（注意：喺呢篇文章裏面，角度嘅單位係弧度。）
以下係一咋三角函數公式。
喺笛卡兒坐標系統裏面，畫一個半徑係 r {\displaystyle r} 嘅圓，對於喺圓上面嘅任何一點 ( x , y ) {\displaystyle (x,y)} ，有：
sin θ = y r {\displaystyle \sin \theta ={\frac {y}{r}}}
cos θ = x r {\displaystyle \cos \theta ={\frac {x}{r}}}
tan θ = y x {\displaystyle \tan \theta ={\frac {y}{x}}}
csc θ = r y {\displaystyle \csc \theta ={\frac {r}{y}}}
sec θ = r x {\displaystyle \sec \theta ={\frac {r}{x}}}
cot θ = x y {\displaystyle \cot \theta ={\frac {x}{y}}}
備注： r = x 2 + y 2 {\displaystyle r={\sqrt {x^{2}+y^{2}}}}
csc θ = 1 sin θ {\displaystyle \csc \theta ={\frac {1}{\sin \theta }}}
sec θ = 1 cos θ {\displaystyle \sec \theta ={\frac {1}{\cos \theta }}}
cot θ = 1 tan θ {\displaystyle \cot \theta ={\frac {1}{\tan \theta }}}
tan θ = sin θ cos θ = sec θ csc θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}={\frac {\sec \theta }{\csc \theta }}}
cot θ = cos θ sin θ = csc θ sec θ {\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}={\frac {\csc \theta }{\sec \theta }}}
sin 2 θ + cos 2 θ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1}
1 + tan 2 θ = sec 2 θ {\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
1 + cot 2 θ = csc 2 θ {\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
sin ( − x ) = − sin ( x ) {\displaystyle \sin(-x)=-\sin(x)}
cos ( − x ) = cos ( x ) {\displaystyle \cos(-x)=\cos(x)}
tan ( − x ) = − tan ( x ) {\displaystyle \tan(-x)=-\tan(x)}
sec ( − x ) = sec ( x ) {\displaystyle \sec(-x)=\sec(x)}
csc ( − x ) = − csc ( x ) {\displaystyle \csc(-x)=-\csc(x)}
cot ( − x ) = − cot ( x ) {\displaystyle \cot(-x)=-\cot(x)}
sin ( x + 2 n π ) = sin ( x ) ∀ n ∈ Z {\displaystyle \sin(x+2n\pi )=\sin(x)\quad \forall n\in \mathbb {Z} }
cos ( x + 2 n π ) = cos ( x ) ∀ n ∈ Z {\displaystyle \cos(x+2n\pi )=\cos(x)\quad \forall n\in \mathbb {Z} }
tan ( x + n π ) = tan ( x ) ∀ n ∈ Z {\displaystyle \tan(x+n\pi )=\tan(x)\quad \forall n\in \mathbb {Z} }
sec ( x + 2 n π ) = sec ( x ) ∀ n ∈ Z {\displaystyle \sec(x+2n\pi )=\sec(x)\quad \forall n\in \mathbb {Z} }
csc ( x + 2 n π ) = csc ( x ) ∀ n ∈ Z {\displaystyle \csc(x+2n\pi )=\csc(x)\quad \forall n\in \mathbb {Z} }
cot ( x + n π ) = cot ( x ) ∀ n ∈ Z {\displaystyle \cot(x+n\pi )=\cot(x)\quad \forall n\in \mathbb {Z} }
sin ( x + π ) = − sin ( x ) {\displaystyle \sin(x+\pi )=-\sin(x)}
cos ( x + π ) = − cos ( x ) {\displaystyle \cos(x+\pi )=-\cos(x)}
sec ( x + π ) = − sec ( x ) {\displaystyle \sec(x+\pi )=-\sec(x)}
