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Triqonometriyada triqonometrik eyniliklər triqonometrik funksiyaların daxil olduğu bərabərliklərdir Həndəsi olaraq isə b

Triqonometriyanın əsas formulları

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

Pifaqorun triqonometrik eynilikləri

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

sin2⁡θ+cos2⁡θ=1,{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}image

burada sin2⁡θ{\displaystyle \sin ^{2}\theta }image – (sin⁡θ)2{\displaystyle (\sin \theta )^{2}}image, cos2⁡θ{\displaystyle \cos ^{2}\theta }image –(cos⁡θ)2{\displaystyle (\cos \theta )^{2}}image deməkdir.

Bu bərabərlikdən sinus və kosinusu tapmaq mümkündür:

sin⁡θ=±1−cos2⁡θ,cos⁡θ=±1−sin2⁡θ.{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}image

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

1+cot2⁡θ=csc2⁡θ1+tan2⁡θ=sec2⁡θsec2⁡θ+csc2⁡θ=sec2⁡θcsc2⁡θ{\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}}}image

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

Triqonometrik funksiyalardan hər birinin digər beşi ilə ifadəsi
sin⁡θ{\displaystyle \sin \theta }image csc⁡θ{\displaystyle \csc \theta }image cos⁡θ{\displaystyle \cos \theta }image sec⁡θ{\displaystyle \sec \theta }image tan⁡θ{\displaystyle \tan \theta }image cot⁡θ{\displaystyle \cot \theta }image
sin⁡θ={\displaystyle \sin \theta =}image sin⁡θ{\displaystyle \sin \theta }image 1csc⁡θ{\displaystyle {\frac {1}{\csc \theta }}}image ±1−cos2⁡θ{\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}image ±sec2⁡θ−1sec⁡θ{\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}image ±tan⁡θ1+tan2⁡θ{\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}image ±11+cot2⁡θ{\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}image
csc⁡θ={\displaystyle \csc \theta =}image 1sin⁡θ{\displaystyle {\frac {1}{\sin \theta }}}image csc⁡θ{\displaystyle \csc \theta }image ±11−cos2⁡θ{\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}image ±sec⁡θsec2⁡θ−1{\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}image ±1+tan2⁡θtan⁡θ{\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}image ±1+cot2⁡θ{\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}image
cos⁡θ={\displaystyle \cos \theta =}image ±1−sin2⁡θ{\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}image ±csc2⁡θ−1csc⁡θ{\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}image cos⁡θ{\displaystyle \cos \theta }image 1sec⁡θ{\displaystyle {\frac {1}{\sec \theta }}}image ±11+tan2⁡θ{\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}image ±cot⁡θ1+cot2⁡θ{\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}image
sec⁡θ={\displaystyle \sec \theta =}image ±11−sin2⁡θ{\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}image ±csc⁡θcsc2⁡θ−1{\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}image 1cos⁡θ{\displaystyle {\frac {1}{\cos \theta }}}image sec⁡θ{\displaystyle \sec \theta }image ±1+tan2⁡θ{\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}image ±1+cot2⁡θcot⁡θ{\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}image
tan⁡θ={\displaystyle \tan \theta =}image ±sin⁡θ1−sin2⁡θ{\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}image ±1csc2⁡θ−1{\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}image ±1−cos2⁡θcos⁡θ{\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}image ±sec2⁡θ−1{\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}image tan⁡θ{\displaystyle \tan \theta }image 1cot⁡θ{\displaystyle {\frac {1}{\cot \theta }}}image
cot⁡θ={\displaystyle \cot \theta =}image ±1−sin2⁡θsin⁡θ{\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}image ±csc2⁡θ−1{\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}image ±cos⁡θ1−cos2⁡θ{\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}image ±1sec2⁡θ−1{\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}image 1tan⁡θ{\displaystyle {\frac {1}{\tan \theta }}}image cot⁡θ{\displaystyle \cot \theta }image

