|
c o s
3
Q°
\ + t g
15
j + b -co s3 0 ^
2
1 + cos 30°
VI.
Trigonometrik funksiyalar ko‘paytmasini yig‘indiga keltirish
formulalari:
1) sin a co s/J = ^-[sin(a + j3) + s in ( a -/} )];
1 - cos 30°
2) cos a cos
p
= -[c o s (a
+ /3) +
cos (a - /})];
3) sin a sin
p = —
[cos(a -
P) -
cos(a +
P)].
Misol.
cosa+cos(a+2/J)+...+cos(a+7r/}) ifodani soddalashtiring.
• P
Y e c h i s h . Berilgan ifodani sm — ga ko'paytiramiz va bo‘lamiz:
1
• P
sm —
2
sin у cos a + sin - j cos(a +
P) +
sin у cos(a + 2/3) +
151
+... + sin у cos(ct + и/
3
)
1
- . p
2 sin —
2
sm| a + j j-sinj a - - j |+
in ^ a + £ j + s i n ^ a + ^ j - s i n ^ a + M j + ...
+ sin |ct + ^ J - s i n
+ sin | a -
■
—
j - sin | a + -y j + sin
+ s i n ( a + ^ ~ p ) - s m ( a + ^ l p
+
a + f j - s i n ^ + M j + . . . +
1
•
P
sin —
2
1
- •
n + l
0
f
n '
I
2
^
2 Я„ —
+
.
И
+ 1 „
/
Л
„
sin —— р cos[ а + — р
s in-
2
2
VII. Trigonometrik funksiyalar yig‘indisi va ayirmasining formulalari:
1
) sina + sin/J =
2
sin
• cos
■
2
2
2) s i n a - s i n /3 = 2sin^—
c o s ^ i - £ -
2
2
3) co sa + cos/3 = 2 c o s ^ - i ^ c o s ^ - ^ -
2
2
4) c o s a - c o s / 3 = 2sin^-iJ® s i n ^ ^ -
2
2
5)
t ga + tgp = ™ L« + P)
,
cos a • cos /?
6) e a - W = ^ 2 ^ ) - ,
cos a • cos p
n
a , P *
— ( 2 л - 1 ) , л е Z
7Г
7) c/ga + dgp = 4 ^ 4 ^ , [a,p * лги.иe Z \,
sin a- sin p
J
152
8)
ctga - c tg f t
=
—— .
sm
a ■
sin /}
Misol.
cos
a
+ cos
p +
cos
у +
cos(a
+ p + y) =
,a + P „„„а + У — Р + У
2
I s b o t i .
- 4 cos—r-^-cos—~ ~ cos —- — ayniyatni isbotlang.
c o sa + cos
p
+ cosy + cos(a +
P + y) =
-
a + P
a ~ P
„
y + a + p + y
y - a - P - y
= 2 cos
— ~ ~
cos-----— + 2 cos - --------- — - • cos -------- -— 7 =
2
2
2
2
, 2coS^
f
c ra £ ^ £ + cos£ ± A ± 2 z ) =
2
I
2
2
J
= 2 c o s ^ — ^ 2 c o s a ~ ^ + a ~ ^ + ^ c o s a
^ ~ a ~ ^ ~ 2^ =
2
4
4
.
a + P
a + y
p + y
= 4 cos-----—cos----- - c o s - —
2
2
2
VII.
Trigonometrik funksiyalami yarim argumentning tangensi orqali
ifodalash:
1) sin
a
=
2fc§
1
+ tg2 ^
[a
*
л(2п + 1 ) , п е Z) ;
1 - & 2 ?
2) cosa = -------- 2.
\ а * к ( 2 п + \),п<= Z }\
1
+ tg
2 a
2 tg l
[ a
n
1
3)
tga =
------
2 _
\ a , ^ ^ ( 2 n
+ l ) , n e Z
1 - f c 2l L 2
2
153
5-§. Trigonometrik ifodalarni soddalashtirishga
doir misollar yechish metodikasi
1-misol.
(1 - si n
a)(\
+ si n
a)
- co s2
a
ifodani soddalashtiring.
Y e c h i s h .
I usul.
(1 - sin
a)(
1 + sin
a
) - c o s 2
a =
1 - sin 2
a
- c o s2
a =
1 - (1 - c o s 2
a)
-
- c o s 2
a =
1 - 1 + c o s2
a
- c o s2
a
= 1 - 1 + c o s2
a -
c o s2
a
=
0
.
II usul.
(1 - sin a ) ( l + sin
a)
- co s2 a =
= 1 - sin 2
a -
co s2
a =
1 - (sin2
a
-Hi
cos2 a ) = 1 - 1 = 0.
sin 4
x
+ cos4
x
- 1
2-misol.
— 7
-----------
7
------ -
ifodani soddalashtiring.
sin
x +
cos
x
- 1
2
sin 4
x +
c o s4
x
- 1 _ (s^n2 x ) + co s
x ~
1 _
Y e c h i s h . sin лг + cos x - 1
(sin 2 xj* + c o s6
x - I
(1 - c o s2 x ) 2 + co s4 x - 1
1 - 2 c o s2 x + c o s4 x + co s4 x - 1
(1 - cos
2
x
)3
+ cos
6
x - 1
1 -3 cos
2
X
+ 3 cos
4
X
- cos
6
x + cos
6
x - 1
2
cos
2
x(cos
2
x -
1
) _
2
3cos
2
x(cos
2
- l )
3
_
. , 1 - ( s i n x - c o s x)
3-misol.
