

${\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}}$

${\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}}$

${\displaystyle \tan(x)=-{\frac {i(e^{ix}-e^{-ix})}{e^{ix}+e^{-ix}}}}$

${\displaystyle \sec(x)={\frac {2}{e^{ix}+e{-ix}}}}$

${\displaystyle \csc(x)={\frac {2i}{e^{ix}-e^{-ix}}}}$

${\displaystyle \cot(x)={\frac {i(e^{ix}+e^{-ix})}{e^{ix}-e^{-ix}}}}$

${\displaystyle \sin ^{-1}(x)=-i\ln(ix+{\sqrt {1-x^{2}}})}$

${\displaystyle \cos ^{-1}(x)=-i\ln(x+{\sqrt {x^{2}-1}})}$

${\displaystyle \tan ^{-1}(x)={\frac {i}{2}}\ln({\frac {i+x}{i-x}})}$

${\displaystyle \csc ^{-1}(x)=-i\ln({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}})}$

${\displaystyle \sec ^{-1}(x)=-i\ln({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}})}$

${\displaystyle \cot ^{-1}(x)={\frac {i}{2}}\ln({\frac {x-i}{x+i}})}$

### 雙曲函數

${\displaystyle \sin(ix)=i\sinh(x)}$

${\displaystyle \cos(ix)=i\cosh(x)}$

${\displaystyle \tan(ix)=i\tanh(x)}$

## 冇變量嘅恆等式

${\displaystyle \tan ^{-1}({\frac {1}{3}})+\tan ^{-1}({\frac {1}{7}})=\tan ^{-1}({\frac {1}{2}})}$

${\displaystyle \tan ^{-1}({\frac {1}{5}})+\tan ^{-1}({\frac {1}{8}})=\tan ^{-1}({\frac {1}{3}})}$

${\displaystyle \tan ^{-1}({\frac {1}{2}})+\tan ^{-1}({\frac {1}{3}})={\frac {\pi }{4}}}$

${\displaystyle 2\tan ^{-1}({\frac {1}{3}})+\tan ^{-1}({\frac {1}{7}})={\frac {\pi }{4}}}$

${\displaystyle \tan ^{-1}({\frac {1}{3}})+\tan ^{-1}({\frac {1}{5}})+\tan ^{-1}({\frac {1}{7}})+\tan ^{-1}({\frac {1}{8}})={\frac {\pi }{4}}}$

${\displaystyle 4\tan ^{-1}({\frac {1}{5}})-\tan ^{-1}({\frac {1}{239}})={\frac {\pi }{4}}}$

${\displaystyle 5\tan ^{-1}({\frac {1}{7}})+2\tan ^{-1}({\frac {3}{79}})={\frac {\pi }{4}}}$歐拉

${\displaystyle \tan ^{-1}{1}+\tan ^{-1}{2}+\tan ^{-1}{3}=\pi }$

${\displaystyle \cos ^{-1}{\frac {4}{5}}+\cos ^{-1}{\frac {5}{13}}+\cos ^{-1}{\frac {16}{65}}=\sin ^{-1}{\frac {3}{5}}+\sin ^{-1}{\frac {12}{13}}+\sin ^{-1}{\frac {63}{65}}=\pi }$

${\displaystyle \cos({\frac {\pi }{9}})\cdot \cos({\frac {2\pi }{9}})\cdot \cos({\frac {4\pi }{9}})={\frac {1}{8}}}$

${\displaystyle \sin({\frac {\pi }{9}})\cdot \sin({\frac {2\pi }{9}})\cdot \sin({\frac {4\pi }{9}})={\frac {\sqrt {3}}{8}}}$

${\displaystyle \sin({\frac {\pi }{18}})\cdot \cos({\frac {5\pi }{18}})\cdot \cos({\frac {7\pi }{18}})={\frac {1}{8}}}$

${\displaystyle \sin({\frac {\pi }{12}})\cdot \sin({\frac {5\pi }{12}})={\frac {1}{4}}}$

${\displaystyle \cos({\frac {\pi }{12}})\cdot \cos({\frac {5\pi }{12}})={\frac {1}{4}}}$

${\displaystyle \tan({\frac {2\pi }{9}})\cdot \tan({\frac {\pi }{6}})\cdot \tan({\frac {\pi }{9}})=\tan({\frac {\pi }{18}})}$

${\displaystyle \tan({\frac {5\pi }{18}})\cdot \tan({\frac {\pi }{3}})\cdot \tan({\frac {7\pi }{18}})=\tan({\frac {4\pi }{9}})}$

${\displaystyle \cos({\frac {2\pi }{15}})+\cos({\frac {4\pi }{15}})+\cos({\frac {8\pi }{15}})+\cos({\frac {14\pi }{15}})={\frac {1}{2}}}$

${\displaystyle \sin ^{2}({\frac {\pi }{10}})+\sin ^{2}({\frac {\pi }{6}})=\sin ^{2}({\frac {\pi }{5}})}$歐幾里得