मुक्त ज्ञानकोश विकिपीडिया से
गणित में त्रिकोणमितीय फलनों के प्रतिलोम फलनों को प्रतिलोम त्रिकोणमितीय फलन (inverse trigonometric functions) कहते हैं। इनके डोमेन समुचित रूप से सीमित करके पारिभाषित किये गये हैं।
इन्हें sin−1 , cos−1 आदि के रूप में निरूपित करते हैं और 'साइन इन्वर्स', 'कॉस इन्वर्स' आदि बोलते हैं।
arcsin
x
=
y
{\displaystyle \operatorname {arcsin} \ x=y}
sin
y
=
x
{\displaystyle \operatorname {sin} \ y=x}
arccos
x
=
y
{\displaystyle \operatorname {arccos} \ x=y}
cos
y
=
x
{\displaystyle \operatorname {cos} \ y=x}
arctg
x
=
y
{\displaystyle \operatorname {arctg} \ x=y}
tg
y
=
x
{\displaystyle \operatorname {tg} \ y=x}
arcctg
x
=
y
{\displaystyle \operatorname {arcctg} \ x=y}
ctg
y
=
x
{\displaystyle \operatorname {ctg} \ y=x}
arcsec
x
=
y
{\displaystyle \operatorname {arcsec} \ x=y}
sec
y
=
x
{\displaystyle \operatorname {sec} \ y=x}
arccsc
x
=
y
{\displaystyle \operatorname {arccsc} \ x=y}
csc
y
=
x
{\displaystyle \operatorname {csc} \ y=x}
उदाहरण:
arcsin
0
=
0
{\displaystyle \operatorname {arcsin} \ 0=0}
arcsin
0.5
=
π
6
{\displaystyle \operatorname {arcsin} \ 0.5={\frac {\pi }{6}}}
arcsin
1
=
π
2
{\displaystyle \operatorname {arcsin} \ 1={\frac {\pi }{2}}}
arccos
0
=
π
2
{\displaystyle \operatorname {arccos} \ 0={\frac {\pi }{2}}}
arccos
0.5
=
π
3
{\displaystyle \operatorname {arccos} \ 0.5={\frac {\pi }{3}}}
arccos
(
−
1
)
=
π
{\displaystyle \operatorname {arccos} (-1)=\pi }
arctg
0
=
0
{\displaystyle \operatorname {arctg} \ 0=0}
arctg
1
=
π
4
{\displaystyle \operatorname {arctg} \ 1={\frac {\pi }{4}}}
arcctg
0
=
π
2
{\displaystyle \operatorname {arcctg} \ 0={\frac {\pi }{2}}}
arcctg
1
=
π
4
{\displaystyle \operatorname {arcctg} \ 1={\frac {\pi }{4}}}
चूँकि कोई भी त्रिकोणमितीय फलन एकैकी (one-to-one) नहीं है, इनके प्रतिलोम फलन तभी सम्भव होंगे यदि इनके डोमेन सीमित रखे जांय।
निम्नांकित सारणी में मुख्य प्रतिलोमों का विवरण दिया गया है-
नाम
सामान्य निरूपण
परिभाषा
वास्तविक परिणाम के लिये x का डोमेन
मुख्य मानों का परास (रेंज) रेडियन )
मुख्य मानों का परास
arcsine y = arcsin x x = sin y −1 ≤ x ≤ 1
−π/2 ≤ y ≤ π/2
−90° ≤ y ≤ 90°
arccosine y = arccos x x = cos y −1 ≤ x ≤ 1
0 ≤ y ≤ π
0° ≤ y ≤ 180°
arctangent y = arctan x x = tan y all real numbers
−π/2 < y < π/2
−90° < y < 90°
arccotangent y = arccot x x = cot y all real numbers
0 < y < π
0° < y < 180°
arcsecant y = arcsec x x = sec y x ≤ −1 or 1 ≤ x 0 ≤ y < π/2 or π/2 < y ≤ π