Çevrilmələr, yerdəyişmələr və dövrilik

Çevrilmələr

Dəyişmələr və dövrilik

image
Dörddə bir dövrdə dəyişmə Yarım dövrdə dəyişmə Tam dövrdə dəyişmə Funksiyanın dövrü
sin⁡(θ±π2)=±cos⁡θ{\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }image sin⁡(θ+π)=−sin⁡θ{\displaystyle \sin(\theta +\pi )=-\sin \theta }image sin⁡(θ+k⋅2π)=+sin⁡θ{\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }image 2π{\displaystyle 2\pi }image
cos⁡(θ±π2)=∓sin⁡θ{\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }image cos⁡(θ+π)=−cos⁡θ{\displaystyle \cos(\theta +\pi )=-\cos \theta }image) cos⁡(θ+k⋅2π)=+cos⁡θ{\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }image 2π{\displaystyle 2\pi }image
csc⁡(θ±π2)=±sec⁡θ{\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }image csc⁡(θ+π)=−csc⁡θ{\displaystyle \csc(\theta +\pi )=-\csc \theta }image csc⁡(θ+k⋅2π)=+csc⁡θ{\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }image 2π{\displaystyle 2\pi }image
sec⁡(θ±π2)=∓csc⁡θ{\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }image sec⁡(θ+π)=−sec⁡θ{\displaystyle \sec(\theta +\pi )=-\sec \theta }image sec⁡(θ+k⋅2π)=+sec⁡θ{\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }image 2π{\displaystyle 2\pi }image
tan⁡(θ±π4)=tan⁡θ±11∓tan⁡θ{\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}image tan⁡(θ+π2)=−cot⁡θ{\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }image tan⁡(θ+k⋅π)=+tan⁡θ{\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }image π{\displaystyle \pi }image
cot⁡(θ±π4)=cot⁡θ∓11±cot⁡θ{\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}image cot⁡(θ+π2)=−tan⁡θ{\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }image cot⁡(θ+k⋅π)=+cot⁡θ{\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }image π{\displaystyle \pi }image

İşarələr

Triqonometrik funksiyaların işarəsi bucağın rübündən asılıdır. Əgər −π<θ≤π{\displaystyle {-\pi }<\theta \leq \pi }image və sgn ifadə edərsə,

sgn⁡(sin⁡θ)=sgn⁡(csc⁡θ)={+1if  0<θ<π−1if  −π<θ<00if  θ∈{0,π}sgn⁡(cos⁡θ)=sgn⁡(sec⁡θ)={+1if  −12π<θ<12π−1if  −π<θ<−12π  or  12π<θ<π0if  θ∈{−12π,12π}sgn⁡(tan⁡θ)=sgn⁡(cot⁡θ)={+1if  −π<θ<−12π  or  0<θ<12π−1if  −12π<θ<0  or  12π<θ<π0if  θ∈{−12π,0,12π,π}{\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}}}image

Bucaqların cəmi və fərqi üçün eyniliklər

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡βsin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡βcos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡βcos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β{\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}}}image

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

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

Sinus sin⁡(α±β){\displaystyle \sin(\alpha \pm \beta )}image ={\displaystyle =}image sin⁡αcos⁡β±cos⁡αsin⁡β{\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }image
Kosinus cos⁡(α±β){\displaystyle \cos(\alpha \pm \beta )}image ={\displaystyle =}image cos⁡αcos⁡β∓sin⁡αsin⁡β{\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }image
Tanqens tan⁡(α±β){\displaystyle \tan(\alpha \pm \beta )}image ={\displaystyle =}image tan⁡α±tan⁡β1∓tan⁡αtan⁡β{\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}image
Kosekans csc⁡(α±β){\displaystyle \csc(\alpha \pm \beta )}image ={\displaystyle =}image sec⁡αsec⁡βcsc⁡αcsc⁡βsec⁡αcsc⁡β±csc⁡αsec⁡β{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}image
Sekans sec⁡(α±β){\displaystyle \sec(\alpha \pm \beta )}image ={\displaystyle =}image sec⁡αsec⁡βcsc⁡αcsc⁡βcsc⁡αcsc⁡β∓sec⁡αsec⁡β{\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}image
Kontanqens cot⁡(α±β){\displaystyle \cot(\alpha \pm \beta )}image ={\displaystyle =}image cot⁡αcot⁡β∓1cot⁡β±cot⁡α{\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}image
Ark-sinus arcsin⁡x±arcsin⁡y{\displaystyle \arcsin x\pm \arcsin y}image ={\displaystyle =}image arcsin⁡(x1−y2±y1−x2){\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}}}\right)}image
Ark-kosinus arccos⁡x±arccos⁡y{\displaystyle \arccos x\pm \arccos y}image ={\displaystyle =}image arccos⁡(xy∓(1−x2)(1−y2)){\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}image
Ark-tanqens arctan⁡x±arctan⁡y{\displaystyle \arctan x\pm \arctan y}image ={\displaystyle =}image arctan⁡(x±y1∓xy){\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}image
Ark-kotanqens arccot⁡x±arccot⁡y{\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}image ={\displaystyle =}image arccot⁡(xy∓1y±x){\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}image

