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Triqonometriyanın əsas düsturları

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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.

Mündəricat

  • 1 Pifaqorun triqonometrik eynilikləri
  • 2 Çevrilmələr, yerdəyişmələr və dövrilik
    • 2.1 Çevrilmələr
    • 2.2 Dəyişmələr və dövrilik
    • 2.3 İşarələr
  • 3 Bucaqların cəmi və fərqi üçün eyniliklər
  • 4 Əsas triqonometrik düsturlar
  • 5 Toplama düsturları
  • 6 İkiqat arqument düsturları
  • 7 Üçqat arqument düsturları
  • 8 Dərəcənin aşağı salma düsturları
  • 9 Hasilin cəmə çevrilməsi düsturla
  • 10 İstinadlar

Pifaqorun triqonometrik eynilikləri

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

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

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

Bu bərabərlikdən sinus və kosinusu tapmaq mümkündür:

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

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

Çevrilmələr, yerdəyişmələr və dövrilik

Çevrilmələr

Dəyişmələr və dövrilik

Unit circle with a swept angle theta plotted at coordinates (a,b). As the swept angle is incremented by one-half pi (90 degrees), the coordinates are transformed to (-b,a). Another increment of one-half pi (180 degrees total) transforms the coordinates to (-a,-b). A final increment of one-half pi (270 degrees total) transforms the coordinates to (b,a).
Dörddə bir dövrdə dəyişmə Yarım dövrdə dəyişmə Tam dövrdə dəyişmə[1] Funksiyanın dövrü
sin ⁡ ( θ ± π 2 ) = ± cos ⁡ θ {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta } {\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta } sin ⁡ ( θ + π ) = − sin ⁡ θ {\displaystyle \sin(\theta +\pi )=-\sin \theta } {\displaystyle \sin(\theta +\pi )=-\sin \theta } sin ⁡ ( θ + k ⋅ 2 π ) = + sin ⁡ θ {\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta } {\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta } 2 π {\displaystyle 2\pi } {\displaystyle 2\pi }
cos ⁡ ( θ ± π 2 ) = ∓ sin ⁡ θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta } {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta } cos ⁡ ( θ + π ) = − cos ⁡ θ {\displaystyle \cos(\theta +\pi )=-\cos \theta } {\displaystyle \cos(\theta +\pi )=-\cos \theta }) cos ⁡ ( θ + k ⋅ 2 π ) = + cos ⁡ θ {\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta } {\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta } 2 π {\displaystyle 2\pi } {\displaystyle 2\pi }
csc ⁡ ( θ ± π 2 ) = ± sec ⁡ θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta } {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta } csc ⁡ ( θ + π ) = − csc ⁡ θ {\displaystyle \csc(\theta +\pi )=-\csc \theta } {\displaystyle \csc(\theta +\pi )=-\csc \theta } csc ⁡ ( θ + k ⋅ 2 π ) = + csc ⁡ θ {\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta } {\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta } 2 π {\displaystyle 2\pi } {\displaystyle 2\pi }
sec ⁡ ( θ ± π 2 ) = ∓ csc ⁡ θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta } {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta } sec ⁡ ( θ + π ) = − sec ⁡ θ {\displaystyle \sec(\theta +\pi )=-\sec \theta } {\displaystyle \sec(\theta +\pi )=-\sec \theta } sec ⁡ ( θ + k ⋅ 2 π ) = + sec ⁡ θ {\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta } {\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta } 2 π {\displaystyle 2\pi } {\displaystyle 2\pi }
tan ⁡ ( θ ± π 4 ) = tan ⁡ θ ± 1 1 ∓ tan ⁡ θ {\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}} {\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}} tan ⁡ ( θ + π 2 ) = − cot ⁡ θ {\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta } {\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta } tan ⁡ ( θ + k ⋅ π ) = + tan ⁡ θ {\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta } {\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta } π {\displaystyle \pi } {\displaystyle \pi }
cot ⁡ ( θ ± π 4 ) = cot ⁡ θ ∓ 1 1 ± cot ⁡ θ {\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}} {\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}} cot ⁡ ( θ + π 2 ) = − tan ⁡ θ {\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta } {\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta } cot ⁡ ( θ + k ⋅ π ) = + cot ⁡ θ {\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta } {\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta } π {\displaystyle \pi } {\displaystyle \pi }

