cos 2 θ + sin 2 θ = 1 {\displaystyle \cos ^{2}\theta +\sin ^{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 ( α ± β ) = sin α cos β ± cos α sin β {\displaystyle \sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }
cos ( α ± β ) = cos α cos β ∓ sin α sin β {\displaystyle \cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
tan ( α ± β ) = tan α ± tan β 1 ∓ tan α tan β {\displaystyle \tan(\alpha \pm \beta )={\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}
tan ( α + β 2 ) = sin α + sin β cos α + cos β = − cos α − cos β sin α − sin β {\displaystyle \tan \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }}}
arctan α ± arctan β = arctan ( α ± β 1 ∓ α β ) {\displaystyle \arctan \alpha \pm \arctan \beta =\arctan \left({\frac {\alpha \pm \beta }{1\mp \alpha \beta }}\right)}
arcsin α ± arcsin β = arcsin ( α 1 − β 2 ± β 1 − α 2 ) {\displaystyle \arcsin \alpha \pm \arcsin \beta =\arcsin \left(\alpha {\sqrt {1-\beta ^{2}}}\pm \beta {\sqrt {1-\alpha ^{2}}}\right)}
arccos α ± arccos β = arccos ( α β ∓ ( 1 − α 2 ) ( 1 − β 2 ) ) {\displaystyle \arccos \alpha \pm \arccos \beta =\arccos \left(\alpha \beta \mp {\sqrt {(1-\alpha ^{2})(1-\beta ^{2})}}\right)}
cos n θ + j sin n θ = ( cos ( θ ) + j sin ( θ ) ) n {\displaystyle \cos n\theta +j\sin n\theta =(\cos(\theta )+j\sin(\theta ))^{n}}
tan ( n + 1 ) θ = tan n θ + tan θ 1 − tan n θ tan θ {\displaystyle \tan \,(n{+}1)\theta ={\frac {\tan n\theta +\tan \theta }{1-\tan n\theta \,\tan \theta }}}
cot ( n + 1 ) θ = cot n θ cot θ − 1 cot n θ + cot θ {\displaystyle \cot \,(n{+}1)\theta ={\frac {\cot n\theta \,\cot \theta -1}{\cot n\theta +\cot \theta }}}
cos n x = 2 ⋅ cos x ⋅ cos ( ( n − 1 ) x ) − cos ( ( n − 2 ) x ) {\displaystyle \cos nx=2\cdot \cos x\cdot \cos((n-1)x)-\cos((n-2)x)}
sin n x = 2 ⋅ cos x ⋅ sin ( ( n − 1 ) x ) − sin ( ( n − 2 ) x ) {\displaystyle \sin nx=2\cdot \cos x\cdot \sin((n-1)x)-\sin((n-2)x)}
sin n θ = ∑ k = 0 n ( n k ) cos k θ sin n − k θ sin ( 1 2 ( n − k ) π ) {\displaystyle \sin n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\sin \left({\frac {1}{2}}(n-k)\pi \right)}
cos n θ = ∑ k = 0 n ( n k ) cos k θ sin n − k θ cos ( 1 2 ( n − k ) π ) {\displaystyle \cos n\theta =\sum _{k=0}^{n}{\binom {n}{k}}\cos ^{k}\theta \,\sin ^{n-k}\theta \,\cos \left({\frac {1}{2}}(n-k)\pi \right)}
cos θ − cos φ = − 2 sin ( θ + φ 2 ) sin ( θ − φ 2 ) {\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\theta +\varphi \over 2}\right)\sin \left({\theta -\varphi \over 2}\right)}
arcsin ( x ) + arccos ( x ) = π / 2 arctan ( x ) + arccot ( x ) = π / 2 arctan ( x ) + arctan ( 1 / x ) = { π / 2 , if x > 0 − π / 2 , if x < 0 {\displaystyle {\begin{aligned}\arcsin(x)+\arccos(x)&=\pi /2\\\arctan(x)+\operatorname {arccot}(x)&=\pi /2\\\arctan(x)+\arctan(1/x)&=\left\{{\begin{matrix}\pi /2,&{\mbox{if }}x>0\\-\pi /2,&{\mbox{if }}x<0\end{matrix}}\right.\\\end{aligned}}}
sin [ arccos ( x ) ] = 1 − x 2 sin [ arctan ( x ) ] = x 1 + x 2 cos [ arctan ( x ) ] = 1 1 + x 2 cos [ arcsin ( x ) ] = 1 − x 2 {\displaystyle \quad {\begin{aligned}\sin[\arccos(x)]&={\sqrt {1-x^{2}}}\\\sin[\arctan(x)]&={\frac {x}{\sqrt {1+x^{2}}}}\\\cos[\arctan(x)]&={\frac {1}{\sqrt {1+x^{2}}}}\\\cos[\arcsin(x)]&={\sqrt {1-x^{2}}}\\\end{aligned}}}