0° ≤ y < 90° or 90° < y ≤ 180°
arccosecant y = arccsc x x = csc y x ≤ −1 or 1 ≤ x −π/2 ≤ y < 0 or 0 < y ≤ π/2
-90° ≤ y < 0° or 0° < y ≤ 90°
यदि x को समिश्र संख्या होने की छूट हो तो y का रेंज केवल इसके वास्तविक भाग (real part) पर ही लागू होगा।
The usual principal values of the arcsin(x ) (red) and arccos(x ) (blue) functions graphed on the cartesian plane. The usual principal values of the arctan(x ) and arccot(x ) functions graphed on the cartesian plane. Principal values of the arcsec(x ) and arccsc(x ) functions graphed on the cartesian plane. Complementary angles:
arccos
x
=
π
2
−
arcsin
x
{\displaystyle \arccos x={\frac {\pi }{2}}-\arcsin x}
arccot
x
=
π
2
−
arctan
x
{\displaystyle \operatorname {arccot} x={\frac {\pi }{2}}-\arctan x}
arccsc
x
=
π
2
−
arcsec
x
{\displaystyle \operatorname {arccsc} x={\frac {\pi }{2}}-\operatorname {arcsec} x}
Negative arguments:
arcsin
(
−
x
)
=
−
arcsin
x
{\displaystyle \arcsin(-x)=-\arcsin x\!}
arccos
(
−
x
)
=
π
−
arccos
x
{\displaystyle \arccos(-x)=\pi -\arccos x\!}
arctan
(
−
x
)
=
−
arctan
x
{\displaystyle \arctan(-x)=-\arctan x\!}
arccot
(
−
x
)
=
π
−
arccot
x
{\displaystyle \operatorname {arccot}(-x)=\pi -\operatorname {arccot} x\!}
arcsec
(
−
x
)
=
π
−
arcsec
x
{\displaystyle \operatorname {arcsec}(-x)=\pi -\operatorname {arcsec} x\!}
arccsc
(
−
x
)
=
−
arccsc
x
{\displaystyle \operatorname {arccsc}(-x)=-\operatorname {arccsc} x\!}
Reciprocal arguments:
arccos
(
1
/
x
)
=
arcsec
x
{\displaystyle \arccos(1/x)\,=\operatorname {arcsec} x\,}
arcsin
(
1
/
x
)
=
arccsc
x
{\displaystyle \arcsin(1/x)\,=\operatorname {arccsc} x\,}
arctan
(
1
/
x
)
=
1
2
π
−
arctan
x
=
arccot
x
,
if
x
>
0
{\displaystyle \arctan(1/x)={\tfrac {1}{2}}\pi -\arctan x=\operatorname {arccot} x,{\text{ if }}x>0\,}
arctan
(
1
/
x
)
=
−
1
2
π
−
arctan
x
=
−
π
+
arccot
x
,
if
x
<
0
{\displaystyle \arctan(1/x)=-{\tfrac {1}{2}}\pi -\arctan x=-\pi +\operatorname {arccot} x,{\text{ if }}x<0\,}
arccot
(
1
/
x
)
=
1
2
π
−
arccot
x
=
arctan
x
,
if
x
>
0
{\displaystyle \operatorname {arccot}(1/x)={\tfrac {1}{2}}\pi -\operatorname {arccot} x=\arctan x,{\text{ if }}x>0\,}
arccot
(
1
/
x
)
=
3
2
π
−
arccot
x
=
π
+
arctan
x
,
if
x
<
0
{\displaystyle \operatorname {arccot}(1/x)={\tfrac {3}{2}}\pi -\operatorname {arccot} x=\pi +\arctan x,{\text{ if }}x<0\,}
arcsec
(
1
/
x
)
=
arccos
x
{\displaystyle \operatorname {arcsec}(1/x)=\arccos x\,}
arccsc
(
