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Трыганаметрычныя формулы

Трыганаметрычныя тоеснасці:

$\{\displaystyle \sin ^\{2\}\alpha +\cos ^\{2\}\alpha =1,\,\}$

$\{\displaystyle \operatorname \{tg\} (\alpha )=\{\frac \{\sin(\alpha )\}\{\cos(\alpha )\}\},\}$

$\{\displaystyle \operatorname \{ctg\} (\alpha )=\{\frac \{\cos(\alpha )\}\{\sin(\alpha )\}\},\}$

$\{\displaystyle 1+\operatorname \{tg\} ^\{2\}\alpha =\{\frac \{1\}\{\cos ^\{2\}\alpha \}\},\,\}$

$\{\displaystyle 1+\operatorname \{ctg\} ^\{2\}\alpha =\{\frac \{1\}\{\sin ^\{2\}\alpha \}\},\,\}$

$\{\displaystyle \operatorname \{tg\} \alpha \cdot \operatorname \{ctg\} \alpha =1.\}$

Формулы складання:

$\{\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \,\cos \beta \pm \cos \alpha \,\sin \beta ,\}$	$\{\displaystyle \operatorname \{tg\} \left(\alpha \pm \beta \right)=\{\frac \{\operatorname \{tg\} \,\alpha \pm \operatorname \{tg\} \,\beta \}\{1\mp \operatorname \{tg\} \,\alpha \,\operatorname \{tg\} \,\beta \}\},\}$
$\{\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \,\cos \beta \mp \sin \alpha \,\sin \beta ,\}$	$\{\displaystyle \operatorname \{ctg\} \left(\alpha \pm \beta \right)=\{\frac \{\operatorname \{ctg\} \,\alpha \,\operatorname \{ctg\} \,\beta \mp 1\}\{\operatorname \{ctg\} \,\beta \pm \operatorname \{ctg\} \,\alpha \}\},\}$

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).}

$\{\displaystyle \sin \,3\alpha =3\sin \alpha -4\sin ^\{3\}\alpha ,\}$

$\{\displaystyle \cos \,3\alpha =4\cos ^\{3\}\alpha -3\cos \alpha ,\}$

$\{\displaystyle \operatorname \{tg\} \,3\alpha =\{\frac \{3\,\operatorname \{tg\} \,\alpha -\operatorname \{tg\} ^\{3\}\,\alpha \}\{1-3\,\operatorname \{tg\} ^\{2\}\,\alpha \}\},\}$

$\{\displaystyle \operatorname \{ctg\} \,3\alpha =\{\frac \{\operatorname \{ctg\} ^\{3\}\,\alpha -3\,\operatorname \{ctg\} \,\alpha \}\{3\,\operatorname \{ctg\} ^\{2\}\,\alpha -1\}\}.\}$

Формулы палавіннага вугла:

$\{\displaystyle \sin \{\frac \{\alpha \}\{2\}\}=\{\sqrt \{\frac \{1-\cos \alpha \}\{2\}\}\},\quad 0\leqslant \alpha \leqslant 2\pi ;\}$	$\{\displaystyle \cos \{\frac \{\alpha \}\{2\}\}=\{\sqrt \{\frac \{1+\cos \alpha \}\{2\}\}\},\quad -\pi \leqslant \alpha \leqslant \pi ;\}$
$\{\displaystyle \operatorname \{tg\} \,\{\frac \{\alpha \}\{2\}\}=\{\frac \{1-\cos \alpha \}\{\sin \alpha \}\}=\{\frac \{\sin \alpha \}\{1+\cos \alpha \}\};\}$	$\{\displaystyle \operatorname \{ctg\} \,\{\frac \{\alpha \}\{2\}\}=\{\frac \{\sin \alpha \}\{1-\cos \alpha \}\}=\{\frac \{1+\cos \alpha \}\{\sin \alpha \}\};\}$
$\{\displaystyle \operatorname \{tg\} \,\{\frac \{\alpha \}\{2\}\}=\{\sqrt \{\frac \{1-\cos \alpha \}\{1+\cos \alpha \}\}\},\quad 0\leqslant \alpha <\pi ;\}$	$\{\displaystyle \operatorname \{ctg\} \,\{\frac \{\alpha \}\{2\}\}=\{\sqrt \{\frac \{1+\cos \alpha \}\{1-\cos \alpha \}\}\},\quad 0<\alpha \leqslant \pi .\}$

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:

$\{\displaystyle \sin \alpha \sin \beta =\{\frac \{\cos(\alpha -\beta )-\cos(\alpha +\beta )\}\{2\}\},\}$

$\{\displaystyle \sin \alpha \cos \beta =\{\frac \{\sin(\alpha -\beta )+\sin(\alpha +\beta )\}\{2\}\},\}$

$\{\displaystyle \cos \alpha \cos \beta =\{\frac \{\cos(\alpha -\beta )+\cos(\alpha +\beta )\}\{2\}\}.\}$

Формулы паніжэння цотнай ступені:

$\{\displaystyle \sin ^\{2\}\alpha =\{\frac \{1-\cos 2\,\alpha \}\{2\}\},\}$

$\{\displaystyle \cos ^\{2\}\alpha =\{\frac \{1+\cos 2\,\alpha \}\{2\}\},\}$

$\{\displaystyle \operatorname \{tg\} ^\{2\}\,\alpha =\{\frac \{1-\cos 2\,\alpha \}\{1+\cos 2\,\alpha \}\},\}$

$\{\displaystyle \operatorname \{ctg\} ^\{2\}\,\alpha =\{\frac \{1+\cos 2\,\alpha \}\{1-\cos 2\,\alpha \}\}.\}$

Тэмы гэтай старонкі (1):
Катэгорыя·Трыганаметрыя