Arkus sinus
Arkus sinus
Osnovne osobine Parnost neparna Domen [-1,1] Kodomen [-π/2,π/2] Specifične vrednosti Nule 0 Specifične osobine Prevoji (0,0) Ulazak u nulu pod uglom π/4
Arkus sinus je funkcija inverzna sinusnoj funkciji na njenom ograničenom intervalu [-π/2,π/2]. Koristi se za određivanje veličine ugla u ovom opsegu, kada je poznata vrednost njegovog sinusa.
Slede neke od formula koje se vezuju za arkus sinus:
arcsin
−
x
=
π
2
−
arccos
x
{\displaystyle \arcsin {-x}={\frac {\pi }{2}-\arccos {x}
arcsin
−
x
=
−
arcsin
x
{\displaystyle \arcsin {-x}=-\arcsin {x}
arcsin
1
x
=
a
r
c
c
o
s
e
c
x
{\displaystyle \arcsin {\frac {1}{x}=arccosec{x}
Preko formule za polovinu ugla se dobija i:
arcsin
x
=
2
a
r
c
t
g
x
1
+
1
−
x
2
{\displaystyle \arcsin x=2arctg{\frac {x}{1+{\sqrt {1-x^{2}
Izvod:
d
d
x
arcsin
x
=
1
1
−
x
2
;
|
x
|
<
1
{\displaystyle {\frac {d}{dx}\arcsin x{}={\frac {1}{\sqrt {1-x^{2};\qquad |x|<1}
Predstavljanje u formi integrala:
arcsin
x
=
∫
0
x
1
1
−
z
2
d
z
,
|
x
|
≤
1
{\displaystyle \arcsin x{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}\,dz,\qquad |x|\leq 1}
Predstavljanje u formi beskonačne sume:
arcsin
z
=
z
+
(
1
2
)
z
3
3
+
(
1
⋅
3
2
⋅
4
)
z
5
5
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
z
7
7
+
⋯
=
∑
n
=
0
∞
(
(
2
n
)
!
2
2
n
(
n
!
)
2
)
z
2
n
+
1
(
2
n
+
1
)
;
|
z
|
≤
1
{\displaystyle {\begin{aligned}\arcsin z&{}=z+\left({\frac {1}{2}\right){\frac {z^{3}{3}+\left({\frac {1\cdot 3}{2\cdot 4}\right){\frac {z^{5}{5}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right){\frac {z^{7}{7}+\cdots \\&{}=\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}\right){\frac {z^{2n+1}{(2n+1)};\qquad |z|\leq 1\end{aligned}
Vanjske veze
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