Əsas triqonometrik düsturlar

Düstur Arqumentin mənası
sin2⁡α+cos2⁡α=1{\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1}image ∀α{\displaystyle \forall \alpha }image
tan2⁡α+1=1cos2⁡α=sec2⁡α{\displaystyle \operatorname {tan} ^{2}\alpha +1={\frac {1}{\cos ^{2}\alpha }}=\operatorname {sec} ^{2}\alpha }image α≠π2+πn,n∈Z{\displaystyle \alpha \neq {\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} }image
cot2⁡α+1=1sin2⁡α=cosec2⁡α{\displaystyle \operatorname {cot} ^{2}\alpha +1={\frac {1}{\sin ^{2}\alpha }}=\operatorname {cosec} ^{2}\alpha }image α≠πn,n∈Z{\displaystyle \alpha \neq \pi n,n\in \mathbb {Z} }image
 tan⁡α⋅cot⁡α=1{\displaystyle ~\operatorname {tan} \alpha \cdot \operatorname {cot} \alpha =1}image α≠πn2,n∈Z{\displaystyle \alpha \neq {\frac {\pi n}{2}},n\in \mathbb {Z} }image
 tan⁡α=sin⁡αcos⁡α{\displaystyle ~\operatorname {tan} \alpha ={\frac {\sin \alpha }{\cos \alpha }}}image

Toplama düsturları

Toplama düsturları
sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β{\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }image
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β{\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }image
tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β{\displaystyle \operatorname {tan} \left(\alpha \pm \beta \right)={\frac {\operatorname {tan} \alpha \pm \operatorname {tan} \beta }{1\mp \operatorname {tan} \alpha \operatorname {tan} \beta }}}image
cot⁡(α±β)=cot⁡αcot⁡β∓1cot⁡β±cot⁡α{\displaystyle \operatorname {cot} \left(\alpha \pm \beta \right)={\frac {\operatorname {cot} \alpha \operatorname {cot} \beta \mp 1}{\operatorname {cot} \beta \pm \operatorname {cot} \alpha }}}image

İkiqat arqument düsturları

İkiqat arqument düsturları
sin⁡2α=2sin⁡αcos⁡α{\displaystyle \sin 2\alpha =2{\sin \alpha }{\cos \alpha }}image
cos⁡2α=cos2⁡α−sin2⁡α{\displaystyle \cos 2\alpha ={\cos ^{2}\alpha }-{\sin ^{2}\alpha }}image
cos⁡2α=2cos2⁡α−1=1−2sin2⁡α{\displaystyle \cos 2\alpha =2{\cos ^{2}\alpha }-1=1-2{\sin ^{2}\alpha }}image
tan⁡2α=2tan⁡α1−tan2⁡α{\displaystyle \operatorname {tan} 2\alpha ={\frac {2\operatorname {tan} \alpha }{1-\operatorname {tan} ^{2}\alpha }}}image
cot⁡2α=cot2⁡α−12cot⁡α{\displaystyle \operatorname {cot} 2\alpha ={\frac {\operatorname {cot} ^{2}\alpha -1}{2\operatorname {cot} \alpha }}}image

Üçqat arqument düsturları

Üçqat arqument düsturları
sin⁡3α=3sin⁡α−4sin3⁡α{\displaystyle \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha \,}image
cos⁡3α=4cos3⁡α−3cos⁡α{\displaystyle \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha \,}image
tan⁡3α=3tan⁡α−tan3⁡α1−3tan2⁡α{\displaystyle \operatorname {tan} 3\alpha ={\frac {3\operatorname {tan} \alpha -\operatorname {tan} ^{3}\alpha }{1-3\operatorname {tan} ^{2}\alpha }}}image
cot⁡3α=3cot⁡α−cot3⁡α1−3cot2⁡α{\displaystyle \operatorname {cot} 3\alpha ={\frac {3\operatorname {cot} \alpha -\operatorname {cot} ^{3}\alpha }{1-3\operatorname {cot} ^{2}\alpha }}}image

