of 18
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1/18
@HTJS]J \S@AOHONJ\@M7
www.te`hf` t.`h
@HTJS]J \S@AOHONJ\S@M DZHM\@OH]
3.7. @H\SOLZM\@OH
Qj egvj rjgl gcout duhmt`ohs, ohj-ohj ohto (c`kjmt`vj) duhmt`ohs ghl hvjrsj od g duhmt`oh. Qj egvj
gbso bjgrht tegt hvjrsj od g duhmt`oh d s ljhotjl cy d7ghl d7jx`sts `d ghl ohby d d s g ohj-ohj ohto
duhmt`oh. \ejrj grj sjvjrgb duhmt`ohs we`me grj hot ohj-ohj ohto ghl ejhmj tej`r hvjrsj lojs hot jx`st.Qj egvj gbso rjgl gcout tr`aohonjtr`m duhmt`ohs grj hot ohj-ohj ohto ovjr tej`r hgturgb long`hs ghl
rghajs ghl ejhmj tej r, hvjrsj lo hot jx`st. Cut d wj rjstr`mt tej`r long`hs ghl rghajs, tejh tejy
w`bb cjmonj ohj-ohj ohto duhmt`ohs ghl tej`r hvjrsj w`bb jx`st. @h te`s megptjr wj w`bb stuly hvjrsjs
od tr`aononjtr`m duhmt`ohs ghl tej`r vgr`ous propjrt`js.
3.3. @HTJS]J OD G DZHM\@OH
Bjt d ; GC @d (cj g duhmt`oh dron G to C) we`me s ohj-ohj ohto. \ejh g duhmt`oh d7; C G (d7dron
C to G) s sg`l to cj tej hvjrsj od tej duhmt`oh d d
y > d(x) d7(y) > x
.j., `ngaj od x uhljr d `s y ngaj od y uhljr d7 s x.
Mbjgrby long`h d7> rghaj d ghl rghaj d 7> long`h d
3.=. YS@HM@YGB CSGHMEJ] OD @HTJS]J \S@AOHONJ\S@M DZHM\@OH]
(`) Ljd`h`t`oh od s`h7x (Yr`hm`pgb crghme od s`h7x) ;
s`h7; V7, 7^
pp-
3
,
3
.j., s`h7`s g duhmt`oh dron V7, 7^ to
pp-
3
,
3
.
]ume tegt s`h7x > q x > s`h q
wejrj 7 x 7 ghl 3
pq2
3
p
(``) Ljd`h`t`oh od mos7x (Yr`hm`pgb crghme od mos7x) ;
mos7; V7, 7^ V9, p^ `s sume tegt mos7x > qx > mos q,
wejrj 7 x 7 ghl 9 qp
(`` ) Ljd`h`t`oh od tgh7x (Yr`hm`pgb crghme od tgh7x) ;
tgh7; (, )
pp-
3,
3sume tegt
tgh7x > qx > tgh q
wejrj 2 x 2 ghl 3
p2 q2
3
p.
8/10/2019 4_Inverse Trigonometic Functions.pdf
2/18
@HTJS]J \S@AOHONJ\@M3
www.te`hf t.`h
(`v) Ljd`h`t`oh od mot7x ; mot7;(, ) (9, p) sume tegt
mot7x > qx > mot q, wejrj 2 x 2 ghl 9 2 q2 p.
(v) Ljd`h`t`oh od sjm7x ; sjm7; (, 7^ V7, ) V9, p^
p
3sume tegt
sjm7x > qx > sjm q
wejrj, 2 x 7 or 7 x 2 ghl 9 q23
por
3
p2 qp
(v`) Ljd`h`t`oh od mosjm7x ;
mosjm7; (, 7^ V7, )
pp-
3,
3 {9}
sume tegt mosjm
7
x > qx > mosjm qwejrj 2 x 7 or 7 x 2 ghl
3
pq2 9 or 9 2 q
3
p
]unngry
Duhmt`oh Long`h od tej duhmt`oh Yr`hm`pgb vgbuj crghme
7. y > s`h7x 7 x 7 3
py
3
p
3. y > mos7
x 7 x 7 9 y p
=. y > tgh7x 2 x 2 3
p2 y 2
3
p
mot7x 2 x 2 9 2 y 2 p
:. y > sjm7x x 7 or x 7 9 y p, y 3
p
4. y > mosjm7x x 7 or x 7 3
py
3
p, y 9
Hotj ;Zhbjss otejrw`sj stgtjl s`h7x, mos7x, tgh7x, mot7, x sjm7x ghl mosjm7x w`bb njgh tej`r
pr hm pgb crghmejs.
