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:
sin 2 θ + cos 2 θ = 1 , {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}
burada sin 2 θ {\displaystyle \sin ^{2}\theta } ( sin θ ) 2 {\displaystyle (\sin \theta )^{2}} cos 2 θ {\displaystyle \cos ^{2}\theta } ( cos θ ) 2 {\displaystyle (\cos \theta )^{2}}
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}}}
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}}}
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 } csc θ {\displaystyle \csc \theta } cos θ {\displaystyle \cos \theta } sec θ {\displaystyle \sec \theta } tan θ {\displaystyle \tan \theta } cot θ {\displaystyle \cot \theta } sin θ = {\displaystyle \sin \theta =} sin θ {\displaystyle \sin \theta } 1 csc θ {\displaystyle {\frac {1}{\csc \theta }}} ± 1 − cos 2 θ {\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}} ± sec 2 θ − 1 sec θ {\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 }}}} ± 1 1 + cot 2 θ {\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}} csc θ = {\displaystyle \csc \theta =} 1 sin θ {\displaystyle {\frac {1}{\sin \theta }}} csc θ {\displaystyle \csc \theta } ± 1 1 − cos 2 θ {\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}}}} ± 1 + tan 2 θ tan θ {\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}} ± 1 + cot 2 θ {\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}} cos θ = {\displaystyle \cos \theta =} ± 1 − sin 2 θ {\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}} ± csc 2 θ − 1 csc θ {\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}} cos θ {\displaystyle \cos \theta } 1 sec θ {\displaystyle {\frac {1}{\sec \theta }}} ± 1 1 + tan 2 θ {\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 }}}} sec θ = {\displaystyle \sec \theta =} ± 1 1 − sin 2 θ {\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}}}} 1 cos θ {\displaystyle {\frac {1}{\cos \theta }}} sec θ {\displaystyle \sec \theta } ± 1 + tan 2 θ {\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}} ± 1 + cot 2 θ cot θ {\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}} tan θ = {\displaystyle \tan \theta =} ± sin θ 1 − sin 2 θ {\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}} ± 1 csc 2 θ − 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 }}} ± sec 2 θ − 1 {\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}} tan θ {\displaystyle \tan \theta } 1 cot θ {\displaystyle {\frac {1}{\cot \theta }}} cot θ = {\displaystyle \cot \theta =} ± 1 − sin 2 θ sin θ {\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}} ± csc 2 θ − 1 {\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}} ± cos θ 1 − cos 2 θ {\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}} ± 1 sec 2 θ − 1 {\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}} 1 tan θ {\displaystyle {\frac {1}{\tan \theta }}} cot θ {\displaystyle \cot \theta }
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 } sin ( θ + π ) = − sin θ {\displaystyle \sin(\theta +\pi )=-\sin \theta } sin ( θ + k ⋅ 2 π ) = + sin θ {\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta } 2 π {\displaystyle 2\pi } cos ( θ ± π 2 ) = ∓ sin θ {\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta } cos ( θ + π ) = − cos θ {\displaystyle \cos(\theta +\pi )=-\cos \theta } cos ( θ + k ⋅ 2 π ) = + cos θ {\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta } 2 π {\displaystyle 2\pi } csc ( θ ± π 2 ) = ± sec θ {\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta } csc ( θ + π ) = − csc θ {\displaystyle \csc(\theta +\pi )=-\csc \theta } csc ( θ + k ⋅ 2 π ) = + csc θ {\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta } 2 π {\displaystyle 2\pi } sec ( θ ± π 2 ) = ∓ csc θ {\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta } sec ( θ + π ) = − sec θ {\displaystyle \sec(\theta +\pi )=-\sec \theta } sec ( θ + k ⋅ 2 π ) = + sec θ {\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta } 2 π {\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 }}} tan ( θ + π 2 ) = − cot θ {\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta } tan ( θ + k ⋅ π ) = + tan θ {\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta } π {\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 }}} cot ( θ + π 2 ) = − tan θ {\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta } cot ( θ + k ⋅ π ) = + cot θ {\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta } π {\displaystyle \pi }
Triqonometrik funksiyaların işarəsi bucağın rübündən asılıdır. Əgər − π < θ ≤ π {\displaystyle {-\pi }<\theta \leq \pi } 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}}}
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}}}
sin ( α − β ) {\displaystyle \sin(\alpha -\beta )} cos ( α − β ) {\displaystyle \cos(\alpha -\beta )} β {\displaystyle \beta } − β {\displaystyle -\beta } sin ( − β ) = − sin ( β ) {\displaystyle \sin(-\beta )=-\sin(\beta )} cos ( − β ) = cos ( β ) {\displaystyle \cos(-\beta )=\cos(\beta )}
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 α cos β ± cos α sin β {\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta } [2] [3] Kosinus cos ( α ± β ) {\displaystyle \cos(\alpha \pm \beta )} = {\displaystyle =} cos α cos β ∓ sin α sin β {\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta } [3] [4] Tanqens tan ( α ± β ) {\displaystyle \tan(\alpha \pm \beta )} = {\displaystyle =} tan α ± tan β 1 ∓ tan α tan β {\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 =} 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 }}} [6] Sekans sec ( α ± β ) {\displaystyle \sec(\alpha \pm \beta )} = {\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 }}} [6] Kontanqens cot ( α ± β ) {\displaystyle \cot(\alpha \pm \beta )} = {\displaystyle =} cot α cot β ∓ 1 cot β ± cot α {\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 1 − y 2 ± y 1 − x 2 ) {\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 y ∓ ( 1 − x 2 ) ( 1 − y 2 ) ) {\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 ± y 1 ∓ x y ) {\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 =} arccot ( x y ∓ 1 y ± x ) {\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}
Düstur Arqumentin mənası sin 2 α + cos 2 α = 1 {\displaystyle \sin ^{2}\alpha +\cos ^{2}\alpha =1} ∀ α {\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 } α ≠ π 2 + π n , n ∈ 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 } α ≠ π n , n ∈ Z {\displaystyle \alpha \neq \pi n,n\in \mathbb {Z} } tan α ⋅ cot α = 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} } tan α = sin α cos α {\displaystyle ~\operatorname {tan} \alpha ={\frac {\sin \alpha }{\cos \alpha }}}
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 } 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 } 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 }}} 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 }}}
İkiqat arqument düsturları sin 2 α = 2 sin α cos α {\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 }} 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 }} 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 }}} 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 }}}
Üçqat arqument düsturları 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 \,} 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 }}} 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 }}}
Sinus Kosinus 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}}} sin 3 α = 3 sin α − sin 3 α 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}}} 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}}} 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}}} 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}}} 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}}}
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}}} 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}}} 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}}} 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}}}
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}}} cos α cos β = cos ( α − β ) + cos ( α + β ) 2 {\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha -\beta )+\cos(\alpha +\beta )}{2}}}
↑ Abramowitz and Stegun, p. 72, 4.3.7–9 ↑ Abramowitz and Stegun, p. 72, 4.3.16 ↑ 1 2 3 4 Weisstein, Eric W. Trigonometric Addition Formulas (ing.) Wolfram MathWorld saytında.↑ Abramowitz and Stegun, p. 72, 4.3.17 ↑ Abramowitz and Stegun, p. 72, 4.3.18 ↑ 1 2 "Angle Sum and Difference Identities" . www.milefoot.com. 2023-04-03 tarixində arxivləşdirilib . İstifadə 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 
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14cc79239a0fcdfb53b132d75a93474e9e6d5ba0)