Dərəcənin aşağı salma düsturları

Sinus Kosinus
sin2⁡α=1−cos⁡2α2{\displaystyle \sin ^{2}\alpha ={\frac {1-\cos 2\alpha }{2}}}image cos2⁡α=1+cos⁡2α2{\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}}image
sin3⁡α=3sin⁡α−sin⁡3α4{\displaystyle \sin ^{3}\alpha ={\frac {3\sin \alpha -\sin 3\alpha }{4}}}image cos3⁡α=3cos⁡α+cos⁡3α4{\displaystyle \cos ^{3}\alpha ={\frac {3\cos \alpha +\cos 3\alpha }{4}}}image
sin4⁡α=3−4cos⁡2α+cos⁡4α8{\displaystyle \sin ^{4}\alpha ={\frac {3-4\cos 2\alpha +\cos 4\alpha }{8}}}image cos4⁡α=3+4cos⁡2α+cos⁡4α8{\displaystyle \cos ^{4}\alpha ={\frac {3+4\cos 2\alpha +\cos 4\alpha }{8}}}image
sin5⁡α=10sin⁡α−5sin⁡3α+sin⁡5α16{\displaystyle \sin ^{5}\alpha ={\frac {10\sin \alpha -5\sin 3\alpha +\sin 5\alpha }{16}}}image cos5⁡α=10cos⁡α+5cos⁡3α+cos⁡5α16{\displaystyle \cos ^{5}\alpha ={\frac {10\cos \alpha +5\cos 3\alpha +\cos 5\alpha }{16}}}image
Düstur
sin2⁡αcos2⁡α=1−cos⁡4α8{\displaystyle \sin ^{2}\alpha \cos ^{2}\alpha ={\frac {1-\cos 4\alpha }{8}}}image
sin3⁡αcos3⁡α=3sin⁡2α−sin⁡6α32{\displaystyle \sin ^{3}\alpha \cos ^{3}\alpha ={\frac {3\sin 2\alpha -\sin 6\alpha }{32}}}image
sin4⁡αcos4⁡α=3−4cos⁡4α+cos⁡8α128{\displaystyle \sin ^{4}\alpha \cos ^{4}\alpha ={\frac {3-4\cos 4\alpha +\cos 8\alpha }{128}}}image
sin5⁡αcos5⁡α=10sin⁡2α−5sin⁡6α+sin⁡10α512{\displaystyle \sin ^{5}\alpha \cos ^{5}\alpha ={\frac {10\sin 2\alpha -5\sin 6\alpha +\sin 10\alpha }{512}}}image

Hasilin cəmə çevrilməsi düsturla

Hasilin cəmə çevrilməsi düsturları
sin⁡αsin⁡β=cos⁡(α−β)−cos⁡(α+β)2{\displaystyle \sin \alpha \sin \beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{2}}}image
cos⁡αcos⁡β=cos⁡(α−β)+cos⁡(α+β)2{\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{2}}}image

İstinadlar

  1. Abramowitz and Stegun, p. 72, 4.3.7–9
  2. Abramowitz and Stegun, p. 72, 4.3.16
  3. Weisstein, Eric W. Trigonometric Addition Formulas (ing.) Wolfram saytında.
  4. Abramowitz and Stegun, p. 72, 4.3.17
  5. Abramowitz and Stegun, p. 72, 4.3.18
  6. "Angle Sum and Difference Identities". www.milefoot.com. 2023-04-03 tarixində . İstifadə tarixi: 2019-10-12.
  7. Abramowitz and Stegun, p. 72, 4.3.19
  8. Abramowitz and Stegun, p. 80, 4.4.32
  9. Abramowitz and Stegun, p. 80, 4.4.33
  10. Abramowitz and Stegun, p. 80, 4.4.34