tan [ arcsin ( x ) ] = x 1 − x 2 tan [ arccos ( x ) ] = 1 − x 2 x cot [ arcsin ( x ) ] = 1 − x 2 x cot [ arccos ( x ) ] = x 1 − x 2 {\displaystyle \quad {\begin{aligned}\tan[\arcsin(x)]&={\frac {x}{\sqrt {1-x^{2}}}}\\\tan[\arccos(x)]&={\frac {\sqrt {1-x^{2}}}{x}}\\\cot[\arcsin(x)]&={\frac {\sqrt {1-x^{2}}}{x}}\\\cot[\arccos(x)]&={\frac {x}{\sqrt {1-x^{2}}}}\\\end{aligned}}}
e i θ = cis θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\operatorname {cis} \,\theta =\cos \theta +i\sin \theta \,}
sin θ = e i θ − e − i θ 2 i {\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}
cos θ = e i θ + e − i θ 2 {\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}
sec θ = 2 e i θ + e − i θ {\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\,}
csc θ = 2 i e i θ − e − i θ {\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\,}
e i π / 2 = i | i 2 = e i π = − 1 | e 2 π i = 1 {\displaystyle e^{i\pi /2}=i\quad \ \vert \ \quad i^{2}=e^{i\pi }=-1\quad \ \vert \ \quad e^{2\pi i}=1} e − i θ = cos ( − θ ) + i sin ( − θ ) = cos ( θ ) − i sin ( θ ) {\displaystyle e^{-i\theta }=\cos(-\theta )+i\sin(-\theta )=\cos(\theta )-i\sin(\theta )}
arcsin x = − i ln ( i x + 1 − x 2 ) {\displaystyle \arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}
arccos x = i ln ( x − i 1 − x 2 ) {\displaystyle \arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\right)\,}
arcsec x = − i ln ( 1 x + 1 − i x 2 ) {\displaystyle \operatorname {arcsec} x=-i\ln \left({\tfrac {1}{x}}+{\sqrt {1-{\tfrac {i}{x^{2}}}}}\right)\,}
arccsc x = − i ln ( i x + 1 − 1 x 2 ) {\displaystyle \operatorname {arccsc} x=-i\ln \left({\tfrac {i}{x}}+{\sqrt {1-{\tfrac {1}{x^{2}}}}}\right)\,}
arccis x = ln x i = − i ln x = arg x {\displaystyle \operatorname {arccis} \,x={\frac {\ln x}{i}}=-i\ln x=\operatorname {arg} \,x\,}
tan θ = e i θ − e − i θ i ( e i θ + e − i θ ) {\displaystyle \tan \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}
cot θ = i ( e i θ + e − i θ ) e i θ − e − i θ {\displaystyle \cot \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}
arctan x = i 2 ln ( i + x i − x ) {\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\,}
arccot x = i 2 ln ( x − i x + i ) {\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\,}
sin x = x ∏ n = 1 ∞ ( 1 − x 2 π 2 n 2 ) {\displaystyle \sin x=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)}
sinh x = x ∏ n = 1 ∞ ( 1 + x 2 π 2 n 2 ) {\displaystyle \sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right)}
cos x = ∏ n = 1 ∞ ( 1 − x 2 π 2 ( n − 1 2 ) 2 ) {\displaystyle \cos x=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)}
cosh x = ∏ n = 1 ∞ ( 1 + x 2 π 2 ( n − 1 2 ) 2 ) {\displaystyle \cosh x=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}(n-{\frac {1}{2}})^{2}}}\right)}
sin x x = ∏ n = 1 ∞ cos ( x 2 n ) {\displaystyle {\frac {\sin x}{x}}=\prod _{n=1}^{\infty }\cos \left({\frac {x}{2^{n}}}\right)}
| sin x | = 1 2 ∏ n = 0 ∞ | tan ( 2 n x ) | 2 n + 1 {\displaystyle |\sin x|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}}
lim x → 0 sin x x = 1 {\displaystyle \lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1}
lim x → 0 1 − cos x x = 0 {\displaystyle \lim _{x\rightarrow 0}{\frac {1-\cos x}{x}}=0}