3.< ASGYE] OD YS@HM@YGB CSGHMEJ] OD @HTJS]J \S@AOHONJ\S@M DZHM\@OH]
(`) Argpe od y > s`h x
Long`h > S > ( , )ghl Sghaj > V7, 7^
P
p 3
p
-
p-3
3
=p-
3
=p
p
3
pO
y > 7
p3U
y > 7
8/10/2019 4_Inverse Trigonometic Functions.pdf
3/18
@HTJS]J \S@AOHONJ\@M=
www.te`hf` t.`h
Argpe od y > s`h7x
Long`h > V7, 7^
Sghaj >
pp-
3,
3
3
p
P
U7
O 7
3
p
-
(``) Argpe od y > mos x
Long`h > S > (, )
Sghaj > V7, 7^
P
p
3
p
-
3
=p-
3
=pp
3
pOU
y > 77
y > 7
Argpe od y > mos7x
Long`h > V7, 7^
Sghaj > V9, p^
3
p
U
p
P
7 O 7
(`` ) Argpe od y > tgh x
Long`h > S
p
+> [h,3
)7h3(x;x
Sghaj > (, ) O3
p
-
3
p
P
U
Argpe od y > tgh7x
Long`h > S > (, )
Sghaj >
pp-
3,
3
P
U
3y
p
>
mot x
Long`h > S {x ; x > h p, h [}
Sghaj > (, )p O
P
U
3
p p
3
p
-
8/10/2019 4_Inverse Trigonometic Functions.pdf
4/18
@HTJS]J \S@AOHONJ\@M mot7x
Long`h > (, )
O
P
U
3
p3
=p
(9, p)
(v) Argpe od y > sjm x
Long`h > S
p
+> [h,3
)7h3(x;x
Sghaj > (, -7^ V7, )
P
3
p
-
3
=pp
3
pO U
Argpe od y > sjm7x
Long`h > (, -7^ V7, )
Sghaj >
p
3,9
p,3
> V9, p^
p
33
p
U
p
P
7 O 7
(v`) Argpe od y > mosjm x
Long`h > S {x ; x > h p, h [}
P
3
p
-
p
3
pOU p
Sghaj > (, 7^ V7, )
Argpe od y > mosjm7x
Long`h > (, -7^ V7, )
Sghaj >
p- 9,
3
p
3
,9 >
pp-
3
,
3
{9}U
3/p
O7
7
P
3/p-
3.:. ]ONJ @NYOS\GH\ YSOYJS\@J] OD @HTJS]J \S@AOHONJ\S@M DZHM\@OH]
Yropjrty @ ;
( ) s`h (s`h7x) > x dor gbb x V7, 7^
( ) mos (mos7x) > x, dor r gbb x V7, 7^
( ` ) tgh (tgh7x) > x, dor gbb x S
( v) mot (mot7x) > x, dor gbb x S.
