|
tg a
__
tg a
, co sa
s m a /ga cos a; —— ; —-----
tg a
; —— + 1;------- .
ctga ctg a
ctga
tga
Bundan keyin o'qituvchi murakkabrok trigonometrik almashtirishlarni
ko'rsatishi maqsadga muvofiqdir.
1-misol. (1—sinoc)(l+sina)—cos2a ifodani soddalashtiring.
I usul.
(1
- sin
a )
(1
+ sin
a )
-c o s
2
a
=
1
- sin
2
a - c o s
2
a =
=
1
- (l - cos
2
a) - cos
2
a =
1
- 1
+ cos
2
a - cos
2
a
=
0
.
I I usul.
(1
- sin
a )
(1
+ sin a ) - cos
2
a
=
1
- sin
2
a
- cos
2
a =
=
1
- (sin
2
a
+ cos
2
a )
=
1
- 1
=
0.
sin4 x + cos4 x - 1
2-misoI.
— 7
---------
7
----- - ifodani soddalashtiring.
sin
X
+ cos
X
- 1
2
2
sin4 x + cos4 x - 1
(s*n2 x ) + cos4
x ~
*
(l “ cos2 *) + cos4 ~ *
sin
x
+ cos
x
1
(sin
2
x)
+ cos
6
x
-
1
(l - cos
2
x)
+ cos
6
x
-
1
_
1 - 2
cos
2
x
+
cos
4
x
+
cos
4
x
-
1
_
2
cos
2
x(cos
2
x
-
1
) _
2
l- 3 c o s 2x + 3cos
4
x - c o s 6 x + cos
6
x - l
3cos
2
x(cos
2
x - 1)
3
cos (a + /?) + c o s ( a - /3)
3-misol.
=
Ctg
ayniyatni isbotlang.
cos(a
+ j3) +
cos(a
-
fi) _
sin(a + /3) + sin(a - /J)
_ cosacosff -sin asin /? + cosacosP
+
sinasin p
_
sinacos/3 + cos a sin [5 + sin a
cos Д - cos a sin /3
2
cosacosS
cosa
= -г-;---------
о
= ------ =
•
2
sin a cos p
sma
10 — S. Alixonov
145
1 + COS /3 + COS
2
P
4-misol. "j
~
f~ r ifodani soddalashtiring.
i + sec p + sec p
1
+ c os p + co s
2
/3 _ 1
+ cosf} + COS
2
/3 _ 1
+ cosp + COS
2
P
_
1
+ sec p + sec2 p
i + _ L _ + _ L _
co s2 p + c osp + 1
cosp
c o s 2 p
(l + cosp + COS
2
P j c o s
2
P
COS
P
COS
P +
COS /3 + 1
=»COS
p .
Yuqoridagilardan ko‘rinadiki, trigonometriya kursida ayniy almashtirishlar
muhim o‘rin egallaydi. 0 ‘quvchilar trigonometrik ayniy shakl almashtirishlami
yaxshi o'zlashtirishlari uchun birinchidan, trigonometrik funksiyalaming birini
ikkinchisi orqali ifodalovchi va asosiy ayniyat kabi formulalarni, ikkinchidan
esa shu formulalarni trigonometrik ifodani berilishiga qarab tatbiq qila olish
malakalariga bog'liqdir. Trigonometrik ayniy *'3hakl almashtirishlami bajarish
uchun quyidagi formulalarni bilishlari kerak:
1. Asosiy trigonometrik ayniyatlar:
1) sin2
a
+ cos2
a =
1;
2) fc a =
s i n a
COSfl
1
%
cosa ,
3
) с
tga = —
----
, ( а * к п ) \
4
)seca
=
sina
’
cos
a
a * — (2n +
1
)
я * у ( 2 л + 1)
, n e Z ;
, n e Z;
5
)
cos
e c a = -
7
^— , ( a * лп); n e Z.
sin a
Bu ayniyatlardan kelib chiqadigan formulalar quyidagilardir:
1
)
tga ■
ctga = I
к
, n e Z .
2
)
1
+
t g a = sec2a,
a
(2w + 1)
n e Z .
3) 1 +
ctg2a
= cos
ec2a , ( a * n ) ,
n e Z.
1-misol. Ayniyatni isbotlang.
cos
a(tga
+
2)(2tga
+1) - 5 sin a cos a = 2,
n
а Ф —
(
2n
+1)
2
146
I s b o t i :
cos2
a(tga
+ 2)(2i
ga +
1) - 5 sin
a
cos
a =
2
(
sin a
. Y 2 sin a
, , c .
= cos a ------- + 2 II----------+ 1 | - 5 s m a c o s a -
I c o s a
Д c o s a
= 2 sin2 a + 4 sin a cos a + 2 cos2
a +
sin a cos a - 5 sin a cos
a
=
= 2(sin2
a
+ cos 2 a ) = 2.