csc ( x + π ) = − csc ( x ) {\displaystyle \csc(x+\pi )=-\csc(x)}
sin ( π − x ) = sin ( x ) {\displaystyle \sin(\pi -x)=\sin(x)}
cos ( π − x ) = − cos ( x ) {\displaystyle \cos(\pi -x)=-\cos(x)}
tan ( π − x ) = − tan ( x ) {\displaystyle \tan(\pi -x)=-\tan(x)}
sec ( π − x ) = − sec ( x ) {\displaystyle \sec(\pi -x)=-\sec(x)}
csc ( π − x ) = csc ( x ) {\displaystyle \csc(\pi -x)=\csc(x)}
cot ( π − x ) = − cot ( x ) {\displaystyle \cot(\pi -x)=-\cot(x)}
sin ( π 2 + x ) = cos ( x ) {\displaystyle \sin({\frac {\pi }{2}}+x)=\cos(x)}
cos ( π 2 + x ) = − sin ( x ) {\displaystyle \cos({\frac {\pi }{2}}+x)=-\sin(x)}
tan ( π 2 + x ) = − cot ( x ) {\displaystyle \tan({\frac {\pi }{2}}+x)=-\cot(x)}
sec ( π 2 + x ) = − csc ( x ) {\displaystyle \sec({\frac {\pi }{2}}+x)=-\csc(x)}
csc ( π 2 + x ) = sec ( x ) {\displaystyle \csc({\frac {\pi }{2}}+x)=\sec(x)}
cot ( π 2 + x ) = − tan ( x ) {\displaystyle \cot({\frac {\pi }{2}}+x)=-\tan(x)}
sin ( π 2 − x ) = cos ( x ) {\displaystyle \sin({\frac {\pi }{2}}-x)=\cos(x)}
cos ( π 2 − x ) = sin ( x ) {\displaystyle \cos({\frac {\pi }{2}}-x)=\sin(x)}
tan ( π 2 − x ) = cot ( x ) {\displaystyle \tan({\frac {\pi }{2}}-x)=\cot(x)}
sec ( π 2 − x ) = csc ( x ) {\displaystyle \sec({\frac {\pi }{2}}-x)=\csc(x)}
csc ( π 2 − x ) = sec ( x ) {\displaystyle \csc({\frac {\pi }{2}}-x)=\sec(x)}
cot ( π 2 − x ) = tan ( x ) {\displaystyle \cot({\frac {\pi }{2}}-x)=\tan(x)}
sin ( π 4 + x ) = 2 2 ( sin ( x ) + cos ( x ) ) {\displaystyle \sin({\frac {\pi }{4}}+x)={\frac {\sqrt {2}}{2}}(\sin(x)+\cos(x))}
cos ( π 4 + x ) = 2 2 ( cos ( x ) − sin ( x ) ) {\displaystyle \cos({\frac {\pi }{4}}+x)={\frac {\sqrt {2}}{2}}(\cos(x)-\sin(x))}
tan ( π 4 + x ) = 1 + tan ( x ) 1 − tan ( x ) {\displaystyle \tan({\frac {\pi }{4}}+x)={\frac {1+\tan(x)}{1-\tan(x)}}}
sec ( π 4 + x ) = 2 sec ( x ) csc ( x ) csc ( x ) − sec ( x ) {\displaystyle \sec({\frac {\pi }{4}}+x)={\frac {{\sqrt {2}}\sec(x)\csc(x)}{\csc(x)-\sec(x)}}}
csc ( π 4 + x ) = 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)}}}
cot ( π 4 + x ) = cot ( x ) − 1 cot ( x ) + 1 {\displaystyle \cot({\frac {\pi }{4}}+x)={\frac {\cot(x)-1}{\cot(x)+1}}}
sin ( x + y ) = sin ( x ) cos ( y ) + cos ( x ) sin ( y ) {\displaystyle \sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)}
sin ( x − y ) = sin ( x ) cos ( y ) − cos ( x ) sin ( y ) {\displaystyle \sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)}
