Triqonometriyanın əsas düsturları

Triqonometriyada triqonometrik eyniliklər triqonometrik funksiyaların daxil olduğu bərabərliklərdir. Həndəsi olaraq isə bu eyniliklər bir və ya bir neçə bucağın müəyyən funksiyalarını ehtiva edən eyniliklərdir.

Sinus və kosinus arasındakı əsas əlaqə Pifaqorun triqonometrik eyniliyi ilə verilir:

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

burada ${\displaystyle \sin ^{2}\theta }$ – ${\displaystyle (\sin \theta )^{2}}$, ${\displaystyle \cos ^{2}\theta }$ –${\displaystyle (\cos \theta )^{2}}$ deməkdir.

Bu bərabərlikdən sinus və kosinusu tapmaq mümkündür:

${\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}$

Bərabərliyin tərəflərini ayrı-ayrılıqda sinusa və kosinusa və ya hər ikisinə böldükdə aşağıdakı eyniliklər alınır:

${\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}$

Bu eyniliklərdən istifadə edərək hər hansı bir triqonometrik funksiyanı digəri ilə ifadə etmək mümkündür:

Triqonometrik funksiyalardan hər birinin digər beşi ilə ifadəsi

	${\displaystyle \sin \theta }$	${\displaystyle \csc \theta }$	${\displaystyle \cos \theta }$	${\displaystyle \sec \theta }$	${\displaystyle \tan \theta }$	${\displaystyle \cot \theta }$
${\displaystyle \sin \theta =}$	${\displaystyle \sin \theta }$	${\displaystyle {\frac {1}{\csc \theta }}}$	${\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}$	${\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}$	${\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}$	${\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}$
${\displaystyle \csc \theta =}$	${\displaystyle {\frac {1}{\sin \theta }}}$	${\displaystyle \csc \theta }$	${\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}$	${\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}$	${\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}$	${\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}$
${\displaystyle \cos \theta =}$	${\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}$	${\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}$	${\displaystyle \cos \theta }$	${\displaystyle {\frac {1}{\sec \theta }}}$	${\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}$	${\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}$
${\displaystyle \sec \theta =}$	${\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}$	${\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}$	${\displaystyle {\frac {1}{\cos \theta }}}$	${\displaystyle \sec \theta }$	${\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}$	${\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}$
${\displaystyle \tan \theta =}$	${\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}$	${\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}$	${\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}$	${\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}$	${\displaystyle \tan \theta }$	${\displaystyle {\frac {1}{\cot \theta }}}$
${\displaystyle \cot \theta =}$	${\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}$	${\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}$	${\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}$	${\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}$	${\displaystyle {\frac {1}{\tan \theta }}}$	${\displaystyle \cot \theta }$

Dörddə bir dövrdə dəyişmə	Yarım dövrdə dəyişmə	Tam dövrdə dəyişmə[1]	Funksiyanın dövrü
${\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }$	${\displaystyle \sin(\theta +\pi )=-\sin \theta }$	${\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }$	${\displaystyle 2\pi }$
${\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }$	${\displaystyle \cos(\theta +\pi )=-\cos \theta }$)	${\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }$	${\displaystyle 2\pi }$
${\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }$	${\displaystyle \csc(\theta +\pi )=-\csc \theta }$	${\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }$	${\displaystyle 2\pi }$
${\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }$	${\displaystyle \sec(\theta +\pi )=-\sec \theta }$	${\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }$	${\displaystyle 2\pi }$
${\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}$	${\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }$	${\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }$	${\displaystyle \pi }$
${\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}$	${\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }$	${\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }$	${\displaystyle \pi }$

Triqonometrik funksiyaların işarəsi bucağın rübündən asılıdır. Əgər ${\displaystyle {-\pi }<\theta \leq \pi }$ və sgn işarə funksiyasını ifadə edərsə,

${\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&{\text{if}}\ \ 0<\theta <\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <0\\0&{\text{if}}\ \ \theta \in \{0,\pi \}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&{\text{if}}\ \ {-{\tfrac {1}{2}}\pi }<\theta <{\tfrac {1}{2}}\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{or}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },{\tfrac {1}{2}}\pi {\bigr \}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{or}}\ \ 0<\theta <{\tfrac {1}{2}}\pi \\-1&{\text{if}}\ \ {-{\tfrac {1}{2}}\pi }<\theta <0\ \ {\text{or}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },0,{\tfrac {1}{2}}\pi ,\pi {\bigr \}}\end{cases}}\end{aligned}}}$

${\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}}$

${\displaystyle \sin(\alpha -\beta )}$ və ${\displaystyle \cos(\alpha -\beta )}$ bucaq fərqlərini "${\displaystyle \beta }$" -nı "${\displaystyle -\beta }$ " ilə əvəz etməklə və ${\displaystyle \sin(-\beta )=-\sin(\beta )}$ və ${\displaystyle \cos(-\beta )=\cos(\beta )}$ faktına əsaslanaraq da tapmaq olar.