(v) sjm (sjm7x) > x, dor gbb x ( , 7^ V7, ) .j., dor gbb x 7 or x 7
(v ) mosjm (mosjm7x) > x, dor gbb x (, 7^ V7, ) .j., dor gbb x 7 or x 7
8/10/2019 4_Inverse Trigonometic Functions.pdf
5/18
@HTJS]J \S@AOHONJ\@M:
www.te`hf` t.`h
Yropjrty @@ ;
( ) s`h7(s`h x) > x, x
pp-
3,
3( ) mos7(mos x) > x, x V9, p^
( ` ) tgh7(tgh x) > x, x
pp-
3,
3( v) mot7(mot x) > x, x (9, p)
(v) sjm7(sjm x) > x, x
p3
,9
p
p,
3 .j., x (9, p)
p
3
(v ) mosjm7(mosjm x) > x, x
p- 9,
3
p3
,9 `.j., x
pp-
3,
3 {9}
s`h7(s`h x) >
p
p-p
p
pp-
p
p-p
p
p-
p-
p--p-
ohsoghl3
1x
3
:`d,x=
3
:x
3
=`d3x
3
=x3`dx
3x
3`d,x
3x
3
=`dx
mos7(mos x) >
pp-pppp-
pp-ppp--
ohsoghl
p22
pp-
p22
pp-
p22
p-
p-22
p-p+
ohsoghl3
:x
3
=,3x
3
=x
3,x
3x
3,x
3x
3
=,x
Yropjrty @@@ ;
( ) s`h
7
x > mosjm
7
x
7
, 7 x 7 ghl x 9mosjm7x > s`h7
x
7, x 7 or x 7
( ) mos7x > sjm7x
7, 7 x 7
sjm7x > mos7x
7, x 7 or x 7
( ` ) tgh7x > mot7
x
7, x 8 9 > p+ mot7
x
7, x 2 9
( v) mot7x > tgh7
x
7, x 8 9 > p+ tgh7
x
7, x 2 9
8/10/2019 4_Inverse Trigonometic Functions.pdf
6/18
@HTJS]J \S@AOHONJ\@M4
www.te`hf t.`h
Yropjrty @T ;
( ) s`h7( x) > s`h7(x), dor gbb x V7, 7^( ) mos7( x) > p mos7x, dor gbb x V7, 7^( ` ) tgh7( x) > tgh7x, dor gbb x S( v) mot7( x) > p mot7x, dor gbb x S(v) sjm7( x) > p sjm7x, dor gbb x (, 7^ V7, ) .j., dor gbb |x| 7
(v ) mosjm7( x) > mosjm7x, dor gbb x (, 7^ V7, ^ .j., dor gbb |x| 7
Yropjrty T ;
( ) s`h7x + mos7x >3
p, dor gbb x V7, 7^
( ) tgh7x + mot7x >3
p, dor gbb x S
( ` ) sjm7x + mosjm7x >3
pdor gbb x (, 7^ V7, ) .j., dor gbb |x| 7
Yropjrty T@ ;
( ) tgh7x + tgh7 y >
822
-
++p-
888
-+
+p
2
-+
-
-
-
7xyghl9y,9x`d,xy7
yxtgh
7xyghl9y,9x`d,xy7
yxtgh
7xy`d,xy7
yxtgh
7
7
7
Yropjrty T@@ ;
tgh7
x tgh7
y > tgh7
xy7
yx
+
-
, `d xy 8 7\e`s rjsubt mgh cj jstgcb`sejl cy putt`ha y `h pbgmj od y `h tej rjsubts od propjrty T us`ha tej
dgmt tegt tgh7( y) > tgh7y.
tgh7x tgh7y >
-282
+
-+p-
-228
+-
+p
-8
+
-
-
-
-
7xyghl9y,9x`d,xy7
yxtgh
7xyghl9y,9x`d,xy7
yxtgh
7xy`d,xy7
yxtgh
7
7
7
Yropjrty T@@@ ;
s`h7x + s`h7y >
8+2--+--p-
8+2-+--p
8+2+--+-
-
-
-
7yxghl9y,x7`d,}x7yy7x{s`h
7yxghl7y,x9`d,}x7yy7x{s`h
7yxghl9xy`dor
7yxghl7y,x7`d,}x7yy7x{s`h
33337
33337
33
33337
8/10/2019 4_Inverse Trigonometic Functions.pdf
7/18
@HTJS]J \S@AOHONJ\@M1
www.te`hf` t.`h
Yropjrty @U ;
s`h7x s`h7y >
8+22-----p-
8+-2----p
8+2+----
-
-
-
7yxghl7y9,9x7`d,}x7yy7x{s`h
7yxghl9y7,7x9`d,}x7yy7x{s`h
7yxghl9xy`dor
7yxghl7y,x7`d,}x7yy7x{s`h
33337
33337
33
33337
Yropjrty U ;
mos7x + mos7y >
+-
----p
+-
---
-
-
9yxghl7y,x7`d,y7x7xymos3
9yxghl7y,x7`d,y7x7xymos
337
337
Yropjrty U@ ;
mos7x mos7y >
2---+-
---+-
-
yxghl7x9,9y7`d,}y7x7xy{mos
yxghl7y,x7`d,}y7x7xy{mos
337
337
\e`s rjsubt mgh cj jstgcb`sejl h tej sgnj wgy gs h propjrty U.