cos ( x + y ) = cos ( x ) cos ( y ) − sin ( x ) sin ( y ) {\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)}
cos ( x − y ) = cos ( x ) cos ( y ) + sin ( x ) sin ( y ) {\displaystyle \cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)}
tan ( x + y ) = tan ( x ) + tan ( y ) 1 − tan ( x ) tan ( y ) {\displaystyle \tan(x+y)={\frac {\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}}}
tan ( x − y ) = tan ( x ) − tan ( y ) 1 + tan ( x ) tan ( y ) {\displaystyle \tan(x-y)={\frac {\tan(x)-\tan(y)}{1+\tan(x)\tan(y)}}}
sec ( x + y ) = 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)}}}
sec ( x − y ) = 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)}}}
csc ( x + y ) = 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)}}}
csc ( x − y ) = 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)}}}
cot ( x + y ) = cot ( x ) cot ( y ) − 1 cot ( x ) + cot ( y ) {\displaystyle \cot(x+y)={\frac {\cot(x)\cot(y)-1}{\cot(x)+\cot(y)}}}
cot ( x − y ) = cot ( x ) cot ( y ) + 1 cot ( y ) − cot ( x ) {\displaystyle \cot(x-y)={\frac {\cot(x)\cot(y)+1}{\cot(y)-\cot(x)}}}
sin ( 2 x ) = 2 sin ( x ) cos ( x ) {\displaystyle \sin(2x)=2\sin(x)\cos(x)}
cos ( 2 x ) = cos 2 ( x ) − sin 2 ( x ) = 2 cos 2 ( x ) − 1 = 1 − 2 sin 2 ( x ) {\displaystyle \cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)}
tan ( 2 x ) = 2 tan ( x ) 1 − tan 2 ( x ) {\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}}
sec ( 2 x ) = sec 2 ( x ) csc 2 ( x ) csc 2 ( x ) − sec 2 ( x ) = sec 2 ( x ) 2 − sec 2 ( x ) = csc 2 ( x ) csc 2 ( x ) − 2 {\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}}}
csc ( 2 x ) = sec ( x ) csc ( x ) 2 {\displaystyle \csc(2x)={\frac {\sec(x)\csc(x)}{2}}}
cot ( 2 x ) = cot 2 ( x ) − 1 2 cot ( x ) {\displaystyle \cot(2x)={\frac {\cot ^{2}(x)-1}{2\cot(x)}}}
sin ( 3 x ) = 3 sin ( x ) − 4 sin 3 ( x ) = 4 sin ( x ) sin ( π 3 − x ) sin ( π 3 + x ) {\displaystyle \sin(3x)=3\sin(x)-4\sin ^{3}(x)=4\sin(x)\sin({\frac {\pi }{3}}-x)\sin({\frac {\pi }{3}}+x)}
cos ( 3 x ) = 4 cos 3 ( x ) − 3 cos ( x ) = 4 cos ( x ) cos ( π 3 − x ) cos ( π 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)}
tan ( 3 x ) = tan 3 ( x ) − 3 tan ( x ) 3 tan 2 ( x ) − 1 = tan ( x ) tan ( π 3 − x ) tan ( π 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)}
sec ( 3 x ) = sec 3 ( x ) 4 − 3 sec 2 ( x ) {\displaystyle \sec(3x)={\frac {\sec ^{3}(x)}{4-3\sec ^{2}(x)}}}