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Triqonometriyada triqonometrik eynilikler triqonometrik funksiyalarin daxil oldugu beraberliklerdir Hendesi olaraq ise bu eynilikler bir ve ya bir nece bucagin mueyyen funksiyalarini ehtiva eden eyniliklerdir Pifaqorun triqonometrik eynilikleriSinus ve kosinus arasindaki esas elaqe Pifaqorun triqonometrik eyniliyi ile verilir sin2 8 cos2 8 1 displaystyle sin 2 theta cos 2 theta 1 burada sin2 8 displaystyle sin 2 theta sin 8 2 displaystyle sin theta 2 cos2 8 displaystyle cos 2 theta cos 8 2 displaystyle cos theta 2 demekdir Bu beraberlikden sinus ve kosinusu tapmaq mumkundur sin 8 1 cos2 8 cos 8 1 sin2 8 displaystyle begin aligned sin theta amp pm sqrt 1 cos 2 theta cos theta amp pm sqrt 1 sin 2 theta end aligned Beraberliyin tereflerini ayri ayriliqda sinusa ve kosinusa ve ya her ikisine boldukde asagidaki eynilikler alinir 1 cot2 8 csc2 81 tan2 8 sec2 8sec2 8 csc2 8 sec2 8csc2 8 displaystyle begin aligned amp 1 cot 2 theta csc 2 theta amp 1 tan 2 theta sec 2 theta amp sec 2 theta csc 2 theta sec 2 theta csc 2 theta end aligned Bu eyniliklerden istifade ederek her hansi bir triqonometrik funksiyani digeri ile ifade etmek mumkundur Triqonometrik funksiyalardan her birinin diger besi ile ifadesi sin 8 displaystyle sin theta csc 8 displaystyle csc theta cos 8 displaystyle cos theta sec 8 displaystyle sec theta tan 8 displaystyle tan theta cot 8 displaystyle cot theta sin 8 displaystyle sin theta sin 8 displaystyle sin theta 1csc 8 displaystyle frac 1 csc theta 1 cos2 8 displaystyle pm sqrt 1 cos 2 theta sec2 8 1sec 8 displaystyle pm frac sqrt sec 2 theta 1 sec theta tan 81 tan2 8 displaystyle pm frac tan theta sqrt 1 tan 2 theta 11 cot2 8 displaystyle pm frac 1 sqrt 1 cot 2 theta csc 8 displaystyle csc theta 1sin 8 displaystyle frac 1 sin theta csc 8 displaystyle csc theta 11 cos2 8 displaystyle pm frac 1 sqrt 1 cos 2 theta sec 8sec2 8 1 displaystyle pm frac sec theta sqrt sec 2 theta 1 1 tan2 8tan 8 displaystyle pm frac sqrt 1 tan 2 theta tan theta 1 cot2 8 displaystyle pm sqrt 1 cot 2 theta cos 8 displaystyle cos theta 1 sin2 8 displaystyle pm sqrt 1 sin 2 theta csc2 8 1csc 8 displaystyle pm frac sqrt csc 2 theta 1 csc theta cos 8 displaystyle cos theta 1sec 8 displaystyle frac 1 sec theta 11 tan2 8 displaystyle pm frac 1 sqrt 1 tan 2 theta cot 81 cot2 8 displaystyle pm frac cot theta sqrt 1 cot 2 theta sec 8 displaystyle sec theta 11 sin2 8 displaystyle pm frac 1 sqrt 1 sin 2 theta csc 8csc2 8 1 displaystyle pm frac csc theta sqrt csc 2 theta 1 1cos 8 displaystyle frac 1 cos theta sec 8 displaystyle sec theta 1 tan2 8 displaystyle pm sqrt 1 tan 2 theta 1 cot2 8cot 8 displaystyle pm frac sqrt 1 cot 2 theta cot theta tan 8 displaystyle tan theta sin 81 sin2 8 displaystyle pm frac sin theta sqrt 1 sin 2 theta 1csc2 8 1 displaystyle pm frac 1 sqrt csc 2 theta 1 1 cos2 8cos 8 displaystyle pm frac sqrt 1 cos 2 theta cos theta sec2 8 1 displaystyle pm sqrt sec 2 theta 1 tan 8 displaystyle tan theta 1cot 8 displaystyle frac 1 cot theta cot 8 displaystyle cot theta 1 sin2 8sin 8 displaystyle pm frac sqrt 1 sin 2 theta sin theta csc2 8 1 