Yropjrty U@@ ;
( ) 3 tgh7x >
-2
-+p-
8
-+p
22-
-
-
-
-
7x`d,x7
x3tgh
7x`d,x7
x3tgh
7x7`d,x7
x3tgh
37
37
37
Yropjrty U@@@ ;
( ) 3 tgh7
x >
-2
+-p-
8
+-p
-
+
-
-
-
7x`d,x7
x3s`h
7x`d,x7
x3
s`h
7x7`d,x7
x3s`h
3
7
3
7
3
7
( ) 3 tgh7x >
2-
+
--
2
+
-
-
-
9x`d,x7
x7mos
x9`d,x7
x7mos
3
37
3
37
8/10/2019 4_Inverse Trigonometic Functions.pdf
8/18
@HTJS]J \S@AOHONJ\@M5
www.te`hf t.`h
Yropjrty U@T ; ( ) s`h7x > mos7 3x7- > tgh7 3x7
x
-
> mot7x
x7 3- > sjm7
- 3x7
7> mosjm7
x
7
Qejrj x 9
( ) mos7x > s`h7 3x7- > tgh7
-x
x7 3
> mot7
- 3x7
x> sjm7
x
7> mosjm7
- 3x7
7
Qejrj x 8 9
( ` ) tgh7x > s`h7
+ 3x7
x
> mos7
+ 3x7
7
> mot7
x
7> sjm7 3x7+ > mosjm7
+x
x7 3
Yropjrty UT ;
( ) 3 s`h7x >
----p-
--p
--
-
-
3
7x7`d),x7x3(s`h
7x3
7`d),x7x3(s`h
3
7x
3
7`d),x7x3(s`h
3
37
37
Yropjrty UT@ ;
( ) 3 mos7x >
---p
--
-
9x7`d,)7x3(mos3
7x9`d,)7x3(mos37
37
Yropjrty UT@@ ;
= s`h7x >
----p-
--p
--
-
-
-
3
7x7`d,)x
8/10/2019 4_Inverse Trigonometic Functions.pdf
9/18
@HTJS]J \S@AOHONJ\@M0
www.te`hf` t.`h
Yropjrty UT@@@ ;
= mos7x >
---+p
---p
-
-
-
-
3
7x7`d,)x=x
8/10/2019 4_Inverse Trigonometic Functions.pdf
10/18
@HTJS]J \S@AOHONJ\@M79
www.te`hf t.`h
Jx.7 Qr`tj jgme od tej dobbo`wha duhmt`ohs `h tej s`npbjst dorn ;
(`) tgh7
33 xg
x, | x | 2 g (``) tgh7
3=
=3
gx=g
xxg=, g 8 96
=
g- x
=
g
]ob. ( ) Bjt x > g s`h q. \ejh,
tgh7
- 33 xg
x > tgh7
q-
q333 s`hgg
s`hg> tgh7
mosg
s`hg> tgh7(tgh q) > q> s`h7
g
x
\eus, tgh7
- 33 xg
x> s`h7
g
x
( ) Bjt x > g tgh q. \ejh
tgh7
--
3=
=3
gx=gxxg= > tgh7
q-q-q
3==
===
tghg=gtghgtghg= > tgh7
q-q-q
3
=
tgh=7tghtgh=
> tgh7(tgh =q) > =q> = tgh7
g
x
\eus, tgh7
-
-3=
==
gx=g
xxg=> = tgh7
g
x
Jx.3 ]`npb`dy ;
(`) tgh7
xs`hgxmosc
xs`hcxmosg, `d
c
gtgh x 8 7
(``) tgh3
7
3
37
3
7
y7
y7mos
x7
x3s`h ,| x | 2 7, y 8 9 ghl xy 2 7
]ob. ( ) Qj egvj
tgh7
+
-
xs`hgxmosc
xs`hcxmosg
> tgh7
+
-
xmosc
xs`hgxmosc
xmosc
xs`hcxmosg
> tgh7
+
-
xtghc
g7
xtghc
g
> tgh7
+-pq7
qp, wejrj p >
c