Bu eyniliklər digər triqonometrik funksiyalar üçün cəm və fərq eyniliklərini ehtiva edən aşağıdakı cədvəldə ümumiləşdirilmişdir:

Sinus	${\displaystyle \sin(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$[2][3]
Kosinus	${\displaystyle \cos(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$[3][4]
Tanqens	${\displaystyle \tan(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}$[3][5]
Kosekans	${\displaystyle \csc(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}$[6]
Sekans	${\displaystyle \sec(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}$[6]
Kontanqens	${\displaystyle \cot(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}$[3][7]
Ark-sinus	${\displaystyle \arcsin x\pm \arcsin y}$			${\displaystyle =}$	${\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}}}\right)}$[8]
Ark-kosinus	${\displaystyle \arccos x\pm \arccos y}$			${\displaystyle =}$	${\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}$[9]
Ark-tanqens	${\displaystyle \arctan x\pm \arctan y}$			${\displaystyle =}$	${\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}$[10]
Ark-kotanqens	${\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}$			${\displaystyle =}$	${\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}$

Düstur	Arqumentin mənası
${\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}$	${\displaystyle \forall \alpha }$
${\displaystyle \operatorname {tan} ^{2}\alpha +1={\frac {1}{\cos ^{2}\alpha }}=\operatorname {sec} ^{2}\alpha }$	${\displaystyle \alpha \neq {\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} }$
${\displaystyle \operatorname {cot} ^{2}\alpha +1={\frac {1}{\sin ^{2}\alpha }}=\operatorname {cosec} ^{2}\alpha }$	${\displaystyle \alpha \neq \pi n,n\in \mathbb {Z} }$
${\displaystyle ~\operatorname {tan} \alpha \cdot \operatorname {cot} \alpha =1}$	${\displaystyle \alpha \neq {\frac {\pi n}{2}},n\in \mathbb {Z} }$
${\displaystyle ~\operatorname {tan} \alpha ={\frac {\sin \alpha }{\cos \alpha }}}$

Toplama düsturları
${\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$
${\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$
${\displaystyle \operatorname {tan} \left(\alpha \pm \beta \right)={\frac {\operatorname {tan} \alpha \pm \operatorname {tan} \beta }{1\mp \operatorname {tan} \alpha \operatorname {tan} \beta }}}$
${\displaystyle \operatorname {cot} \left(\alpha \pm \beta \right)={\frac {\operatorname {cot} \alpha \operatorname {cot} \beta \mp 1}{\operatorname {cot} \beta \pm \operatorname {cot} \alpha }}}$

İkiqat arqument düsturları
${\displaystyle \sin 2\alpha =2{\sin \alpha }{\cos \alpha }}$
${\displaystyle \cos 2\alpha ={\cos ^{2}\alpha }-{\sin ^{2}\alpha }}$ ${\displaystyle \cos 2\alpha =2{\cos ^{2}\alpha }-1=1-2{\sin ^{2}\alpha }}$
${\displaystyle \operatorname {tan} 2\alpha ={\frac {2\operatorname {tan} \alpha }{1-\operatorname {tan} ^{2}\alpha }}}$
${\displaystyle \operatorname {cot} 2\alpha ={\frac {\operatorname {cot} ^{2}\alpha -1}{2\operatorname {cot} \alpha }}}$

Üçqat arqument düsturları
${\displaystyle \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha \,}$
${\displaystyle \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha \,}$
${\displaystyle \operatorname {tan} 3\alpha ={\frac {3\operatorname {tan} \alpha -\operatorname {tan} ^{3}\alpha }{1-3\operatorname {tan} ^{2}\alpha }}}$
${\displaystyle \operatorname {cot} 3\alpha ={\frac {3\operatorname {cot} \alpha -\operatorname {cot} ^{3}\alpha }{1-3\operatorname {cot} ^{2}\alpha }}}$

Sinus	Kosinus
${\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\alpha }{2}}}$	${\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}}$
${\displaystyle \sin ^{3}\alpha ={\frac {3\sin \alpha -\sin 3\alpha }{4}}}$	${\displaystyle \cos ^{3}\alpha ={\frac {3\cos \alpha +\cos 3\alpha }{4}}}$
${\displaystyle \sin ^{4}\alpha ={\frac {3-4\cos 2\alpha +\cos 4\alpha }{8}}}$	${\displaystyle \cos ^{4}\alpha ={\frac {3+4\cos 2\alpha +\cos 4\alpha }{8}}}$
${\displaystyle \sin ^{5}\alpha ={\frac {10\sin \alpha -5\sin 3\alpha +\sin 5\alpha }{16}}}$	${\displaystyle \cos ^{5}\alpha ={\frac {10\cos \alpha +5\cos 3\alpha +\cos 5\alpha }{16}}}$

Düstur
${\displaystyle \sin ^{2}\alpha \cos ^{2}\alpha ={\frac {1-\cos 4\alpha }{8}}}$
${\displaystyle \sin ^{3}\alpha \cos ^{3}\alpha ={\frac {3\sin 2\alpha -\sin 6\alpha }{32}}}$
${\displaystyle \sin ^{4}\alpha \cos ^{4}\alpha ={\frac {3-4\cos 4\alpha +\cos 8\alpha }{128}}}$
${\displaystyle \sin ^{5}\alpha \cos ^{5}\alpha ={\frac {10\sin 2\alpha -5\sin 6\alpha +\sin 10\alpha }{512}}}$

Hasilin cəmə çevrilməsi düsturları
${\displaystyle \sin \alpha \sin \beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{2}}}$
${\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{2}}}$