133
T e o r e m a . Ixtiyoriy
a
va
b
uchun quyidagi tenglik o‘rinli
bo‘ladi:
cos(
a + b
)
=
cos
a
cos
b -
sin
a
sin
b.
(1)
M
0
(1; 0) nuqtani koordinatalar boshi atrofida
a
,
-b
,
a
+ b
radian
burchaklarga burish natijasida mos ravishda
M
a
,
M
-b
va
M
a+b
nuqtalar
hosil bo‘ladi, deylik (69- rasm).
Sinus va kosinusning ta’rifiga ko‘ra, bu nuqtalar quyidagi koordi-
natalarga ega:
M
a
(cos
a
; sin
a
),
M
-b
(cos(
-b
); sin(
-b
)),
M
a+b
(cos(
a
+ b
); sin(
a
+ b
)).
Ð
M
0
OM
a+b
=Ð
M
-b
OM
a
bo‘lgani uchun
M
0
OM
a+b
va
M
-b
OM
a
teng yonli
uchburchaklar teng va, demak, ularning
M
0
M
a+b
va
M
-b
M
a
asoslari
ham teng.
Shuning uchun
(
M
0
M
a+b
)
2
=
(
M
-b
M
a
)
2
.
Geometriya kursidan ma’lum bo‘lgan ikki nuqta orasidagi masofa
formulasidan foydalanib, hosil qilamiz:
(1
-
cos(
a + b
))
2
+
(sin(
a + b
))
2
=
(cos(
-b
)
-
cos
a
)
2
+
(sin(
-b
)
-
sin
a
)
2
.
25- § dagi (1) formuladan foydalanib, bu tenglikning shaklini almash-
tiramiz:
1
-
2cos(
a + b
)
+
cos
2
(
a + b
)
+
sin
2
(
a + b
)
=
=
cos
2
b -
2cos
b
cos
a +
cos
2
a +
sin
2
b +
2sin
b
sin
a +
sin
2
a
.
Asosiy trigonometrik ayniyatdan foydalanib,
hosil qilamiz:
2
-
2cos(
a + b
)
=
2
-
2cos
a
cos
b +
2sin
a
sin
b
,
bundan cos(
a + b
)
=
cos
a
cos
b -
sin
a
sin
b
.
1 - m a s a l a .
cos75
°
ni hisoblang.
(1) formula bo‘yicha topamiz:
cos75
° =
cos(45
° +
30
°
)
=
=
cos45
°
cos30
° -
sin45
°
sin30
° =
=
-
=
×
×
-
2
2
3
2
2
2
1
2
6
2
4
.
!
69- rasm.
134
(1)
formulada
b
ni
-b
ga almashtirib, hosil qilamiz:
cos(
a - b
)
=
cos
a
cos(
-b
)
-
sin
a
sin(
-b
),
bundan
cos(
a - b
)
=
cos
a
cos
b +
sin
a
sin
b
.
(2)
2- m a s a l a .
cos15
°
ni hisoblang.
(2) formulaga ko‘ra, hosil qilamiz:
cos15
° =
cos(45
° -
30
°
)
=
cos45
°
cos30
° +
sin45
°
sin30
° =
=
+
=
×
×
+
2
2
3
2
2
2
1
2
6
2
4
.
3- m a s a l a .
Ushbu formulalarni isbotlang:
(
)
(
)
cos
sin , sin
cos
p
p
a
a
a
a
2
2
-
=
-
=
. (3)
a
p
=
2
bo‘lganda (2) formulaga asosan:
( )
cos
cos cos
sin sin
sin ,
p
p
p
b
b
b
b
2
2
2
-
=
+
=
ya’ni
( )
cos
sin
p
b
b
2
-
=
. (4)
Bu formulada
b
ni
a
ga almashtirib, hosil qilamiz:
(
)
cos
sin
p
a
a
2
-
=
.
(4) formulada
b
a
p
= -
2
deb faraz qilsak:
(
)
sin
cos
p
a
a
2
-
=
.
(1)—(4) formulalardan foydalanib,
sinus uchun qo‘shish formulasini
keltirib chiqaramiz:
(
)
(
)
( )
(
)
sin
cos
(
)
cos
a b
a b
a
b
p
p
+
=
- +
=
-
-
=
2
2
( )
( )
=
-
+
-
=
+
cos
cos
sin
sin
sin cos
cos sin
p
p
a
b
a
b
a
b
a
b
2
2
.
Shunday qilib,
sin(
a + b
)
=
sin
a
cos
b +
cos
a
sin
b
.
(5)
!
!
135
(5) formulada
b
ni
-b
ga almashtirib, hosil qilamiz:
sin(
a - b
)
=
sin
a
cos(
-b
)
+
cos
a
sin(
-b
)
,
bundan
sin(
a - b
)
=
sin
a
cos
b -
cos
a
sin
b
.
(6)
4- m a s a l a .
sin210
°
ni hisoblang.
sin210
° =
sin(180
° +
30
°
)
=
=
sin180
°
cos30
° +
cos180
°
sin30
° =
+ -
= -
×
×
0
1
3
2
1
2
1
2
( )
.
5- m a s a l a .
Hisoblang:
sin
cos
sin cos
8
7
7
7
8
7
p
p
p
p
-
.
(
)
sin
cos
sin cos
sin
sin
8
7
7
7
8
7
8
7
7
0
p
p
p
p
p
p
p
-
=
-
=
=
.