İşarələr

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

sgn ⁡ ( sin ⁡ θ ) = sgn ⁡ ( csc ⁡ θ ) = { + 1 if     0 < θ < π − 1 if     − π < θ < 0 0 if     θ ∈ { 0 , π } sgn ⁡ ( cos ⁡ θ ) = sgn ⁡ ( sec ⁡ θ ) = { + 1 if     − 1 2 π < θ < 1 2 π − 1 if     − π < θ < − 1 2 π     or     1 2 π < θ < π 0 if     θ ∈ { − 1 2 π , 1 2 π } sgn ⁡ ( tan ⁡ θ ) = sgn ⁡ ( cot ⁡ θ ) = { + 1 if     − π < θ < − 1 2 π     or     0 < θ < 1 2 π − 1 if     − 1 2 π < θ < 0     or     1 2 π < θ < π 0 if     θ ∈ { − 1 2 π , 0 , 1 2 π , 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}}} {\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}}}

Bucaqların cəmi və fərqi üçün eyniliklər

sin ⁡ ( α + β ) = sin ⁡ α cos ⁡ β + cos ⁡ α sin ⁡ β sin ⁡ ( α − β ) = sin ⁡ α cos ⁡ β − cos ⁡ α sin ⁡ β cos ⁡ ( α + β ) = cos ⁡ α cos ⁡ β − sin ⁡ α sin ⁡ β cos ⁡ ( α − β ) = cos ⁡ α cos ⁡ β + sin ⁡ α sin ⁡ 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}}} {\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}}}

sin ⁡ ( α − β ) {\displaystyle \sin(\alpha -\beta )} {\displaystyle \sin(\alpha -\beta )} və cos ⁡ ( α − β ) {\displaystyle \cos(\alpha -\beta )} {\displaystyle \cos(\alpha -\beta )} bucaq fərqlərini " β {\displaystyle \beta } {\displaystyle \beta }" -nı " − β {\displaystyle -\beta } {\displaystyle -\beta } " ilə əvəz etməklə və sin ⁡ ( − β ) = − sin ⁡ ( β ) {\displaystyle \sin(-\beta )=-\sin(\beta )} {\displaystyle \sin(-\beta )=-\sin(\beta )} və cos ⁡ ( − β ) = cos ⁡ ( β ) {\displaystyle \cos(-\beta )=\cos(\beta )} {\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 sin ⁡ ( α ± β ) {\displaystyle \sin(\alpha \pm \beta )} {\displaystyle \sin(\alpha \pm \beta )} = {\displaystyle =} {\displaystyle =} sin ⁡ α cos ⁡ β ± cos ⁡ α sin ⁡ β {\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta } {\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }[2][3]
Kosinus cos ⁡ ( α ± β ) {\displaystyle \cos(\alpha \pm \beta )} {\displaystyle \cos(\alpha \pm \beta )} = {\displaystyle =} {\displaystyle =} cos ⁡ α cos ⁡ β ∓ sin ⁡ α sin ⁡ β {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta } {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }[3][4]
Tanqens tan ⁡ ( α ± β ) {\displaystyle \tan(\alpha \pm \beta )} {\displaystyle \tan(\alpha \pm \beta )} = {\displaystyle =} {\displaystyle =} tan ⁡ α ± tan ⁡ β 1 ∓ tan ⁡ α tan ⁡ β {\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}} {\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}[3][5]
Kosekans csc ⁡ ( α ± β ) {\displaystyle \csc(\alpha \pm \beta )} {\displaystyle \csc(\alpha \pm \beta )} = {\displaystyle =} {\displaystyle =} sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β sec ⁡ α csc ⁡ β ± csc ⁡ α sec ⁡ β {\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}} {\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}[6]
Sekans sec ⁡ ( α ± β ) {\displaystyle \sec(\alpha \pm \beta )} {\displaystyle \sec(\alpha \pm \beta )} = {\displaystyle =} {\displaystyle =} sec ⁡ α sec ⁡ β csc ⁡ α csc ⁡ β csc ⁡ α csc ⁡ β ∓ sec ⁡ α sec ⁡ β {\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}} {\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}[6]
Kontanqens cot ⁡ ( α ± β ) {\displaystyle \cot(\alpha \pm \beta )} {\displaystyle \cot(\alpha \pm \beta )} = {\displaystyle =} {\displaystyle =} cot ⁡ α cot ⁡ β ∓ 1 cot ⁡ β ± cot ⁡ α {\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}} {\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}[3][7]
Ark-sinus arcsin ⁡ x ± arcsin ⁡ y {\displaystyle \arcsin x\pm \arcsin y} {\displaystyle \arcsin x\pm \arcsin y} = {\displaystyle =} {\displaystyle =} arcsin ⁡ ( x 1 − y 2 ± y 1 − x 2 ) {\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}}}\right)} {\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}}}\right)}[8]
Ark-kosinus arccos ⁡ x ± arccos ⁡ y {\displaystyle \arccos x\pm \arccos y} {\displaystyle \arccos x\pm \arccos y} = {\displaystyle =} {\displaystyle =} arccos ⁡ ( x y ∓ ( 1 − x 2 ) ( 1 − y 2 ) ) {\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)} {\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}[9]
Ark-tanqens arctan ⁡ x ± arctan ⁡ y {\displaystyle \arctan x\pm \arctan y} {\displaystyle \arctan x\pm \arctan y} = {\displaystyle =} {\displaystyle =} arctan ⁡ ( x ± y 1 ∓ x y ) {\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)} {\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}[10]
Ark-kotanqens arccot ⁡ x ± arccot ⁡ y {\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y} {\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y} = {\displaystyle =} {\displaystyle =} arccot ⁡ ( x y ∓ 1 y ± x ) {\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)} {\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}