csc ( 3 x ) = csc 3 ( x ) 3 csc 2 ( x ) − 4 {\displaystyle \csc(3x)={\frac {\csc ^{3}(x)}{3\csc ^{2}(x)-4}}}
cot ( 3 x ) = cot 3 ( x ) − 3 cot ( x ) 3 cot 2 ( x ) − 1 {\displaystyle \cot(3x)={\frac {\cot ^{3}(x)-3\cot(x)}{3\cot ^{2}(x)-1}}}
sin ( x ) cos ( y ) = 1 2 ( sin ( x + y ) + sin ( x − y ) ) {\displaystyle \sin(x)\cos(y)={\frac {1}{2}}(\sin(x+y)+\sin(x-y))}
cos ( x ) sin ( y ) = 1 2 ( sin ( x + y ) − sin ( x − y ) ) {\displaystyle \cos(x)\sin(y)={\frac {1}{2}}(\sin(x+y)-\sin(x-y))}
cos ( x ) cos ( y ) = 1 2 ( cos ( x + y ) + cos ( x − y ) ) {\displaystyle \cos(x)\cos(y)={\frac {1}{2}}(\cos(x+y)+\cos(x-y))}
sin ( x ) sin ( y ) = 1 2 ( cos ( x − y ) − cos ( x + y ) ) {\displaystyle \sin(x)\sin(y)={\frac {1}{2}}(\cos(x-y)-\cos(x+y))}
sin ( x ) + sin ( y ) = 2 sin ( x + y 2 ) cos ( x − y 2 ) {\displaystyle \sin(x)+\sin(y)=2\sin({\frac {x+y}{2}})\cos({\frac {x-y}{2}})}
sin ( x ) − sin ( y ) = 2 cos ( x + y 2 ) sin ( x − y 2 ) {\displaystyle \sin(x)-\sin(y)=2\cos({\frac {x+y}{2}})\sin({\frac {x-y}{2}})}
cos ( x ) + cos ( y ) = 2 cos ( x + y 2 ) cos ( x − y 2 ) {\displaystyle \cos(x)+\cos(y)=2\cos({\frac {x+y}{2}})\cos({\frac {x-y}{2}})}
cos ( x ) − cos ( y ) = − 2 sin ( x + y 2 ) sin ( x − y 2 ) {\displaystyle \cos(x)-\cos(y)=-2\sin({\frac {x+y}{2}})\sin({\frac {x-y}{2}})}
tan ( x ) + tan ( y ) = sin ( x + y ) cos ( x ) cos ( y ) {\displaystyle \tan(x)+\tan(y)={\frac {\sin(x+y)}{\cos(x)\cos(y)}}}
tan ( x ) − tan ( y ) = sin ( x − y ) cos ( x ) cos ( y ) {\displaystyle \tan(x)-\tan(y)={\frac {\sin(x-y)}{\cos(x)\cos(y)}}}
如果有三隻角 x {\displaystyle x} ， y {\displaystyle y} 同 z {\displaystyle z} ，令到 x + y + z = π {\displaystyle x+y+z=\pi } ，噉就會有：
tan ( x ) + tan ( y ) + tan ( z ) = tan ( x ) tan ( y ) tan ( z ) {\displaystyle \tan(x)+\tan(y)+\tan(z)=\tan(x)\tan(y)\tan(z)}
cot ( x 2 ) + cot ( y 2 ) + cot ( z 2 ) = cot ( x 2 ) cot ( y 2 ) cot ( z 2 ) {\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}})}
tan ( x 2 ) tan ( y 2 ) + tan ( x 2 ) tan ( z 2 ) + tan ( y 2 ) tan ( z 2 ) = 1 {\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}
cot ( x ) cot ( y ) + cot ( x ) cot ( z ) + cot ( y ) cot ( z ) = 1 {\displaystyle \cot(x)\cot(y)+\cot(x)\cot(z)+\cot(y)\cot(z)=1}
sin ( x ) + sin ( y ) + sin ( z ) = 4 cos ( x 2 ) cos ( y 2 ) cos ( z 2 ) {\displaystyle \sin(x)+\sin(y)+\sin(z)=4\cos({\frac {x}{2}})\cos({\frac {y}{2}})\cos({\frac {z}{2}})}