displaystyle pm sqrt csc 2 theta 1 cos 81 cos2 8 displaystyle pm frac cos theta sqrt 1 cos 2 theta 1sec2 8 1 displaystyle pm frac 1 sqrt sec 2 theta 1 1tan 8 displaystyle frac 1 tan theta cot 8 displaystyle cot theta Cevrilmeler yerdeyismeler ve dovrilikCevrilmeler Deyismeler ve dovrilik Dordde bir dovrde deyisme Yarim dovrde deyisme Tam dovrde deyisme Funksiyanin dovrusin 8 p2 cos 8 displaystyle sin theta pm tfrac pi 2 pm cos theta sin 8 p sin 8 displaystyle sin theta pi sin theta sin 8 k 2p sin 8 displaystyle sin theta k cdot 2 pi sin theta 2p displaystyle 2 pi cos 8 p2 sin 8 displaystyle cos theta pm tfrac pi 2 mp sin theta cos 8 p cos 8 displaystyle cos theta pi cos theta cos 8 k 2p cos 8 displaystyle cos theta k cdot 2 pi cos theta 2p displaystyle 2 pi csc 8 p2 sec 8 displaystyle csc theta pm tfrac pi 2 pm sec theta csc 8 p csc 8 displaystyle csc theta pi csc theta csc 8 k 2p csc 8 displaystyle csc theta k cdot 2 pi csc theta 2p displaystyle 2 pi sec 8 p2 csc 8 displaystyle sec theta pm tfrac pi 2 mp csc theta sec 8 p sec 8 displaystyle sec theta pi sec theta sec 8 k 2p sec 8 displaystyle sec theta k cdot 2 pi sec theta 2p displaystyle 2 pi tan 8 p4 tan 8 11 tan 8 displaystyle tan theta pm tfrac pi 4 tfrac tan theta pm 1 1 mp tan theta tan 8 p2 cot 8 displaystyle tan theta tfrac pi 2 cot theta tan 8 k p tan 8 displaystyle tan theta k cdot pi tan theta p displaystyle pi cot 8 p4 cot 8 11 cot 8 displaystyle cot theta pm tfrac pi 4 tfrac cot theta mp 1 1 pm cot theta cot 8 p2 tan 8 displaystyle cot theta tfrac pi 2 tan theta cot 8 k p cot 8 displaystyle cot theta k cdot pi cot theta p displaystyle pi Isareler Triqonometrik funksiyalarin isaresi bucagin rubunden asilidir Eger p lt 8 p displaystyle pi lt theta leq pi ve sgn ifade ederse sgn sin 8 sgn csc 8 1if 0 lt 8 lt p 1if p lt 8 lt 00if 8 0 p sgn cos 8 sgn sec 8 1if 12p lt 8 lt 12p 1if p lt 8 lt 12p or 12p lt 8 lt p0if 8 12p 12p sgn tan 8 sgn cot 8 1if p lt 8 lt 12p or 0 lt 8 lt 12p 1if 12p lt 8 lt 0 or 12p lt 8 lt p0if 8 12p 0 12p p displaystyle begin aligned operatorname sgn sin theta operatorname sgn csc theta amp begin cases 1 amp text if 0 lt theta lt pi 1 amp text if pi lt theta lt 0 0 amp text if theta in 0 pi end cases 5mu operatorname sgn cos theta operatorname sgn sec theta amp begin cases 1 amp text if tfrac 1 2 pi lt theta lt tfrac 1 2 pi 1 amp text if pi lt theta lt tfrac 1 2 pi text or tfrac 1 2 pi lt theta lt pi 0 amp 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 amp begin cases 1 amp text if pi lt theta lt tfrac 1 2 pi text or 0 lt theta lt tfrac 1 2 pi 1 amp text if tfrac 1 2 pi lt theta lt 0 text or tfrac 1 2 pi lt theta lt pi 0 amp text if theta in bigl tfrac 1 2 pi 0 tfrac 1 2 pi pi bigr end cases end aligned Bucaqlarin cemi ve ferqi ucun eyniliklersin a b sin acos b cos asin bsin a b sin acos b cos asin bcos a b cos acos b sin asin bcos a b cos acos b sin asin b displaystyle begin aligned sin alpha beta amp sin alpha cos beta cos alpha sin beta sin alpha beta amp sin alpha cos beta cos alpha sin beta cos alpha beta amp cos alpha cos beta sin alpha sin beta cos alpha beta amp cos alpha cos beta sin alpha