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Трыганаметрычныя формулы
Трыганаметрычныя
тоеснасці:
sin
2
α
+
cos
2
α
=
1
,
{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1,\,}
tg
(
α
)
=
sin
(
α
)
cos
(
α
)
,
{\displaystyle \operatorname {tg} (\alpha )={\frac {\sin(\alpha )}{\cos(\alpha )},}
ctg
(
α
)
=
cos
(
α
)
sin
(
α
)
,
{\displaystyle \operatorname {ctg} (\alpha )={\frac {\cos(\alpha )}{\sin(\alpha )},}
1
+
tg
2
α
=
1
cos
2
α
,
{\displaystyle 1+\operatorname {tg} ^{2}\alpha ={\frac {1}{\cos ^{2}\alpha },\,}
1
+
ctg
2
α
=
1
sin
2
α
,
{\displaystyle 1+\operatorname {ctg} ^{2}\alpha ={\frac {1}{\sin ^{2}\alpha },\,}
tg
α
⋅
ctg
α
=
1.
{\displaystyle \operatorname {tg} \alpha \cdot \operatorname {ctg} \alpha =1.}
Формулы складання:
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 ,}
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 },}
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 ,}
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 },}
Формулы кратных вуглоў:
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).}
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}.}
Формулы палавіннага вугла:
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 .}
Формулы сумы і рознасці функцый:
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}.}
Здабыткаў функцый:
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}.}
Формулы паніжэння цотнай ступені:
sin
2
α
=
1
−
cos
2
α
2
,
{\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\,\alpha }{2},}
cos
2
α
=
1
+
cos
2
α
2
,
{\displaystyle \cos ^{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 },}
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 }.}