− sin ( x ) + sin ( y ) + sin ( z ) = 4 cos ( x 2 ) sin ( y 2 ) sin ( z 2 ) {\displaystyle -\sin(x)+\sin(y)+\sin(z)=4\cos({\frac {x}{2}})\sin({\frac {y}{2}})\sin({\frac {z}{2}})}
cos ( x ) + cos ( y ) + cos ( z ) = 4 sin ( x 2 ) sin ( y 2 ) sin ( z 2 ) + 1 {\displaystyle \cos(x)+\cos(y)+\cos(z)=4\sin({\frac {x}{2}})\sin({\frac {y}{2}})\sin({\frac {z}{2}})+1}
− cos ( x ) + cos ( y ) + cos ( z ) = 4 sin ( x 2 ) cos ( y 2 ) cos ( z 2 ) − 1 {\displaystyle -\cos(x)+\cos(y)+\cos(z)=4\sin({\frac {x}{2}})\cos({\frac {y}{2}})\cos({\frac {z}{2}})-1}
sin ( 2 x ) + sin ( 2 y ) + sin ( 2 z ) = 4 sin ( x ) sin ( y ) sin ( z ) {\displaystyle \sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\sin(y)\sin(z)}
− sin ( 2 x ) + sin ( 2 y ) + sin ( 2 z ) = 4 sin ( x ) cos ( y ) cos ( z ) {\displaystyle -\sin(2x)+\sin(2y)+\sin(2z)=4\sin(x)\cos(y)\cos(z)}
cos ( 2 x ) + cos ( 2 y ) + cos ( 2 z ) = − 4 cos ( x ) cos ( y ) cos ( z ) − 1 {\displaystyle \cos(2x)+\cos(2y)+\cos(2z)=-4\cos(x)\cos(y)\cos(z)-1}
− cos ( 2 x ) + cos ( 2 y ) + cos ( 2 z ) = − 4 cos ( x ) sin ( y ) sin ( z ) + 1 {\displaystyle -\cos(2x)+\cos(2y)+\cos(2z)=-4\cos(x)\sin(y)\sin(z)+1}
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)\cos(y)\cos(z)+2}
− sin 2 ( x ) + sin 2 ( y ) + sin 2 ( z ) = 2 cos ( x ) sin ( y ) sin ( z ) {\displaystyle -\sin ^{2}(x)+\sin ^{2}(y)+\sin ^{2}(z)=2\cos(x)\sin(y)\sin(z)}
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)\cos(y)\cos(z)+1}
− cos 2 ( x ) + cos 2 ( y ) + cos 2 ( z ) = − 2 cos ( x ) sin ( y ) sin ( z ) + 1 {\displaystyle -\cos ^{2}(x)+\cos ^{2}(y)+\cos ^{2}(z)=-2\cos(x)\sin(y)\sin(z)+1}
− sin 2 ( 2 x ) + sin 2 ( 2 y ) + sin 2 ( 2 z ) = − 2 cos ( 2 x ) sin ( 2 y ) sin ( 2 z ) {\displaystyle -\sin ^{2}(2x)+\sin ^{2}(2y)+\sin ^{2}(2z)=-2\cos(2x)\sin(2y)\sin(2z)}
− cos 2 ( 2 x ) + cos 2 ( 2 y ) + cos 2 ( 2 z ) = 2 cos ( 2 x ) sin ( 2 y ) sin ( 2 z ) + 1 {\displaystyle -\cos ^{2}(2x)+\cos ^{2}(2y)+\cos ^{2}(2z)=2\cos(2x)\sin(2y)\sin(2z)+1}
sin 2 ( x 2 ) + sin 2 ( y 2 ) + sin 2 ( z 2 ) + 2 sin ( x 2 ) sin ( y 2 ) sin ( z 2 ) = 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}
sin ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ {\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 }
cos ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ {\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 }