Əsas triqonometrik düsturlar

Düstur Arqumentin mənası
sin 2 ⁡ α + cos 2 ⁡ α = 1 {\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1} {\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1} ∀ α {\displaystyle \forall \alpha } {\displaystyle \forall \alpha }
tan 2 ⁡ α + 1 = 1 cos 2 ⁡ α = sec 2 ⁡ α {\displaystyle \operatorname {tan} ^{2}\alpha +1={\frac {1}{\cos ^{2}\alpha }}=\operatorname {sec} ^{2}\alpha } {\displaystyle \operatorname {tan} ^{2}\alpha +1={\frac {1}{\cos ^{2}\alpha }}=\operatorname {sec} ^{2}\alpha } α ≠ π 2 + π n , n ∈ Z {\displaystyle \alpha \neq {\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} } {\displaystyle \alpha \neq {\frac {\pi }{2}}+\pi n,n\in \mathbb {Z} }
cot 2 ⁡ α + 1 = 1 sin 2 ⁡ α = cosec 2 ⁡ α {\displaystyle \operatorname {cot} ^{2}\alpha +1={\frac {1}{\sin ^{2}\alpha }}=\operatorname {cosec} ^{2}\alpha } {\displaystyle \operatorname {cot} ^{2}\alpha +1={\frac {1}{\sin ^{2}\alpha }}=\operatorname {cosec} ^{2}\alpha } α ≠ π n , n ∈ Z {\displaystyle \alpha \neq \pi n,n\in \mathbb {Z} } {\displaystyle \alpha \neq \pi n,n\in \mathbb {Z} }
  tan ⁡ α ⋅ cot ⁡ α = 1 {\displaystyle ~\operatorname {tan} \alpha \cdot \operatorname {cot} \alpha =1} {\displaystyle ~\operatorname {tan} \alpha \cdot \operatorname {cot} \alpha =1} α ≠ π n 2 , n ∈ Z {\displaystyle \alpha \neq {\frac {\pi n}{2}},n\in \mathbb {Z} } {\displaystyle \alpha \neq {\frac {\pi n}{2}},n\in \mathbb {Z} }
  tan ⁡ α = sin ⁡ α cos ⁡ α {\displaystyle ~\operatorname {tan} \alpha ={\frac {\sin \alpha }{\cos \alpha }}} {\displaystyle ~\operatorname {tan} \alpha ={\frac {\sin \alpha }{\cos \alpha }}}

Toplama düsturları

Toplama düsturları
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 } {\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 } {\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
tan ⁡ ( α ± β ) = tan ⁡ α ± tan ⁡ β 1 ∓ tan ⁡ α tan ⁡ 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 }}} {\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 ⁡ ( α ± β ) = cot ⁡ α cot ⁡ β ∓ 1 cot ⁡ 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 }}} {\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