d d x sin x = cos x {\displaystyle {\mathrm {d} \over \mathrm {d} x}\sin x=\cos x}
d d x cos x = − sin x {\displaystyle {\mathrm {d} \over \mathrm {d} x}\cos x=-\sin x}
d d x arcsin x = 1 1 − x 2 {\displaystyle {\mathrm {d} \over \mathrm {d} x}\arcsin x={1 \over {\sqrt {1-x^{2}}}}}
d d x arccos x = − 1 1 − x 2 {\displaystyle {\mathrm {d} \over \mathrm {d} x}\arccos x={-1 \over {\sqrt {1-x^{2}}}}}
d d x tan x = sec 2 x {\displaystyle {\mathrm {d} \over \mathrm {d} x}\tan x=\sec ^{2}x}
d d x cot x = − csc 2 x {\displaystyle {\mathrm {d} \over \mathrm {d} x}\cot x=-\csc ^{2}x}
d d x arctan x = 1 1 + x 2 {\displaystyle {\mathrm {d} \over \mathrm {d} x}\arctan x={1 \over 1+x^{2}}}
d d x arccot x = − 1 1 + x 2 {\displaystyle {\mathrm {d} \over \mathrm {d} x}\operatorname {arccot} x={-1 \over 1+x^{2}}}
d d x sec x = tan x sec x {\displaystyle {\mathrm {d} \over \mathrm {d} x}\sec x=\tan x\sec x}
d d x csc x = − csc x cot x {\displaystyle {\mathrm {d} \over \mathrm {d} x}\csc x=-\csc x\cot x}
d d x arcsec x = 1 | x | x 2 − 1 {\displaystyle {\mathrm {d} \over \mathrm {d} x}\operatorname {arcsec} x={1 \over |x|{\sqrt {x^{2}-1}}}}
d d x arccsc x = − 1 | x | x 2 − 1 {\displaystyle {\mathrm {d} \over \mathrm {d} x}\operatorname {arccsc} x={-1 \over |x|{\sqrt {x^{2}-1}}}}
∫ d u a 2 − u 2 = sin − 1 ( u a ) + C {\displaystyle \int {\frac {\mathrm {d} u}{\sqrt {a^{2}-u^{2}}}}=\sin ^{-1}\left({\frac {u}{a}}\right)+C}
∫ d u a 2 + u 2 = 1 a tan − 1 ( u a ) + C {\displaystyle \int {\frac {\mathrm {d} u}{a^{2}+u^{2}}}={\frac {1}{a}}\tan ^{-1}\left({\frac {u}{a}}\right)+C}
∫ d u u u 2 − a 2 = 1 a sec − 1 | u a | + C {\displaystyle \int {\frac {\mathrm {d} u}{u{\sqrt {u^{2}-a^{2}}}}}={\frac {1}{a}}\sec ^{-1}\left|{\frac {u}{a}}\right|+C}
t ≜ tan x 2 . {\displaystyle t\triangleq \tan {\frac {x}{2}}.}
sin x = 2 t 1 + t 2 {\displaystyle \sin x={\frac {2t}{1+t^{2}}}}
cos x = 1 − t 2 1 + t 2 {\displaystyle \cos x={\frac {1-t^{2}}{1+t^{2}}}}
e i x = 1 + i t 1 − i t {\displaystyle e^{ix}={\frac {1+it}{1-it}}}
d x = 2 d t 1 + t 2 {\displaystyle \mathrm {d} x={\frac {2\,\mathrm {d} t}{1+t^{2}}}}
![{\displaystyle \quad {\begin{aligned}\sin[\arccos(x)]&={\sqrt {1-x^{2}}}\\\sin[\arctan(x)]&={\frac {x}{\sqrt {1+x^{2}}}}\\\cos[\arctan(x)]&={\frac {1}{\sqrt {1+x^{2}}}}\\\cos[\arcsin(x)]&={\sqrt {1-x^{2}}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9300063a6fdc617371427997d63d6859f1c8aa50)
![{\displaystyle \quad {\begin{aligned}\tan[\arcsin(x)]&={\frac {x}{\sqrt {1-x^{2}}}}\\\tan[\arccos(x)]&={\frac {\sqrt {1-x^{2}}}{x}}\\\cot[\arcsin(x)]&={\frac {\sqrt {1-x^{2}}}{x}}\\\cot[\arccos(x)]&={\frac {x}{\sqrt {1-x^{2}}}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65bfaa544db814dae4746c0a198ee7ca1e5842e7)
![{\displaystyle |\sin x|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\tan \left(2^{n}x\right)\right|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71abcbf6e9d8704e6030ef84b53d6de4a37681ce)
![{\displaystyle [{\text{Phasor}}]\cdot [{\text{complex const}}]:\ \operatorname {Re} \{(Ae^{i\theta }\cdot e^{i\omega t})\cdot Be^{i\phi }\}=\operatorname {Re} \{(ABe^{i(\theta +\phi )})\cdot e^{i\omega t}\}=AB\cos(\omega t+(\theta +\phi ))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fda320e8dd4aa2070c9c7628aa6a2fbabe187d93)