sin − 1 ( x ) = ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 = x + x 3 6 + 3 x 5 40 + ⋯ {\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 }
cos − 1 ( x ) = π 2 − ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 = π 2 − x − x 3 6 − 3 x 5 40 − ⋯ {\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 }
tan − 1 ( x ) = ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 x 2 n + 1 = x − x 3 3 + x 5 5 − ⋯ {\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 }
由歐拉恆等式可以得到：
( cos ( x ) + i sin ( x ) ) n = cos ( n x ) + i sin ( n x ) {\displaystyle (\cos(x)+i\sin(x))^{n}=\cos(nx)+i\sin(nx)} （狄默夫公式）
cos ( x ) = e i x + e − i x 2 {\displaystyle \cos(x)={\frac {e^{ix}+e^{-ix}}{2}}}
sin ( x ) = e i x − e − i x 2 i {\displaystyle \sin(x)={\frac {e^{ix}-e^{-ix}}{2i}}}
tan ( x ) = − i ( e i x − e − i x ) e i x + e − i x {\displaystyle \tan(x)=-{\frac {i(e^{ix}-e^{-ix})}{e^{ix}+e^{-ix}}}}
sec ( x ) = 2 e i x + e − i x {\displaystyle \sec(x)={\frac {2}{e^{ix}+e{-ix}}}}
csc ( x ) = 2 i e i x − e − i x {\displaystyle \csc(x)={\frac {2i}{e^{ix}-e^{-ix}}}}
cot ( x ) = i ( e i x + e − i x ) e i x − e − i x {\displaystyle \cot(x)={\frac {i(e^{ix}+e^{-ix})}{e^{ix}-e^{-ix}}}}
sin − 1 ( x ) = − i ln ( i x + 1 − x 2 ) {\displaystyle \sin ^{-1}(x)=-i\ln(ix+{\sqrt {1-x^{2}}})}
cos − 1 ( x ) = − i ln ( x + x 2 − 1 ) {\displaystyle \cos ^{-1}(x)=-i\ln(x+{\sqrt {x^{2}-1}})}
tan − 1 ( x ) = i 2 ln ( i + x i − x ) {\displaystyle \tan ^{-1}(x)={\frac {i}{2}}\ln({\frac {i+x}{i-x}})}
csc − 1 ( x ) = − i ln ( i x + 1 − 1 x 2 ) {\displaystyle \csc ^{-1}(x)=-i\ln({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}})}
sec − 1 ( x ) = − i ln ( 1 x + i 1 − 1 x 2 ) {\displaystyle \sec ^{-1}(x)=-i\ln({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}})}
cot − 1 ( x ) = i 2 ln ( x − i x + i ) {\displaystyle \cot ^{-1}(x)={\frac {i}{2}}\ln({\frac {x-i}{x+i}})}
sin ( i x ) = i sinh ( x ) {\displaystyle \sin(ix)=i\sinh(x)}
cos ( i x ) = cosh ( x ) {\displaystyle \cos(ix)=\cosh(x)}
tan ( i x ) = i tanh ( x ) {\displaystyle \tan(ix)=i\tanh(x)}
下面係一啲只有常數嘅恆等式：
tan − 1 ( 1 3 ) + tan − 1 ( 1 7 ) = tan − 1 ( 1 2 ) {\displaystyle \tan ^{-1}({\frac {1}{3}})+\tan ^{-1}({\frac {1}{7}})=\tan ^{-1}({\frac {1}{2}})}
tan − 1 ( 1 5 ) + tan − 1 ( 1 8 ) = tan − 1 ( 1 3 ) {\displaystyle \tan ^{-1}({\frac {1}{5}})+\tan ^{-1}({\frac {1}{8}})=\tan ^{-1}({\frac {1}{3}})}
tan − 1 ( 1 2 ) + tan − 1 ( 1 3 ) = π 4 {\displaystyle \tan ^{-1}({\frac {1}{2}})+\tan ^{-1}({\frac {1}{3}})={\frac {\pi }{4}}}