gghl q > tgh x
> tgh7p tgh7q > tgh7
c
g tgh7(tgh x) > tgh7
c
g x
( ) Bjt x > tgh qghl y > tgh d. \ejh
tgh37
+-+
+--
3
37
37
y7y7mos
x7x3s`h
]OBTJL YSOCBJ]N
8/10/2019 4_Inverse Trigonometic Functions.pdf
11/18
@HTJS]J \S@AOHONJ\@M77
www.te`hf` t.`h
> tgh
+
-+
+--
3
37
3
7
y7
y7mos
3
7
x7
x3s`h
3
7> tgh
+ -- )^y(tgh3V
3
7)xtgh3(
3
7 77
> tgh (tgh7x + tgh7y) > tgh (q+ d) > dq-d+q
tghtgh7
tghtgh>
xy7
yx
-+
Jx.= Yrovj tegt
(`) tgh777
3+ tgh7
3 tgh7
3
7(``) 3 tgh7
3
7+ tgh7
1
7> tgh7
71
=7
]ob. ( ) Qj egvj
tgh777
3+ tgh7
3 tgh7
-
+
3 tgh7
3
7
( ) Qj egvj
tgh73
7+ tgh7
1
7> tgh7
-
+
1
7
3
77
1
7
3
7
> tgh7
7=
0..............(7)
How, 3 tgh73
7+ tgh7
1
7> tgh7
3
7+
+ --
1
7tgh
3
7tgh 77
> tgh73
7+ tgh7
7=
0> tgh7
-
+
7=
0
3
77
7=
0
3
7
> tgh7
71
=7
Jx.< D`hl tej vgbuj od ;
(`) s`h7
p
=
3s`h (``) tgh7
p
x
R s`h7
p=
3s`h >
=
3p
Cut, =
3p
pp
- 3,3 , we`me s tej pr`hm`pgb crghme od s`h7x
Eowjvjr, s`h
p=
3> s`h
p-p
=
3> s`h
=
pghl
=
p
pp-
3,
3
Ejhmj, s`h7
p=
3s`h >
=
p
( ) Qj fhow tegt tgh7(tgh x) > x
tgh7
p
mos
4
:pghl
4
:p(9, p)
Ejhmj, mos7
p4
1mos >
4
:p
Jx.: Yrovj tegt
tgh7
x7x7
x7x7>
3
p
3
7mos7x, dor
3
7 x 7
]ob. Bjt x > mos 3 q. \ejh, x7+ > q+ 3mos7 > q3mos3 > 3 mos q
ghl x7- > q- 3mos7 > q3s`h3 > 3 s`h q
R BE] > tgh7
-++
--+
x7x7
x7x7
> tgh7
q+qq-q
s`h3mos3s`h3mos3 > tgh7
q+qq-q
s`hmoss`hmos > tgh7
q+q-
tgh7tgh7 > tgh7
q-
p
SE]
Jx.4 Yrovj tegt
tgh7 x >3
7mos7
x7
x7, x (9, 7)
]ob. Bjt x > tgh3q. \ejh,
x > tgh q6 ghl so tgh7( x ) > tgh7(tgh q) >q
Durtejr, SE] >3
7mos7
+-
x7
x7>
3
7mos7
q+
q-3
3
tgh7
tgh7>
3
7mos7(mos 3q) >
3
7 3q> q
Ejhmj, BE] > SE]
Jx.1 ]obvj jgme od tej dobbow`ha dor x ;
(`) s`h7
x
5+ s`h7
x
7:>
3
p
(``) s`h7(x) mos7(x) > s`h7(=x 3), `d x 8 9
(```) tgh7(x + 7) + tgh7(x 7) > tgh7
=7
5(`v) tgh73x + tgh7=x >
3
p
s`h7
x
7:>
3
p s`h7
x
5
x7:
> s`h
-
p -x
5
s`h3
7
> mos
-x
5s`h
7
x
7:>
3
x
57
- 3x
33:> 7 3x
4 7 x > 71
Cut, x > 71 mghhot sgt`sdy tej a`vjh jqugt`oh.
