|
2
cos 2x -
1
.
Javobi:
x = 2
k n ~ .
2
sin
4
x + cos
4
x -
2
s in
2
x + sin
2
2
x =
0
.
Javobi: x
=
arcsin (2 -
J 2
) + ^ .
216
12
. sin
3
x cos x - c o s
3
s in x = cos4 — .
Javobi: x '■
13. sin
4
x
+ cos
4
x =
cos
4x.
14.
tg(
40° +
x)ctg(
5° - x ) = - .
Javobi: x ~
1
(~
1)*+1
arcsin - +
kn
Javobi: — n.
n k
- 35° + (-l)* +l arcsin —
v
’
10
15.(sinx+cosx) =tgx+ctgx.
16. sin3x-sin
3
x+cos3xxos
3
x ;
17. sinx+sm3x+sin7x=3,
18.
tg7x+tg3x-0,
19. 7+sinx+cosx=Q,
29
20
. sm,
0
x+cosl
0
x
es7 7
cos
42
x.
lo
8
'
К
Javobi: x*=2kn+— .
4
Javobi: x= n k ± ^
•
о
Javobi: x=0.
Javobi: x ■
n k
10
'
n
Javobi: х}= кк~— , x2=n(2n+l)
T
A -
2 k + l
Javobi: x -
— -— rt.
21
. Itgx+ctg^*
- j
’ .
22, 13sitU'-12cosx+13sin3;c=30,
n
Javobi: х шап к ± — .
о
Javobi: x
1
n k
> - - —
2
’
23, l+2cos2x+2cos4x+2cos6x®,0.
Javobi: x m—r-
+
x - ± ~ ± ~ .
4
2
3
3
24. cos^+cos
4
(x
”
4
)
,
wlr
Javobi: х = - ^ (k+ l).
217
25.2sin2x+sinx+co&x= 1.
26. 3sinx+sin(x + — ) =1—3 sinxcosx.
Javobi:
(2&+1)
n, keZ.
17-§. Parametrli trigonometrik tenglamalarni yechish
•
Param etrli trigonom etrik tenglam ani yechish param etrlam ing m u m -
kin b o ‘lgan h ar bir qiym atlari sistemasi u ch u n berilgan tenglam aning
h am m a yechim lari to 'p la m in i aniqlash dem akdir. Param etrik k o ‘ri-
nishdagi trigonom etrik tenglam alarni yechish quyidagi ketm a-ketliklar
asosida am alga oshiriladi.
1. P aram etrlam ing m um kin bo'lgan h ar bir qiym atlari sistem asi
u c h u n yechim lar sonini aniqlash.
2. H osil qilingan param etrli yechim form ulalarini topish.
3. T englam ani p aram etrlar asosidagi qiym atlar sistem asini aniqlash.
a
1
-misol. sin(a+x)+sirLx=cos— tenglamani yeching.
Y e c h i s h . Berilgan tenglam aning chap tom onida turgan ifodaga
trigonometrik funksiyalar yig‘indisini ko'paytmaga keltirish formulasini tatbiq
qilsak, u quyidagi ko'rinishni oladi:
27. 3-co s
2
jc—3sinx=0.
и
Javobi: — +2kn, keZ.
_
_
я
28. cos
3
x+4sin
2
2
'Cos
2
j
co&x=0.
Javobi: — +2kn, keZ.
29.
cos
2
2x-sin
2
2 x = - —■
x
x
30. sin
3
.v+2sin 'J cos
2
=0.
x
Javobi: ± ^ + ^ k , keZ.
о
2
Javobi: nk, keZ.
X
31. sinJx cos
2
— sin2xsin
2
= 0
Javobi: nk, keZ.
- . , a
4
a
a
2
sm (— +
x)
cos — = cos —.
2
2
2
Bu tenglamada quyidagi ikki hoi bo‘lishi mumkin.
218
ос
1. Agar c o s - * 0,
a
* (2
n + \)
k
bo'lsa, u holda tenglamani quyidagicha
yozish mumkin:
2 sin(“ +
x)
= 1,
x
=
+ (-1)* ^ +
kn, к
e
Z.
2
2
6
(X
2. Agar
а=(2п+\)к
bo'lsa, u holda cos — = 0 bo'ladi. Bu holda ham
tenglik o'rinli bo'ladi:
Javobi: x
- ~ +
( - l ) fc
+
kn, a
* (
2
« +
l)n,
2
6
-ixtiyoriy
son ,
a = (2 n + l)n.
n
2
-misoI. sin
2
x+asm
22
x=sin — tenglamani yeching.
