Galvenās vienādības Tā kā sinuss un kosinuss ir attiecīgi punkta ordināta un abscisa, kas atbilst leņķa α riņķim, tad, atbilstoši Pitagora teorēmai
sin 2 α + cos 2 α = 1. {\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1.\qquad \qquad \,} Dalot šīs vienādības abas puses ar sinusa kvadrātu vai kosinusa kvadrātu, iegūstam:
1 + t g 2 α = 1 cos 2 α , {\displaystyle 1+\mathop {\mathrm {tg} } \,^{2}\alpha ={\frac {1}{\cos ^{2}\alpha }},\qquad \qquad \,} 1 + c t g 2 α = 1 sin 2 α . {\displaystyle 1+\mathop {\mathrm {ctg} } \,^{2}\alpha ={\frac {1}{\sin ^{2}\alpha }}.\qquad \qquad \,} Nepārtrauktība Sinuss un kosinuss ir nepārtrauktas funkcijas, bet tangensam, kotangensam, sekansam un kosekansam ir pārtraukuma punkti ± π 2 , ± π , ± 3 π 2 , … {\displaystyle \pm {\frac {\pi }{2}},\;\pm \pi ,\;\pm {\frac {3\pi }{2}},\;\dots } 0 , ± π , ± 2 π , … {\displaystyle 0,\;\pm \pi ,\;\pm 2\pi ,\;\dots }
Paritāte Kosinuss un sekanss ir funkcijas, kurām ir simetrija attiecībā uz funkcijas zīmes maiņu. Pārējām četrām funkcijām tādas īpašības nav, t.i.:
sin ( − α ) = − sin α , {\displaystyle \sin \left(-\alpha \right)=-\sin \alpha \,,} cos ( − α ) = cos α , {\displaystyle \cos \left(-\alpha \right)=\cos \alpha \,,} t g ( − α ) = − t g α , {\displaystyle \mathop {\mathrm {tg} } \,\left(-\alpha \right)=-\mathop {\mathrm {tg} } \,\alpha \,,} c t g ( − α ) = − c t g α , {\displaystyle \mathop {\mathrm {ctg} } \,\left(-\alpha \right)=-\mathop {\mathrm {ctg} } \,\alpha \,,} sec ( − α ) = sec α , {\displaystyle \sec \left(-\alpha \right)=\sec \alpha \,,} c o s e c ( − α ) = − c o s e c α . {\displaystyle \mathop {\mathrm {cosec} } \,\left(-\alpha \right)=-\mathop {\mathrm {cosec} } \,\alpha \,.} Periodiskums Funkcijas y = sin α {\displaystyle y=\sin \alpha } y = cos α {\displaystyle y=\cos \alpha } y = sec α {\displaystyle y=\sec \alpha } y = csc α {\displaystyle y=\csc \alpha } 2 π {\displaystyle 2\pi } y = tan α {\displaystyle y=\tan \alpha } y = cot α {\displaystyle y=\cot \alpha } π {\displaystyle \pi }
Saskaitīšanas formulas Summas trigonometriskās funkcijas nozīme un divu leņķu starpība:
sin ( α ± β ) = sin α cos β ± cos α sin β , {\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \,\cos \beta \pm \cos \alpha \,\sin \beta ,} cos ( α ± β ) = cos α cos β ∓ sin α sin β , {\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \,\cos \beta \mp \sin \alpha \,\sin \beta ,} tg ( α ± β ) = tg α ± tg β 1 ∓ tg α tg β , {\displaystyle \operatorname {tg} \left(\alpha \pm \beta \right)={\frac {\operatorname {tg} \,\alpha \pm \operatorname {tg} \,\beta }{1\mp \operatorname {tg} \,\alpha \,\operatorname {tg} \,\beta }},} ctg ( α ± β ) = ctg α ctg β ∓ 1 ctg β ± ctg α . {\displaystyle \operatorname {ctg} \left(\alpha \pm \beta \right)={\frac {\operatorname {ctg} \,\alpha \,\operatorname {ctg} \,\beta \mp 1}{\operatorname {ctg} \,\beta \pm \operatorname {ctg} \,\alpha }}.} Līdzīgas formulas trim leņķiem:
sin ( α + β + γ ) = sin α cos β cos γ + cos α sin β cos γ + cos α cos β sin γ − sin α sin β sin γ , {\displaystyle \sin \left(\alpha +\beta +\gamma \right)=\sin \alpha \cos \beta \cos \gamma +\cos \alpha \sin \beta \cos \gamma +\cos \alpha \cos \beta \sin \gamma -\sin \alpha \sin \beta \sin \gamma ,} cos ( α + β + γ ) = cos α cos β cos γ − sin α sin β cos γ − sin α cos β sin γ − cos α sin β sin γ . {\displaystyle \cos \left(\alpha +\beta +\gamma \right)=\cos \alpha \cos \beta \cos \gamma -\sin \alpha \sin \beta \cos \gamma -\sin \alpha \cos \beta \sin \gamma -\cos \alpha \sin \beta \sin \gamma .} Formulas leņķu daudzkārtņiem Divkārša leņķa formulas:
sin 2 α = 2 sin α cos α = 2 tg α 1 + tg 2 α , {\displaystyle \sin 2\alpha =2\sin \alpha \cos \alpha ={\frac {2\,\operatorname {tg} \,\alpha }{1+\operatorname {tg} ^{2}\alpha }},} cos 2 α = cos 2 α − sin 2 α = 2 cos 2 α − 1 = 1 − 2 sin 2 α = 1 − tg 2 α 1 + tg 2 α = ctg α − tg α ctg α + tg α , {\displaystyle \cos 2\alpha =\cos ^{2}\alpha \,-\,\sin ^{2}\alpha =2\cos ^{2}\alpha \,-\,1=1\,-\,2\sin ^{2}\alpha ={\frac {1-\operatorname {tg} ^{2}\alpha }{1+\operatorname {tg} ^{2}\alpha }}={\frac {\operatorname {ctg} \,\alpha -\operatorname {tg} \,\alpha }{\operatorname {ctg} \,\alpha +\operatorname {tg} \,\alpha }},} tg 2 α = 2 tg α 1 − tg 2 α , {\displaystyle \operatorname {tg} \,2\alpha ={\frac {2\,\operatorname {tg} \,\alpha }{1-\operatorname {tg} ^{2}\alpha }},} ctg 2 α = ctg 2 α − 1 2 ctg α = 1 2 ( ctg α − tg α ) . {\displaystyle \operatorname {ctg} \,2\alpha ={\frac {\operatorname {ctg} ^{2}\alpha -1}{2\,\operatorname {ctg} \,\alpha }}={\frac {1}{2}}\left(\operatorname {ctg} \,\alpha -\operatorname {tg} \,\alpha \right).} Trīskārša leņķa formulas:
sin 3 α = 3 sin α − 4 sin 3 α , {\displaystyle \sin \,3\alpha =3\sin \alpha -4\sin ^{3}\alpha ,} cos 3 α = 4 cos 3 α − 3 cos α , {\displaystyle \cos \,3\alpha =4\cos ^{3}\alpha -3\cos \alpha ,} tg 3 α = 3 tg α − tg 3 α 1 − 3 tg 2 α , {\displaystyle \operatorname {tg} \,3\alpha ={\frac {3\,\operatorname {tg} \,\alpha -\operatorname {tg} ^{3}\,\alpha }{1-3\,\operatorname {tg} ^{2}\,\alpha }},} ctg 3 α = ctg 3 α − 3 ctg α 3 ctg 2 α − 1 . {\displaystyle \operatorname {ctg} \,3\alpha ={\frac {\operatorname {ctg} ^{3}\,\alpha -3\,\operatorname {ctg} \,\alpha }{3\,\operatorname {ctg} ^{2}\,\alpha -1}}.} Citas leņķu daudzkārtņu formulas:
sin 4 α = cos α ( 4 sin α − 8 sin 3 α ) , {\displaystyle \sin \,4\alpha =\cos \alpha \left(4\sin \alpha -8\sin ^{3}\alpha \right),} cos 4 α = 8 cos 4 α − 8 cos 2 α + 1 , {\displaystyle \cos \,4\alpha =8\cos ^{4}\alpha -8\cos ^{2}\alpha +1,} tg 4 α = 4 tg α − 4 tg 3 α 1 − 6 tg 2 α + tg 2 α , {\displaystyle \operatorname {tg} \,4\alpha ={\frac {4\,\operatorname {tg} \,\alpha -4\,\operatorname {tg} ^{3}\,\alpha }{1-6\,\operatorname {tg} ^{2}\,\alpha +\operatorname {tg} ^{2}\,\alpha }},} ctg 4 α = ctg 4 α − 6 ctg 2 α + 1 4 ctg 3 α − 4 ctg α , {\displaystyle \operatorname {ctg} \,4\alpha ={\frac {\operatorname {ctg} ^{4}\,\alpha -6\,\operatorname {ctg} ^{2}\,\alpha +1}{4\,\operatorname {ctg} ^{3}\,\alpha -4\,\operatorname {ctg} \,\alpha }},} sin 5 α = 16 sin 5 α − 20 sin 3 α + 5 sin α {\displaystyle \sin \,5\alpha =16\sin ^{5}\alpha -20\sin ^{3}\alpha +5\sin \alpha } cos 5 α = 16 cos 5 α − 20 cos 3 α + 5 cos α {\displaystyle \cos \,5\alpha =16\cos ^{5}\alpha -20\cos ^{3}\alpha +5\cos \alpha } tg 5 α = tg α tg 4 α − 10 tg 2 α + 5 5 tg 4 α − 10 tg 2 α + 1 {\displaystyle \operatorname {tg} \,5\alpha =\operatorname {tg} \alpha {\frac {\operatorname {tg} ^{4}\alpha -10\operatorname {tg} ^{2}\alpha +5}{5\operatorname {tg} ^{4}\alpha -10\operatorname {tg} ^{2}\alpha +1}}} sin ( n α ) = 2 n − 1 ∏ k = 0 n − 1 sin ( α + π k n ) {\displaystyle \sin(n\alpha )=2^{n-1}\prod _{k=0}^{n-1}\sin \left(\alpha +{\frac {\pi k}{n}}\right)} Pusleņķa formulas:
sin α 2 = 1 − cos α 2 , 0 ⩽ α ⩽ 2 π , {\displaystyle \sin {\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{2}}},\quad 0\leqslant \alpha \leqslant 2\pi ,} cos α 2 = 1 + cos α 2 , − π ⩽ α ⩽ π , {\displaystyle \cos {\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{2}}},\quad -\pi \leqslant \alpha \leqslant \pi ,} tg α 2 = 1 − cos α sin α = sin α 1 + cos α , {\displaystyle \operatorname {tg} \,{\frac {\alpha }{2}}={\frac {1-\cos \alpha }{\sin \alpha }}={\frac {\sin \alpha }{1+\cos \alpha }},} ctg α 2 = sin α 1 − cos α = 1 + cos α sin α , {\displaystyle \operatorname {ctg} \,{\frac {\alpha }{2}}={\frac {\sin \alpha }{1-\cos \alpha }}={\frac {1+\cos \alpha }{\sin \alpha }},} tg α 2 = 1 − cos α 1 + cos α , 0 ⩽ α < π , {\displaystyle \operatorname {tg} \,{\frac {\alpha }{2}}={\sqrt {\frac {1-\cos \alpha }{1+\cos \alpha }}},\quad 0\leqslant \alpha <\pi ,} ctg α 2 = 1 + cos α 1 − cos α , 0 < α ⩽ π . {\displaystyle \operatorname {ctg} \,{\frac {\alpha }{2}}={\sqrt {\frac {1+\cos \alpha }{1-\cos \alpha }}},\quad 0<\alpha \leqslant \pi .} Reizināšana Formulas divu leņķu reizināšanai:
sin α sin β = cos ( α − β ) − cos ( α + β ) 2 , {\displaystyle \sin \alpha \sin \beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{2}},} sin α cos β = sin ( α − β ) + sin ( α + β ) 2 , {\displaystyle \sin \alpha \cos \beta ={\frac {\sin(\alpha -\beta )+\sin(\alpha +\beta )}{2}},} cos α cos β = cos ( α − β ) + cos ( α + β ) 2 , {\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{2}},} tg α tg β = cos ( α − β ) − cos ( α + β ) cos ( α − β ) + cos ( α + β ) , {\displaystyle \operatorname {tg} \,\alpha \,\operatorname {tg} \,\beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{\cos(\alpha -\beta )+\cos(\alpha +\beta )}},} tg α ctg β = sin ( α − β ) + sin ( α + β ) sin ( α + β ) − sin ( α − β ) , {\displaystyle \operatorname {tg} \,\alpha \,\operatorname {ctg} \,\beta ={\frac {\sin(\alpha -\beta )+\sin(\alpha +\beta )}{\sin(\alpha +\beta )-\sin(\alpha -\beta )}},} ctg α ctg β = cos ( α − β ) + cos ( α + β ) cos ( α − β ) − cos ( α + β ) . {\displaystyle \operatorname {ctg} \,\alpha \,\operatorname {ctg} \,\beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{\cos(\alpha -\beta )-\cos(\alpha +\beta )}}.} Līdzīgas formulas triju leņķu sinusu un kosinusu reizināšanai:
sin α sin β sin γ = sin ( α + β − γ ) + sin ( β + γ − α ) + sin ( α − β + γ ) − sin ( α + β + γ ) 4 , {\displaystyle \sin \alpha \sin \beta \sin \gamma ={\frac {\sin(\alpha +\beta -\gamma )+\sin(\beta +\gamma -\alpha )+\sin(\alpha -\beta +\gamma )-\sin(\alpha +\beta +\gamma )}{4}},} sin α sin β cos γ = − cos ( α + β − γ ) + cos ( β + γ − α ) + cos ( α − β + γ ) − cos ( α + β + γ ) 4 , {\displaystyle \sin \alpha \sin \beta \cos \gamma ={\frac {-\cos(\alpha +\beta -\gamma )+\cos(\beta +\gamma -\alpha )+\cos(\alpha -\beta +\gamma )-\cos(\alpha +\beta +\gamma )}{4}},} sin α cos β cos γ = sin ( α + β − γ ) − sin ( β + γ − α ) + sin ( α − β + γ ) − sin ( α + β + γ ) 4 , {\displaystyle \sin \alpha \cos \beta \cos \gamma ={\frac {\sin(\alpha +\beta -\gamma )-\sin(\beta +\gamma -\alpha )+\sin(\alpha -\beta +\gamma )-\sin(\alpha +\beta +\gamma )}{4}},} cos α cos β cos γ = cos ( α + β − γ ) + cos ( β + γ − α ) + cos ( α − β + γ ) + cos ( α + β + γ ) 4 . {\displaystyle \cos \alpha \cos \beta \cos \gamma ={\frac {\cos(\alpha +\beta -\gamma )+\cos(\beta +\gamma -\alpha )+\cos(\alpha -\beta +\gamma )+\cos(\alpha +\beta +\gamma )}{4}}.} Attiecīgās formulas triju leņķu tangensiem un kotangensiem var iegūt, izdalot augstāk minēto vienādojumu labās puses ar kreisajām.
Pakāpes sin 2 α = 1 − cos 2 α 2 , {\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\,\alpha }{2}},} tg 2 α = 1 − cos 2 α 1 + cos 2 α , {\displaystyle \operatorname {tg} ^{2}\,\alpha ={\frac {1-\cos 2\,\alpha }{1+\cos 2\,\alpha }},} cos 2 α = 1 + cos 2 α 2 , {\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\,\alpha }{2}},} ctg 2 α = 1 + cos 2 α 1 − cos 2 α , {\displaystyle \operatorname {ctg} ^{2}\,\alpha ={\frac {1+\cos 2\,\alpha }{1-\cos 2\,\alpha }},} sin 3 α = 3 sin α − sin 3 α 4 , {\displaystyle \sin ^{3}\alpha ={\frac {3\sin \alpha -\sin 3\,\alpha }{4}},} tg 3 α = 3 sin α − sin 3 α 3 cos α + cos 3 α , {\displaystyle \operatorname {tg} ^{3}\,\alpha ={\frac {3\sin \alpha -\sin 3\,\alpha }{3\cos \alpha +\cos 3\,\alpha }},} cos 3 α = 3 cos α + cos 3 α 4 , {\displaystyle \cos ^{3}\alpha ={\frac {3\cos \alpha +\cos 3\,\alpha }{4}},} ctg 3 α = 3 cos α + cos 3 α 3 sin α − sin 3 α , {\displaystyle \operatorname {ctg} ^{3}\,\alpha ={\frac {3\cos \alpha +\cos 3\,\alpha }{3\sin \alpha -\sin 3\,\alpha }},} sin 4 α = cos 4 α − 4 cos 2 α + 3 8 , {\displaystyle \sin ^{4}\alpha ={\frac {\cos 4\alpha -4\cos 2\,\alpha +3}{8}},} tg 4 α = cos 4 α − 4 cos 2 α + 3 cos 4 α + 4 cos 2 α + 3 , {\displaystyle \operatorname {tg} ^{4}\,\alpha ={\frac {\cos 4\alpha -4\cos 2\,\alpha +3}{\cos 4\alpha +4\cos 2\,\alpha +3}},} cos 4 α = cos 4 α + 4 cos 2 α + 3 8 , {\displaystyle \cos ^{4}\alpha ={\frac {\cos 4\alpha +4\cos 2\,\alpha +3}{8}},} ctg 4 α = cos 4 α + 4 cos 2 α + 3 cos 4 α − 4 cos 2 α + 3 . {\displaystyle \operatorname {ctg} ^{4}\,\alpha ={\frac {\cos 4\alpha +4\cos 2\,\alpha +3}{\cos 4\alpha -4\cos 2\,\alpha +3}}.}