sin beta end aligned sin a b displaystyle sin alpha beta ve cos a b displaystyle cos alpha beta bucaq ferqlerini b displaystyle beta ni b displaystyle beta ile evez etmekle ve sin b sin b displaystyle sin beta sin beta ve cos b cos b displaystyle cos beta cos beta faktina esaslanaraq da tapmaq olar Bu eynilikler diger triqonometrik funksiyalar ucun cem ve ferq eyniliklerini ehtiva eden asagidaki cedvelde umumilesdirilmisdir Sinus sin a b displaystyle sin alpha pm beta displaystyle sin acos b cos asin b displaystyle sin alpha cos beta pm cos alpha sin beta Kosinus cos a b displaystyle cos alpha pm beta displaystyle cos acos b sin asin b displaystyle cos alpha cos beta mp sin alpha sin beta Tanqens tan a b displaystyle tan alpha pm beta displaystyle tan a tan b1 tan atan b displaystyle frac tan alpha pm tan beta 1 mp tan alpha tan beta Kosekans csc a b displaystyle csc alpha pm beta displaystyle sec asec bcsc acsc bsec acsc b csc asec b displaystyle frac sec alpha sec beta csc alpha csc beta sec alpha csc beta pm csc alpha sec beta Sekans sec a b displaystyle sec alpha pm beta displaystyle sec asec bcsc acsc bcsc acsc b sec asec b displaystyle frac sec alpha sec beta csc alpha csc beta csc alpha csc beta mp sec alpha sec beta Kontanqens cot a b displaystyle cot alpha pm beta displaystyle cot acot b 1cot b cot a displaystyle frac cot alpha cot beta mp 1 cot beta pm cot alpha Ark sinus arcsin x arcsin y displaystyle arcsin x pm arcsin y displaystyle arcsin x1 y2 y1 x2 displaystyle arcsin left x sqrt 1 y 2 pm y sqrt 1 x 2 right Ark kosinus arccos x arccos y displaystyle arccos x pm arccos y displaystyle arccos xy 1 x2 1 y2 displaystyle arccos left xy mp sqrt left 1 x 2 right left 1 y 2 right right Ark tanqens arctan x arctan y displaystyle arctan x pm arctan y displaystyle arctan x y1 xy displaystyle arctan left frac x pm y 1 mp xy right Ark kotanqens arccot x arccot y displaystyle operatorname arccot x pm operatorname arccot y displaystyle arccot xy 1y x displaystyle operatorname arccot left frac xy mp 1 y pm x right Esas triqonometrik dusturlarDustur Arqumentin menasisin2 a cos2 a 1 displaystyle sin 2 alpha cos 2 alpha 1 a displaystyle forall alpha tan2 a 1 1cos2 a sec2 a displaystyle operatorname tan 2 alpha 1 frac 1 cos 2 alpha operatorname sec 2 alpha a p2 pn n Z displaystyle alpha neq frac pi 2 pi n n in mathbb Z cot2 a 1 1sin2 a cosec2 a displaystyle operatorname cot 2 alpha 1 frac 1 sin 2 alpha operatorname cosec 2 alpha a pn n Z displaystyle alpha neq pi n n in mathbb Z tan a cot a 1 displaystyle operatorname tan alpha cdot operatorname cot alpha 1 a pn2 n Z displaystyle alpha neq frac pi n 2 n in mathbb Z tan a sin acos a displaystyle operatorname tan alpha frac sin alpha cos alpha Toplama dusturlariToplama dusturlarisin a b sin acos b cos asin b displaystyle sin left alpha pm beta right sin alpha cos beta pm cos alpha sin beta cos a b cos acos b sin asin b displaystyle cos left alpha pm beta right cos alpha cos beta mp sin alpha sin beta tan a b tan a tan b1 tan atan b displaystyle operatorname tan left alpha pm beta right frac operatorname tan alpha pm operatorname tan beta 1 mp