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 \,}
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 }
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 \,.}
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 }
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 .}
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 .}
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.
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}}.}
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}}}}.}
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}}}}}