^---------
5
— ifodani soddalashtiring.
1 + sin x - cos x
,
1
- (sin x -co s x
)2
1
-s in 2x +
2
sin x co sx -co s2 x
Y e c h ish . —
; ^
2
—
•
2
2
■
2
2
1
+ sin x - cos x
sm x + cos x + sin x - cos x
l- ( s in 2x + cos
2
x) +
2
sinxcosx
2
sinxcosx
cosx
= ----------------------------------------= ---------------= —-----= ctgx.
2
sin x
2
sin x
sinx
1
1
sin
2
x
4-misol.
2
T
~2
------71
ifodani soddalashtiring.
cos x ctg x
tg x
®
154
Y e c h i s h .
1
sin2
x
sin2
x
sin2
x
cos2
x
cos2
x
ctg2x
tg2x
cos2
x
cos2
x
sin2
x
cos2
x
cos2
x
sin2
X
2
cos
X =
1 - sin2
x -
cos4
x
cos2
X
-
cos4
X
cos2
X
cos2
X
COS
x(l
- COS x )
. 2
= -------
— j
-------- - = sm
x.
COS X
[cos(-a) + sin(-tf
)]2
- 1
5-misol.
tz
— :---- — :— r ifodani soddalashtiring.
cos (-o) + sin ( - a ) - l
[cos(-a) + sin(-fl)} - 1 _ (cos
a -
sin
a)2
- 1 _
cos2
(- a )
+ sin2
( - a )
- 1
cos2
a -
sin2
a
- 1
_ cos2
a - 2
cos
a
sin
a
+ sin2
a
- 1 _ -2 cos
a
sin
a
cos2
a -
sin2
a
- (cos2
a
+ sin2
a)
-2 sin2
a
=
ctga.
6-misol. l+sina+cosa ifodani ko'paytma shakliga keltiring.
Y e c h i s h . l + s in a + c o sa = (l + co sa) + sin a =
„
2 a _ . a
a
л
a t
a
. a ,
= 2cos —+ 2sin —cos— = 2 cos— cos —+ sm — 1 =
-
a
= 2 cos — sin [ 90° - — |+ sin — |= 2 cos
% ■
2
sin 45° x
I .
2 J
2
2
xcos^45°
=
2y/2
cos у • cos ^45°
7-misol. V 3 - 2 s i n a
ifodani ayniy almashtirish orqali ko‘paytma
shakliga keltiring.
Y e c h i s h .
= 4 sin
л/3 - 2 sin a = 2
j
- sin a
3 0 ° - § ) с < к ( з 0 ° + “ }.
= 2 (sin60° - s i n a ) =
155
MUSTAQIL YECHISH UCHUN MlSOLLAR
4 - 2 i * 4 5 '+ « 6 0 *
1.
:
:
г
Javohi■
-
^
3sin90 -4co s60
+ 4ctg45
*
с
.
2
Я
4
Я
4
-tg — + ctg
—
3
112
2» _ . ч
к
2
я
к
.
Javohi'
---
3sin'—+ cos —
+ ctg—
1SV
2
3
4
3 { 4 s i n 7 ) ' ( 2® l ) " ( 2 c o s i )
~ { 2ctg
7
) •
Javobi:
- i -
4. sin
2 k
+ cos
4 n
+
t g 2 n 4
b
- 4
a c
.
Javobi:
1.
5.
ctg
| + cos ec | + sec 0°.
^
Javobi:
2
6.
a 2
sin
+ 2
ab
cos
к - b2
sin -
к.
2
2
Javobi: (a ~ b )\
3
3
7.
1 0 tg 2 K
+ 3cos —л
- 4 t g K
- 5 s i n —к.
°
2
2
Javobi:
5.
8- 4 sin 90° + 3 cos 720° - 3 sin 630° + 5 cos 900°.
Javobi:
S.
9. 5/g540° + 2 cos 1170° + 4 sin 990° - 3 cos 540°.
Javobi:
-1 .
10.100c/g2990° + 25/g2540“ - 3cos2 900.
Javobi:
-3 .
11. #900° - sin(-1095°) + cos(-1460°).
Javobi:
л Д д
12. sin(-1125°) + cos2(-900°) + £1710°.
Javobi:
2
- V
2
2
13.
cos20°
+
cos40°
+ cos60° +... +
cosl6Q°
+ coj180°.
Javobi:
-1 .
5 + sin 30° cos 60°
- t g —
15____________________ i .
a + b c o s 2 n - s i a n
Javobi:
17
4(0 + 6 ) ’
л
. it
„
m
cos
— +
и sin —
tgn
4
4
16
.
л
к
mn - mtg -
7
-ctg —
4
2
2
2
17. (sin
1 + cos
В
+ cos2
В
18- 1--------«-------
T 7 -
1 + sec
p +
sec'1
p
■ 2 Of ,
• 2 «
2 a
4 «
19. sm — +
sin — cos —
+
cos
—.
2
2
2
2
20
.
21.
1
cos2
2 a
-------
-
--------- 1--------- -—
cos
ec 2 a - 1
1 - sin
a
1 -
fg 2P
+ sin2
p
1 +
tg2P
22. (l - cos2 x )
ctg2x -
1.
23 • cos4
x -
sin4
x
+ sin2
x.
1 - sin4 2
a -
cos4 2
a
,
24.
7------------+ 1.
2 sin
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