2-misol. Ayniyatni isbotlang:
7t
(l + s in a ) ( /g a + c / g a ) ( l - s i n a ) =
ctga.\
a
Ф
у ’ ” 6
^
I s b o t i :
(1 + sin a )
(tga + ctga)(
1 - sin a ) = (1 + sin a )
_ (1 - sin2 a )(sin 2 a + cos2 a ) _
cos2 a
_
s m a
c o s a
4
------- + -------- (1 - sin a ) =
c o s a
sin a
=
ctga.
sin
a ■
cos a
sin a • cos a
II. Ikki burchak yig'indisi va ayirmasining trigonometrik funksiyalari.
1 ) s i n ( a ± / } ) = s in a c o s j3 ± c o s a s in /J ;
2) c o s ( a ± /? ) = c o s a c o s /? + sin a sin /3 ;
3)
’
1 +
tga ■
tgP
it
a , / 3 , a ±
p * — (2n + l ) , n e Z
4
)
c t g ( a ± p ) =
ctga ctgp
+1
ctga
±
ctgp
1-misol. cos!5° ni hisoblang.
( a, p, a = P * лп,пе Z).
Y e c h i s h .
cos!5°= cos(45°—30°) = cos45° cos 30*+sin 45° sin 30°=
= A
. A
+ J L . L = L U
6 + 7 2 ) = 0 , 9 6 5 9 .
2
2
2
2
4 '
’
147
2
-misol.
sinl5° ni hisoblang .
Y e c h i s h .
sinl5°= sin(45°—30°) =sins45° cos 30°— cos 45° sin 30°=
e - ж
2
2
2 2
4 '
’
Shuningdek, tgl5°=2-V 3 , cgl5°=2+V3,
secl5°='/6
- 4 2
hisoblash mumkin.
—
В
3-misol. -----
2
-----
2
~o ~
+
~ P) ayniyatni isbotlang.
1 -
tgLa ■
tgl p
■
tg2a - t g 2p = tga
+
tgP
t g a - tg P _
s b o t i .
i _ t g 2a tg2/}
l - t g a tgfi 1 + tga tgP
=
tg(a
+
P )tg(a - P).
4-misol.
sin(I s b o t i . sin(a +
P) ■
sin(a
- P) =
(sin a cos
p +
cosasin
p ) x
x
(sin a cos
P -
cos
a
sin
p
) = sin2
a
cos2
p
- sin2
p
cos2
a
=
= sin2
a(l
- sin2 j3) - sin2 /3(1 - sin2 a ) = sin2
a -
sin2
a
sin2
- s in 2
p
+ sin2 a sin2
p
= sin2 a - s i n 2
p.
Keltirish formulalari:
6)
c t g \ ^ ± a
=
+tga, ctg(n ± a ) = ±c1ga.
J
IV. Ikkilangan va uchlangan burchakning trigonometrik funksiyalari:
1) sin 2 a = 2 sin a cos a; 2) cos 2 a = cos2 a - sin2 a ;
3) e 2 a = J S “
1
- t g 2a
2a, a
ф
^ \2 n + l ) , n e Z
4)
ctglcc =
—f ——-
(2 a ,a * n n ,n
e Z );
2ctga
5) sin 3a = 3 s i n a - 4 s i n 3a; 6) cos3a = 4cos3 a - 3 c o s a ;
7) i&3a =
1 - 3
ig2a
3tga - tg3a
2 a , а Ф ^ ( 2 n + l), n e Z
8) С 83а= -З« а - « 3“ f,
1 - 3(g a
^
a * ^ , n 6 Z | ;
.
2
1 - cos 2a
9) sin
a =
------------
114
2
l + cos2a
; 11) cos a = ------------ ;
10) sin
a =
з л _ 3 s in a - s in 3 a
__ 3
_ c o s 3 a + 3 co sa
12) cos a =
1-misol. sino-sin(60°—a)sin(60°+a)= — sin3a ayniyatni isbotlang.
I s b o t i : s in a sin(600 - a ) • sin(60° + a ) = sin a(sin 2 60° - sin2 a ) =
= s i na l ^ - sin2
2-misol. cosa-cos(60°-a)cos(60°+a)= — cos3a ayniyatni isbotlang.
4
3-misol.
tgatg(60 °-d )tg (6 0 °+ a )= tg 3 a
ayniyatni isbotlang.
Bu ayniyatlardan foydalanib, quyidagi trigonometrik ifodalarni osonlikcha
hisoblash mumkin:
a)
sin 20° sin 40° sin 80° = 1 sin 3 - 20 ° = 1 sin60° =
-
• — = — ;
4
4
4
2
8
149
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