Summas sin α ± sin β = 2 sin α ± β 2 cos α ∓ β 2 {\displaystyle \sin \alpha \pm \sin \beta =2\sin {\frac {\alpha \pm \beta }{2}}\cos {\frac {\alpha \mp \beta }{2}}} cos α + cos β = 2 cos α + β 2 cos α − β 2 {\displaystyle \cos \alpha +\cos \beta =2\cos {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}} cos α − cos β = − 2 sin α + β 2 sin α − β 2 {\displaystyle \cos \alpha -\cos \beta =-2\sin {\frac {\alpha +\beta }{2}}\sin {\frac {\alpha -\beta }{2}}} tg α ± tg β = sin ( α ± β ) cos α cos β {\displaystyle \operatorname {tg} \alpha \pm \operatorname {tg} \beta ={\frac {\sin(\alpha \pm \beta )}{\cos \alpha \cos \beta }}} 1 ± sin 2 α = ( sin α ± cos α ) 2 . {\displaystyle 1\pm \sin {2\alpha }=(\sin \alpha \pm \cos \alpha )^{2}.} Funkcijām ar argumentu x {\displaystyle x}
A sin x + B cos x = A 2 + B 2 sin ( x + ϕ ) , {\displaystyle A\sin x+B\cos x={\sqrt {A^{2}+B^{2}}}\sin(x+\phi ),} kur leņķi ϕ {\displaystyle \phi }
sin ϕ = B A 2 + B 2 , cos ϕ = A A 2 + B 2 . {\displaystyle \sin \phi ={\frac {B}{\sqrt {A^{2}+B^{2}}}},\cos \phi ={\frac {A}{\sqrt {A^{2}+B^{2}}}}.} Tangensa vienādības Jebkuru trigonometrisko funkciju var izteikt kā pusleņķa tangensu.
sin x = sin x 1 = 2 sin x 2 cos x 2 sin 2 x 2 + cos 2 x 2 = 2 tg x 2 1 + tg 2 x 2 {\displaystyle \sin x={\frac {\sin x}{1}}={\frac {2\sin {\frac {x}{2}}\cos {\frac {x}{2}}}{\sin ^{2}{\frac {x}{2}}+\cos ^{2}{\frac {x}{2}}}}={\frac {2\operatorname {tg} {\frac {x}{2}}}{1+\operatorname {tg} ^{2}{\frac {x}{2}}}}}
cos x = cos x 1 = cos 2 x 2 − sin 2 x 2 cos 2 x 2 + sin 2 x 2 = 1 − tg 2 x 2 1 + tg 2 x 2 {\displaystyle \cos x={\frac {\cos x}{1}}={\frac {\cos ^{2}{\frac {x}{2}}-\sin ^{2}{\frac {x}{2}}}{\cos ^{2}{\frac {x}{2}}+\sin ^{2}{\frac {x}{2}}}}={\frac {1-\operatorname {tg} ^{2}{\frac {x}{2}}}{1+\operatorname {tg} ^{2}{\frac {x}{2}}}}}
tg x = sin x cos x = 2 tg x 2 1 − tg 2 x 2 {\displaystyle \operatorname {tg} ~x={\frac {\sin x}{\cos x}}={\frac {2\operatorname {tg} {\frac {x}{2}}}{1-\operatorname {tg} ^{2}{\frac {x}{2}}}}}
ctg x = cos x sin x = 1 − tg 2 x 2 2 tg x 2 {\displaystyle \operatorname {ctg} ~x={\frac {\cos x}{\sin x}}={\frac {1-\operatorname {tg} ^{2}{\frac {x}{2}}}{2\operatorname {tg} {\frac {x}{2}}}}}
sec x = 1 cos x = 1 + tg 2 x 2 1 − tg 2 x 2 {\displaystyle \sec x={\frac {1}{\cos x}}={\frac {1+\operatorname {tg} ^{2}{\frac {x}{2}}}{1-\operatorname {tg} ^{2}{\frac {x}{2}}}}}
cosec x = 1 sin x = 1 + tg 2 x 2 2 tg x 2 {\displaystyle \operatorname {cosec} ~x={\frac {1}{\sin x}}={\frac {1+\operatorname {tg} ^{2}{\frac {x}{2}}}{2\operatorname {tg} {\frac {x}{2}}}}}