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

Üçqat arqument düsturları

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

Dərəcənin aşağı salma düsturları

Sinus Kosinus
cos 2 ⁡ α = 1 + cos ⁡ 2 α 2 {\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}} {\displaystyle \cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}}
sin 3 ⁡ α = 3 sin ⁡ α − sin ⁡ 3 α 4 {\displaystyle \sin ^{3}\alpha ={\frac {3\sin \alpha -\sin 3\alpha }{4}}} {\displaystyle \sin ^{3}\alpha ={\frac {3\sin \alpha -\sin 3\alpha }{4}}} cos 3 ⁡ α = 3 cos ⁡ α + cos ⁡ 3 α 4 {\displaystyle \cos ^{3}\alpha ={\frac {3\cos \alpha +\cos 3\alpha }{4}}} {\displaystyle \cos ^{3}\alpha ={\frac {3\cos \alpha +\cos 3\alpha }{4}}}
sin 4 ⁡ α = 3 − 4 cos ⁡ 2 α + cos ⁡ 4 α 8 {\displaystyle \sin ^{4}\alpha ={\frac {3-4\cos 2\alpha +\cos 4\alpha }{8}}} {\displaystyle \sin ^{4}\alpha ={\frac {3-4\cos 2\alpha +\cos 4\alpha }{8}}} cos 4 ⁡ α = 3 + 4 cos ⁡ 2 α + cos ⁡ 4 α 8 {\displaystyle \cos ^{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}}}
sin 5 ⁡ α = 10 sin ⁡ α − 5 sin ⁡ 3 α + sin ⁡ 5 α 16 {\displaystyle \sin ^{5}\alpha ={\frac {10\sin \alpha -5\sin 3\alpha +\sin 5\alpha }{16}}} {\displaystyle \sin ^{5}\alpha ={\frac {10\sin \alpha -5\sin 3\alpha +\sin 5\alpha }{16}}} cos 5 ⁡ α = 10 cos ⁡ α + 5 cos ⁡ 3 α + cos ⁡ 5 α 16 {\displaystyle \cos ^{5}\alpha ={\frac {10\cos \alpha +5\cos 3\alpha +\cos 5\alpha }{16}}} {\displaystyle \cos ^{5}\alpha ={\frac {10\cos \alpha +5\cos 3\alpha +\cos 5\alpha }{16}}}
Düstur
sin 2 ⁡ α cos 2 ⁡ α = 1 − cos ⁡ 4 α 8 {\displaystyle \sin ^{2}\alpha \cos ^{2}\alpha ={\frac {1-\cos 4\alpha }{8}}} {\displaystyle \sin ^{2}\alpha \cos ^{2}\alpha ={\frac {1-\cos 4\alpha }{8}}}
sin 3 ⁡ α cos 3 ⁡ α = 3 sin ⁡ 2 α − sin ⁡ 6 α 32 {\displaystyle \sin ^{3}\alpha \cos ^{3}\alpha ={\frac {3\sin 2\alpha -\sin 6\alpha }{32}}} {\displaystyle \sin ^{3}\alpha \cos ^{3}\alpha ={\frac {3\sin 2\alpha -\sin 6\alpha }{32}}}
sin 4 ⁡ α cos 4 ⁡ α = 3 − 4 cos ⁡ 4 α + cos ⁡ 8 α 128 {\displaystyle \sin ^{4}\alpha \cos ^{4}\alpha ={\frac {3-4\cos 4\alpha +\cos 8\alpha }{128}}} {\displaystyle \sin ^{4}\alpha \cos ^{4}\alpha ={\frac {3-4\cos 4\alpha +\cos 8\alpha }{128}}}
sin 5 ⁡ α cos 5 ⁡ α = 10 sin ⁡ 2 α − 5 sin ⁡ 6 α + sin ⁡ 10 α 512 {\displaystyle \sin ^{5}\alpha \cos ^{5}\alpha ={\frac {10\sin 2\alpha -5\sin 6\alpha +\sin 10\alpha }{512}}} {\displaystyle \sin ^{5}\alpha \cos ^{5}\alpha ={\frac {10\sin 2\alpha -5\sin 6\alpha +\sin 10\alpha }{512}}}

Hasilin cəmə çevrilməsi düsturla

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

İstinadlar

  1. ↑ Abramowitz and Stegun, p. 72, 4.3.7–9
  2. ↑ Abramowitz and Stegun, p. 72, 4.3.16
  3. 1 2 3 4 Weisstein, Eric W. Trigonometric Addition Formulas (ing.) Wolfram MathWorld saytında.
  4. ↑ Abramowitz and Stegun, p. 72, 4.3.17
  5. ↑ Abramowitz and Stegun, p. 72, 4.3.18
  6. 1 2 "Angle Sum and Difference Identities". www.milefoot.com. 3 aprel 2023 tarixində arxivləşdirilib. İstifadə tarixi: 12 oktyabr 2019.
  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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