operatorname tan alpha operatorname tan beta cot a b cot acot b 1cot b cot a displaystyle operatorname cot left alpha pm beta right frac operatorname cot alpha operatorname cot beta mp 1 operatorname cot beta pm operatorname cot alpha Ikiqat arqument dusturlariIkiqat arqument dusturlarisin 2a 2sin acos a displaystyle sin 2 alpha 2 sin alpha cos alpha cos 2a cos2 a sin2 a displaystyle cos 2 alpha cos 2 alpha sin 2 alpha cos 2a 2cos2 a 1 1 2sin2 a displaystyle cos 2 alpha 2 cos 2 alpha 1 1 2 sin 2 alpha tan 2a 2tan a1 tan2 a displaystyle operatorname tan 2 alpha frac 2 operatorname tan alpha 1 operatorname tan 2 alpha cot 2a cot2 a 12cot a displaystyle operatorname cot 2 alpha frac operatorname cot 2 alpha 1 2 operatorname cot alpha Ucqat arqument dusturlariUcqat arqument dusturlarisin 3a 3sin a 4sin3 a displaystyle sin 3 alpha 3 sin alpha 4 sin 3 alpha cos 3a 4cos3 a 3cos a displaystyle cos 3 alpha 4 cos 3 alpha 3 cos alpha tan 3a 3tan a tan3 a1 3tan2 a displaystyle operatorname tan 3 alpha frac 3 operatorname tan alpha operatorname tan 3 alpha 1 3 operatorname tan 2 alpha cot 3a 3cot a cot3 a1 3cot2 a displaystyle operatorname cot 3 alpha frac 3 operatorname cot alpha operatorname cot 3 alpha 1 3 operatorname cot 2 alpha Derecenin asagi salma dusturlariSinus Kosinussin2 a 1 cos 2a2 displaystyle sin 2 alpha frac 1 cos 2 alpha 2 cos2 a 1 cos 2a2 displaystyle cos 2 alpha frac 1 cos 2 alpha 2 sin3 a 3sin a sin 3a4 displaystyle sin 3 alpha frac 3 sin alpha sin 3 alpha 4 cos3 a 3cos a cos 3a4 displaystyle cos 3 alpha frac 3 cos alpha cos 3 alpha 4 sin4 a 3 4cos 2a cos 4a8 displaystyle sin 4 alpha frac 3 4 cos 2 alpha cos 4 alpha 8 cos4 a 3 4cos 2a cos 4a8 displaystyle cos 4 alpha frac 3 4 cos 2 alpha cos 4 alpha 8 sin5 a 10sin a 5sin 3a sin 5a16 displaystyle sin 5 alpha frac 10 sin alpha 5 sin 3 alpha sin 5 alpha 16 cos5 a 10cos a 5cos 3a cos 5a16 displaystyle cos 5 alpha frac 10 cos alpha 5 cos 3 alpha cos 5 alpha 16 Dustursin2 acos2 a 1 cos 4a8 displaystyle sin 2 alpha cos 2 alpha frac 1 cos 4 alpha 8 sin3 acos3 a 3sin 2a sin 6a32 displaystyle sin 3 alpha cos 3 alpha frac 3 sin 2 alpha sin 6 alpha 32 sin4 acos4 a 3 4cos 4a cos 8a128 displaystyle sin 4 alpha cos 4 alpha frac 3 4 cos 4 alpha cos 8 alpha 128 sin5 acos5 a 10sin 2a 5sin 6a sin 10a512 displaystyle sin 5 alpha cos 5 alpha frac 10 sin 2 alpha 5 sin 6 alpha sin 10 alpha 512 Hasilin ceme cevrilmesi dusturlaHasilin ceme cevrilmesi dusturlarisin asin b cos a b cos a b 2 displaystyle sin alpha sin beta frac cos alpha beta cos alpha beta 2 cos acos b cos a b cos a b 2 displaystyle cos alpha cos beta frac cos alpha beta cos alpha beta 2 IstinadlarAbramowitz and Stegun p 72 4 3 7 9 Abramowitz and Stegun p 72 4 3 16 Weisstein Eric W Trigonometric Addition Formulas ing Wolfram saytinda Abramowitz and Stegun p 72 4 3 17 Abramowitz and Stegun p 72 4 3 18 Angle Sum and Difference Identities www milefoot com 2023 04 03 tarixinde Istifade tarixi 2019 10 12 Abramowitz and Stegun p 72 4 3 19 Abramowitz and Stegun p 80 4 4 32 Abramowitz and Stegun p 80 4 4 33 Abramowitz and Stegun p 80 4 4 34

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