R x > 71( ) Bjt s`h7x > gghl mos7x > c. \ejh
x > s`h gghl x > mos c mos g> 3x7- > s`h cDron tej a`vjh mohl`t`oh, wj egvj
g c> s`h7(=x 3) =x 3 > s`h (g c)
> s`h gmos c mos gs`h c =x 3 > x3 (7 x3) =x 3 > 3x3 7 3x3 =x + 7 > 9 (3x 7) (x 7) > 9
x > 7,37
( ` ) Qj egvj, tgh7(x + 7) + tgh7(x 7) > tgh7=7
5
tgh7
-+--++
)7x)(7x(7
)7x()7x(> tgh7
=7
5
)7x(7
x33 --
>=7
5, gssun`ha tegt (x + 7) (x 7) 2 7
`.j., x3 7 2 7
3x3x3
->
=7
5 `.j., x32 3
43x > 74 5x3
5x3+ 43x 74 > 9 9
x >5
735047=7 +->
5
===7->
5, x3> 4< 2/ 3. Ejhmj, wj rjkjmt te`s vgbuj od x. Ejhmj, tej rjqu`rjl vgbuj s x > 9
x > 733 73
1:-> 7 or 4
7
How tej dornubg,
tgh7x + tgh7y > tgh7xy7
yx
-+
eobls ohby wejh xy 2 7
\eus, wejh x > 7, (3x) (=x) > (3) (=) > 4 8 7
]o, wj rjkjmt x > 7 ghl gmmjpt x >4
7
Jx.5 ]obvj dor x ; 3 tgh7x > s`h7 3
g7
g3
mos7 3
3
c7
c7
]ob. Bjt g > tgh qghl c > tgh d.
\ejh, s`h7
+ 3g7
g3> s`h7
q+
q3tgh7
tgh3
> s`h7(s`h 3q) > 3q
ghl mos7
+
-3
3
c7
c7> mos7
d+
d-3
3
tgh7
tgh7
> mos7(mos 3d) > 3dEjhmj, SE] > 3q 3d> 3 (q d)
> 3 (tgh7g tgh7c)]o, tej a`vjh jqugt`oh npb`js
3 tgh7x > 3 (tgh7g tgh7c), 7 2 g 2 7, 7 2 c 2 7
tgh7x > tgh7g tgh7c
> tgh7
+-gc7
cg6 gc 8 7
x >gc7
cg
+-
6 gc 8 7
Jx.0 ]obvj ;
(`) 3 tgh7(mos x) > tgh7(3 mosjm x)
(``) tgh7x7
x7
>3
7tgh7x (x 8 9)
(```) s`h V3 mos7{mot (3 tgh7x)}^ > 9
]ob. ( ) Qj egvj 3 tgh7(mos x) > tgh7(3 mosjm x)
tgh7
- xmos7
xmos33 > tgh
7
xs`h
3
xmos7
xmos33-
>xs`h
3 s`h x mos x > 7 mos3x > s`h3x
s`h x (mos x s`h x) > 9 s`h x > 9 or mos x s`h x > 9wejh s`h x > 9, x > hp, h [
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@HTJS]J \S@AOHONJ\@M7:
www.te`hf` t.`h
wejh mos x s h x > 9 or tgh x > 7, x > hp+3
7tgh7x (x 8 9)
tgh77 tgh7x >3
7tgh7x (W tgh7
xy7
yx
+-
> tgh7x tgh7y)
tgh7x +
3
7tgh7x >
3
=tgh7x
tgh7x >=
3
4
px > tgh
4
p>
=
7
( ` ) Qj egvj, s`h V3 mos7{mot (3 tgh7x)}^ > 9
s`h
---
3
77
x7
x3tghmotmos3 > 9 (W 3 tgh-7x > tgh7 3x7
x3
-)
s`h
---
x3
x7motmotmos3
377 > 9 (Wmot7x > tgh7
x
7)
s`h
--x3
x7mos3
37 > 9
s`h
--
--3
337
x3
x77
x3
x73s`h > 9 (W 3 mos7x > s`h7(3x 3x7- )^
-
x
x7 3
33
x3
x7
7
-
- > 9 xx7 3-
> 9 or
33
x3
x77
-- > 9
7 x3> 9 or
33
x3
x7
-> 7 x > 7 or
(7 x3)3> 9 (7 x3 3x) (7 x3+ 3x) > 9 7 x3 3x > 9 or 7 x3+ 3x > 9 x3+ 3x 7 > 9 or x3 3x 7 > 9
x > 7 3 or x > 7 3
