Karakteristike
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Grafik funkcije sekans Grafik funkcije kosekans Definiciono područje
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Sekans:
− ∞ < x < + ∞ ; x ≠ ( n + 1 2 ) ⋅ π ; n ∈ Z {\displaystyle -\infty <x<+\infty \quad ;\quad x\neq \left(n+{\frac {1}{2}}\right)\cdot \pi \,;\,n\in \mathbb {Z} }
Kosekans:
− ∞ < x < + ∞ ; x ≠ n ⋅ π ; n ∈ Z {\displaystyle -\infty <x<+\infty \quad ;\quad x\neq n\cdot \pi \ ;\,n\in \mathbb {Z} }
Područje vrijednosti
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− ∞ < f ( x ) ≤ − 1 ; 1 ≤ f ( x ) < + ∞ {\displaystyle -\infty <f(x)\leq -1\quad ;\quad 1\leq f(x)<+\infty } Periodičnost
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Dužina perioda 2 ⋅ π : f ( x + 2 π ) = f ( x ) {\displaystyle 2\cdot \pi \,:\,f(x+2\pi )=f(x)}
Sekans:
Osa simetrije: f ( x ) = f ( − x ) {\displaystyle f(x)=f(-x)}
Kosekans:
Tačka simetrije: f ( − x ) = − f ( x ) {\displaystyle f(-x)=-f(x)}
Sekans:
x = ( n + 1 2 ) ⋅ π ; n ∈ Z {\displaystyle x=\left(n+{\frac {1}{2}}\right)\cdot \pi \,;\,n\in \mathbb {Z} }
Kosekans:
x = n ⋅ π ; n ∈ Z {\displaystyle x=n\cdot \pi \ ;\quad n\in \mathbb {Z} }
Sekans:
Minimum:
x = 2 n ⋅ π ; n ∈ Z {\displaystyle x=2n\cdot \pi \,;\,n\in \mathbb {Z} }
Maksimum:
x = ( 2 n − 1 ) ⋅ π ; n ∈ Z {\displaystyle x=(2n-1)\cdot \pi \ ;\,n\in \mathbb {Z} }
Kosekans:
Minimum:
x = ( 2 n + 1 2 ) ⋅ π ; n ∈ Z {\displaystyle x=\left(2n+{\frac {1}{2}}\right)\cdot \pi \ ;\,n\in \mathbb {Z} }
Maksimum:
x = ( 2 n − 1 2 ) ⋅ π ; n ∈ Z {\displaystyle x=\left(2n-{\frac {1}{2}}\right)\cdot \pi \ ;\,n\in \mathbb {Z} }
Nijedna funkcija nema nula.
Nijedna funkcija nema horizontalnih asimptota .
Nijedna funkcija nema prekida.
Prevojne tačke
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Nijedna funkcija nema prevojnih tačaka.
Inverzne funkcije
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Sekans:
U pola perioda dužine, npr. x ∈ [ 0 , π ] {\displaystyle x\in [0,\pi ]} je reverzibilna funkcija (arkussekans): Kosekans
U pola perioda dužine, npr. x ∈ [ − π 2 , π 2 ] {\displaystyle x\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]} je reverzibilna funkcija (arkuskosekans): Proširene serije
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Sekans:
sec ( x ) = 4 π ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ( 2 k + 1 ) 2 π 2 − 4 x 2 {\displaystyle \sec(x)=4\pi \,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k+1)}{(2k+1)^{2}\pi ^{2}-4x^{2}}}} Kosekans:
csc ( x ) = 1 x − 2 x ∑ k = 1 ∞ ( − 1 ) k k 2 π 2 − x 2 = ∑ k = − ∞ ∞ ( − 1 ) k x x 2 − k 2 π 2 {\displaystyle \csc(x)={\frac {1}{x}}-2x\,\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k^{2}\pi ^{2}-x^{2}}}=\sum _{k=-\infty }^{\infty }{\frac {(-1)^{k}\,x}{x^{2}-k^{2}\pi ^{2}}}} Izvod (derivacija)
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Sekans:
d d x sec ( x ) = sec ( x ) ⋅ tan ( x ) = sec 2 ( x ) csc ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\sec(x)=\sec(x)\cdot \tan(x)={\frac {\sec ^{2}(x)}{\csc(x)}}} Kosekans
d d x csc ( x ) = d d x 1 sin ( x ) = − csc ( x ) ⋅ cot ( x ) = − csc 2 ( x ) sec ( x ) = − cos ( x ) sin 2 ( x ) {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\csc(x)={\frac {\mathrm {d} }{\mathrm {d} x}}{\frac {1}{\sin(x)}}=-\csc(x)\cdot \cot(x)=-{\frac {\csc ^{2}(x)}{\sec(x)}}=-{\frac {\cos(x)}{\sin ^{2}(x)}}}
Sekans:
∫ sec ( x ) d x = ln | 1 + sin ( x ) cos ( x ) | = ln | sec ( x ) + tan ( x ) | {\displaystyle \int \sec(x)\,\mathrm {d} x=\ln \left|{\frac {1+\sin(x)}{\cos(x)}}\right|=\ln {\Big |}\sec(x)+\tan(x){\Big |}} Kosekans
∫ csc ( x ) d x = ln | sin ( x ) 1 + cos ( x ) | = ln | tan ( x 2 ) | {\displaystyle \int \csc(x)\,\mathrm {d} x=\ln \left|{\frac {\sin(x)}{1+\cos(x)}}\right|=\ln \left|\tan \left({\frac {x}{2}}\right)\right|} Kompleksni argumenti
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Također pogledajte
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^ Konstantin A. Semendjajew: Taschenbuch der Mathematik
Vanjski linkovi
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