6- m a s a l a .
Tenglikni isbotlang:
tg(
)
tg
tg
tg tg
a b
a
b
a
b
+
=
+
-
1
.
(7)
tg(
)
sin(
)
cos(
)
sin cos
cos sin
cos cos
sin sin
a b
a b
a b
a
b
a
b
a
b
a
b
+
=
=
+
+
+
-
.
Bu kasrning surat va maxrajini cos
a
cos
b
ga bo‘lib, (7)
formulani
hosil qilamiz.
(7) formula hisoblashlarda foydali bo‘lishi mumkin.
Masalan, shu formula bo‘yicha topamiz:
tg
tg(
)
tg
tg
tg
tg
225
180
45
1
180
45
1
180
45
o
o
o
o
o
o
o
=
+
=
=
+
-
.
M a s h q l a r
Qo‘shish formulalari yordamida hisoblang
(325–326)
:
325.
1) cos135
°
; 2) cos120
°
; 3) cos150
°
; 4) cos240
°
.
326.
1) cos57
°
30
¢
cos27
°
30
¢ +
sin57
°
30
¢
sin27
°
30
¢
;
2) cos19
°
30
¢
cos25
°
30
¢ -
sin19
°
30
¢
sin25
°
30
¢
;
3)
p
p
p
p
-
7
11
7
11
9
9
9
9
cos
cos
sin
sin
; 4)
p
p
p
p
+
8
8
7
7
7
7
cos
cos
sin
sin .
!
136
327.
1)
(
)
cos
,
sin
p
p
a
a
a
3
1
3
2
0
+
=
<
<
bunda
va
;
2)
(
)
cos
,
cos
a
a
a p
p
p
-
= -
<
<
4
1
3
2
bunda
va
.
Ifodani
soddalashtiring
(328–329)
:
328.
1) cos3
a
cos
a -
sin
a
sin3
a
;
2) cos5
b
cos2
b +
sin5
b
sin2
b
;
3)
(
)
(
)
(
)
(
)
cos
cos
sin
sin
p
p
p
p
a
a
a
a
7
5
14
7
5
14
+
-
-
+
-
;
4)
(
) (
)
(
) (
)
cos
cos
sin
sin
7
5
2
5
7
5
2
5
p
p
p
p
a
a
a
a
+
+
+
+
+
.
329.
1)
(
)
(
)
( )
cos
cos
cos
a b
a
b
p
p
+
+
-
-
2
2
;
2)
(
) ( )
(
)
sin
sin
cos
p
p
a
b
a b
2
2
-
-
-
-
.
Qo‘shish formulalari yordamida hisoblang
(330–331)
:
330.
1) sin73
°
cos17
° +
cos73
°
sin17
°
;
2) sin73
°
cos13
° -
cos73
°
sin13
°
;
3) sin
cos
sin
cos
5
12
12
12
5
12
p
p
p
p
+
;
4) sin
cos
sin
cos
7
12
12
12
7
12
p
p
p
p
-
.
331.
1)
(
)
sin
,
cos
a
a
p a
p
p
+
= -
<
<
6
3
5
3
2
bunda
va
;
2)
(
)
sin
,
p
p
a
a
a p
4
2
3
2
-
=
<
<
bunda sin
va
.
332.
Ifodani soddalashtiring:
1) sin(
a + b
)
+
sin(
-a
)cos(
-b
);
2) cos(
-a
)sin(
-b
)
-
sin(
a - b
);
3)
(
)
( )
(
)
cos
sin
sin
p
p
a
b
a b
2
2
-
-
-
-
;
4)
(
)
(
)
( )
sin
sin
sin
a b
a
b
p
+
+
-
-
2
.
333.
Agar sin
a = -
3
5
,
3
2
2
p a
p
<
<
va sin
b =
8
17
, 0
2
< <
b
p
bo‘lsa,
cos(
a + b
) va cos(
a - b
) ni hisoblang.
137
334.
Agar
cos
,
a = -
0 8 ,
p
a p
2
<
<
va sin
b = -
12
13
,
p b
p
< <
3
2
bo‘lsa,
sin(
a - b
) ni hisoblang.
335.
Ifodani soddalashtiring:
1)
(
)
( )
cos
cos
2
3
3
p a
a
p
-
+
+
;
2)
(
)
( )
sin
sin
a
p
a
p
+
-
-
2
3
3
;
3)
2
2
cos sin
sin(
)
cos cos
cos(
)
a
b
a b
a
b
a b
+
-
-
-
;
4)
cos
cos
cos(
)
cos(
) sin
sin
a
b
a b
a b
a
b
-
+
- -
.
336 .
Ayniyatni isbotlang:
1) sin(
) sin(
)
sin
sin
a b
a b
a
b
-
+
=
-
2
2
;
2) cos(
) cos(
)
cos
sin
a b
a b
a
b
-
+
=
-
2
2
;
3)
( )
( )
2
2
4
2
6
3
2
cos
cos
sin
sin
tg
a
p a
p a
a
a
-
-
+ -
= -
;
4)
( )
( )
cos
cos
sin
sin
tg
a
p a
a p
a
a
-
+
- -
= -
2
3
2
6
3
3
.
337 .
Ifodani soddalashtiring: 1)
o
o
o
o
1
3
tg
29
tg
1
31
tg
29
tg
-
+
; 2)
p
×
p
+
p
-
p
16
3
16
7
16
3
16
7
tg
tg
1
tg
tg
.
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