Ejhmj, x > 7, 7 3 , 7 3
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@HTJS]J \S@AOHONJ\@M74
www.te`hf t.`h
W.7 Jvgbugtj tej dobbow`ha ;
( ) mos7
p=
tgh7
+
-==
==
cg7
cg+ tgh7
+
-==
==
mc7
mc+ tgh7
+
-==
==
gm7
gm
W.= Yrovj tegt
s`h77=
73+ mos7
:
p
W.< Yrovj tegt
3 tgh7
+-
3
xtgh
cg
cg> mos7
++
xmoscg
xmosgcdor 9 2 c g ghl x 9
W.: @d tgh7x + tgh7y + tgh7z >3
p, provj tegt
xy + yz + zx > 7
W.4 @d mos7x + mos7y + mos7z > p, provj tegtx3+ y3+ z3+ 3xyz > 7
W.1 @d s`h7x + s`h7y + s`h7z > p, provj tegt
x 3x7- + y 3y7- + z 3z7- > 3xyz
W.5 ]obvj ; tgh73x + tgh7=x >=3p
mos7x mos7y >=
p
W.79 @d s`h V3 mos7{mot (3 tgh7x)}^ > 9, d`hl x.
W.77 @d 7 x, y, z 7, sume tegt s`h7x + s`h7y + s`h7z >3
=p,
d`hl tej vgbuj od
x3999+ y3997+ z3993 399339973999 zyx0
++
ZH]OBTJL YSOCBJN]
JUJSM@]J @
8/10/2019 4_Inverse Trigonometic Functions.pdf
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@HTJS]J \S@AOHONJ\@M71
www.te`hf` t.`h
W.73 Yrovj tegt
tgh7=
7+ tgh7
1
7+ ........... + tgh7
7hh
73 ++
> tgh7
+3hh
W.7= ]un to h tjrns tej sjr`js
tgh7
+ 3x3.77
x+ tgh7
+ 3x=.37
x+ tgh7
+ 3x y > z
W.7: @d tgh7x + tgh7y + tgh7z > 3
pghl x + y + z > = , tejh provj tegt x > y > z
W.74 ]obvj tegt jqugt`oh dor x 6
= s`h7 3x7
x3
+ < mos7
3
3
x7
x7
+
-+ 3tgh7 3x7
x3
->
=
p
W.71 @d s`h (pmos q) > mos(ps`h q), provj tegt
q> 3
7s`h7
3
p,
wejrj x3+ y3+ z3> r3
W.70 Yrovj tegt
tgh7
+-
xyg
yxg
7
7+ tgh7
+-
7g,g
gg
3
73+ tgh7
+-
7gg
gg
=3
3= + .......+ tgh7
+-
-
-
7gg
gg
7hh
7hh+ tgh7
hg
7> tgh
7
y
x
W.39 @d g7, g
3, g
=, .............. dorn gh G.Y. w`te monnoh l`ddjrjhmj l (g 8 9, l 8 9) provj tegt
tgh7
37gg7
l
+ + tgh7
=3gg7
l
+ + .......... + tgh7
7hh gg7
l
++ > tgh7
7h7
h7h
gg7
gg
+
+
+
-
8/10/2019 4_Inverse Trigonometic Functions.pdf
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@HTJS]J \S@AOHONJ\@M75
e` f ` `
W7. Yrovj tegt dobbow`ha ; tgh7
=
7+ tgh7
:
7+ tgh7
1
7+ tgh7
5
7>
3
x, x
p tgh7(3 mosjm x) VMC]J 3990^
W tgh7
-
-3
=
x=7
xx=
OS
Yrovj tej dobbow`ha ; mos Vtgh7(s`h {mot7x)}^ > 3
3
x3
x7
+
+VMC]J 3979^
W4. D`hl tej vgbuj od s`h7
p:
+ --
VM.C.].J. 3973^W0. ]eow tegt ;
tgh=
1 s`h
-
3
7, y > 7 79. x > 7, (7 3 )
77. 9 7=. ] > tgh7
++ 3x)7h(7
hx74. x >
=
7
JUJS@M@]J 3 (COGSL YSOCBJN])
W3. p W=.