2 tan − 1 ( 1 3 ) + tan − 1 ( 1 7 ) = π 4 {\displaystyle 2\tan ^{-1}({\frac {1}{3}})+\tan ^{-1}({\frac {1}{7}})={\frac {\pi }{4}}}
tan − 1 ( 1 3 ) + tan − 1 ( 1 5 ) + tan − 1 ( 1 7 ) + tan − 1 ( 1 8 ) = π 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}}}
4 tan − 1 ( 1 5 ) − tan − 1 ( 1 239 ) = π 4 {\displaystyle 4\tan ^{-1}({\frac {1}{5}})-\tan ^{-1}({\frac {1}{239}})={\frac {\pi }{4}}}
5 tan − 1 ( 1 7 ) + 2 tan − 1 ( 3 79 ) = π 4 {\displaystyle 5\tan ^{-1}({\frac {1}{7}})+2\tan ^{-1}({\frac {3}{79}})={\frac {\pi }{4}}} （歐拉）
tan − 1 1 + tan − 1 2 + tan − 1 3 = π {\displaystyle \tan ^{-1}{1}+\tan ^{-1}{2}+\tan ^{-1}{3}=\pi }
cos − 1 4 5 + cos − 1 5 13 + cos − 1 16 65 = sin − 1 3 5 + sin − 1 12 13 + sin − 1 63 65 = π {\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 }
cos ( π 9 ) ⋅ cos ( 2 π 9 ) ⋅ cos ( 4 π 9 ) = 1 8 {\displaystyle \cos({\frac {\pi }{9}})\cdot \cos({\frac {2\pi }{9}})\cdot \cos({\frac {4\pi }{9}})={\frac {1}{8}}}
sin ( π 9 ) ⋅ sin ( 2 π 9 ) ⋅ sin ( 4 π 9 ) = 3 8 {\displaystyle \sin({\frac {\pi }{9}})\cdot \sin({\frac {2\pi }{9}})\cdot \sin({\frac {4\pi }{9}})={\frac {\sqrt {3}}{8}}}
sin ( π 18 ) ⋅ cos ( 5 π 18 ) ⋅ cos ( 7 π 18 ) = 1 8 {\displaystyle \sin({\frac {\pi }{18}})\cdot \cos({\frac {5\pi }{18}})\cdot \cos({\frac {7\pi }{18}})={\frac {1}{8}}}
sin ( π 12 ) ⋅ sin ( 5 π 12 ) = 1 4 {\displaystyle \sin({\frac {\pi }{12}})\cdot \sin({\frac {5\pi }{12}})={\frac {1}{4}}}
cos ( π 12 ) ⋅ cos ( 5 π 12 ) = 1 4 {\displaystyle \cos({\frac {\pi }{12}})\cdot \cos({\frac {5\pi }{12}})={\frac {1}{4}}}
tan ( 2 π 9 ) ⋅ tan ( π 6 ) ⋅ tan ( π 9 ) = tan ( π 18 ) {\displaystyle \tan({\frac {2\pi }{9}})\cdot \tan({\frac {\pi }{6}})\cdot \tan({\frac {\pi }{9}})=\tan({\frac {\pi }{18}})}
tan ( 5 π 18 ) ⋅ tan ( π 3 ) ⋅ tan ( 7 π 18 ) = tan ( 4 π 9 ) {\displaystyle \tan({\frac {5\pi }{18}})\cdot \tan({\frac {\pi }{3}})\cdot \tan({\frac {7\pi }{18}})=\tan({\frac {4\pi }{9}})}
cos ( 2 π 15 ) + cos ( 4 π 15 ) + cos ( 8 π 15 ) + cos ( 14 π 15 ) = 1 2 {\displaystyle \cos({\frac {2\pi }{15}})+\cos({\frac {4\pi }{15}})+\cos({\frac {8\pi }{15}})+\cos({\frac {14\pi }{15}})={\frac {1}{2}}}
sin 2 ( π 10 ) + sin 2 ( π 6 ) = sin 2 ( π 5 ) {\displaystyle \sin ^{2}({\frac {\pi }{10}})+\sin ^{2}({\frac {\pi }{6}})=\sin ^{2}({\frac {\pi }{5}})} （歐幾里得）
嘅圓，對於喺圓上面嘅任何一點 
，有：
嘅條件恆等式
，
同 
，令到 
，噉就會有：
（狄默夫公式）
（歐拉）
（歐幾里得）