1
/
x
)
=
arcsin
x
{\displaystyle \operatorname {arccsc}(1/x)=\arcsin x\,}
If you only have a fragment of a sine table:
arccos
x
=
arcsin
1
−
x
2
,
if
0
≤
x
≤
1
{\displaystyle \arccos x=\arcsin {\sqrt {1-x^{2}}},{\text{ if }}0\leq x\leq 1}
arctan
x
=
arcsin
x
x
2
+
1
{\displaystyle \arctan x=\arcsin {\frac {x}{\sqrt {x^{2}+1}}}}
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
From the half-angle formula
tan
θ
2
=
sin
θ
1
+
cos
θ
{\displaystyle \tan {\frac {\theta }{2}}={\frac {\sin \theta }{1+\cos \theta }}}
arcsin
x
=
2
arctan
x
1
+
1
−
x
2
{\displaystyle \arcsin x=2\arctan {\frac {x}{1+{\sqrt {1-x^{2}}}}}}
arccos
x
=
2
arctan
1
−
x
2
1
+
x
,
if
−
1
<
x
≤
+
1
{\displaystyle \arccos x=2\arctan {\frac {\sqrt {1-x^{2}}}{1+x}},{\text{ if }}-1<x\leq +1}
arctan
x
=
2
arctan
x
1
+
1
+
x
2
{\displaystyle \arctan x=2\arctan {\frac {x}{1+{\sqrt {1+x^{2}}}}}}
[ संपादित करें ]
sin
(
arccos
x
)
=
cos
(
arcsin
x
)
=
1
−
x
2
{\displaystyle \sin(\arccos x)=\cos(\arcsin x)={\sqrt {1-x^{2}}}}
sin
(
arctan
x
)
=
x
1
+
x
2
{\displaystyle \sin(\arctan x)={\frac {x}{\sqrt {1+x^{2}}}}}
cos
(
arctan
x
)
=
1
1
+
x
2
{\displaystyle \cos(\arctan x)={\frac {1}{\sqrt {1+x^{2}}}}}
tan
(
arcsin
x
)
=
x
1
−
x
2
{\displaystyle \tan(\arcsin x)={\frac {x}{\sqrt {1-x^{2}}}}}
tan
(
arccos
x
)
=
1
−
x
2
x
{\displaystyle \tan(\arccos x)={\frac {\sqrt {1-x^{2}}}{x}}}
निम्नलिखित में k कोई पूर्णांक है।
sin
(
y
)
=
x
⇔
y
=
arcsin
(
x
)
+
2
k
π
or
y
=
π
−
arcsin
(
x
)
+
2
k
π
{\displaystyle \sin(y)=x\ \Leftrightarrow \ y=\arcsin(x)+2k\pi {\text{ or }}y=\pi -\arcsin(x)+2k\pi }
cos
(
y
)
=
x
⇔
y
=
arccos
(
x
)
+
2
k
π
or
y
=
2
π
−
arccos
(
x
)
+
2
k
π
{\displaystyle \cos(y)=x\ \Leftrightarrow \ y=\arccos(x)+2k\pi {\text{ or }}y=2\pi -\arccos(x)+2k\pi }
tan
(
y
)
=
x
⇔
y
=
arctan
(
x
)
+
k
π
{\displaystyle \tan(y)=x\ \Leftrightarrow \ y=\arctan(x)+k\pi }
cot
(
y
)
=
x
⇔
y
=
arccot
(
x
)
+
k
π
{\displaystyle \cot(y)=x\ \Leftrightarrow \ y=\operatorname {arccot}(x)+k\pi }
sec
(
y
)
=
x
⇔
y
=
arcsec
(
x
)
+
2
k
π
or
y
=
2
π
−
arcsec
(
x
)
+
2
k
π
{\displaystyle \sec(y)=x\ \Leftrightarrow \ y=\operatorname {arcsec}(x)+2k\pi {\text{ or }}y=2\pi -\operatorname {arcsec}(x)+2k\pi }
csc
(
y
)
=
x
⇔
y
=
arccsc
(
x
)
+
2
k
π
or
y
=
π
−
arccsc
(
x
)
+
2
k
π
{\displaystyle \csc(y)=x\ \Leftrightarrow \ y=\operatorname {arccsc}(x)+2k\pi {\text{ or }}y=\pi -\operatorname {arccsc}(x)+2k\pi }