6
. . .
l - c o s
2
x
2
^ ч
1
Y e c h i s h . ----- ------ +
a(l
- cosz
2x)
= - ,
1
- cos
2
x+
2
a —
2
acos
22
x = l,
2
acos
22
x+cos
2
x-
2
a=
0
,
(cos
2
x) = ~ 1+- ^ + 16— ;
(cos
2
x) =
~ l ~
A -t -l6a—,
4
a
4 a
\
т
—
1
+
"v/l
+
16a2
, ,
I —
1
+
"\/l
+
16a2
I .
a) c o s
2
x = -------- -----------,
l+ 1 6 a2>0,
| --------- -----------|s l,
4a
4a ,
—1+
yji
+ ]бд
2
S4a,
l+ 16a2<4a2+ 8 a+ l,
12
a
2
-
8
a<
0
; a (
12
a -
8)^0
1
) a =
0
,
12
a—
8
*
0
,
2
2
2
) a*
0
,
12
o -
8
=
0
, a = - , bundan a s -
bo'ladi.
0
-
1
- V l + I
6
a
2
, - l - > / l + 16a2
b)
cos
2
x = -------- ---------- ;
| -------------------- |£
1
,
4a
4a
tengsizlik a ning hech qanday qiymatlarida bajarilmaydi. Agar a=0 bo'lsa
f
•
2
О
,
П
,
sin X = - => xw = ± - +
kn, keZ .
219
Javobi:
a*0 da jc,
=izk ±
2
arccos
(
- 1 + Vl + 16a2
4
a
к
4
3-misol.
cosx+sim =a tenglamani yeching.
a
= 0
da jc34= ± y +Atc, &eZ
Y e c h is h . cosx+siiu=
^2
cos(x~ ^ ); agar
\a\< ^2
bo'lsa, bu tenglama
i
7t
d
yechimga ega bo'ladi. Agar
\a\<4l
bo‘lsa, *
=
±
arccos
—
j= + 2kn,
agar
\a\> s/2
bo'lsa, yechim yo'q.
4-misol.
sin2x+3cos2x=a tenglamani yeching.
Y e c h is h . 2smxcosx+3cos
2
jt—3sinbc=a,
2tgx+3-3tg
2
jc
=a(
1 +tg
2
x),
tg2x(a$-3)-2tgx+(a-3)=0,
1 ± VlO - a
2
te 4 , 2 =
a +
3
I
-------
j
Agar M< VlO bo'lsa, a, =
arctg
— —^
a
+
kn, k e Z .
Agar
a==—3
a +
3
bo'lsa, sin2x+30052*=-3 bo'ladi.
2simccosx+3cos
2
x+3cos
2
;c—3=—3,
2sinjccosjc+6cos
2
jc=0,
cosx(sinjc+3co&x)=0,
1)
cosx=0,
siruc+3co&x*0,
jc
=
+kn, keZ;
2)
cosx/0,
siruc+3cosx=0,
x
2
=arctg(-3)+fai,
keZ.
I
-------
2
Javobi:
Agar |а
|<л/10
(a*- 3) bo'lsa,
a
=
arctg
— ——— — + k n ,k e Z ,
a +
3
agar ( a * - 3) bo'lsa,
x = — +kn, keZ.;
agar |a|>VlO
bo'lsa, tenglam a
yechimga ega emas.
5- misol.
sin
6
x+cos
6
x=a(sin
4
jc+cos
4
jc) tenglamani yeching.
Y e c h i s h . sin
4
x+cos‘,x=(sin
2
x+cos
2
x)
2
—2sin
2
xcos
2
x=
1
—2sin
2
xcos
2
x,
sin
6
x+cos
6
x=(sin
2
x+cos
2
x)
3
—3sin
4
xcos
2
x -3 sin
2
x-cos
4
x ) = l—3sin
2
jccos
2
jc,
1 - 3sin
2
xcos
2
x=a( 1 -2 sin
2
xcos
2
x),
1—3sin
2
Jccos
2
Jc=a—2asin
2
jccos
2
Jc,
220
sin
2
x co s2x
a
- 1
2 a - 3 ’
sin
2
2x=4-
f l
- 1
2 a - 3
a
- 1
Agar bunda 0<4-„— - <1 bo'lsa, tenglama m a’noga ega bo'ladi. Bu
l a — 5
1
3
3
tengsizlikni yechsak,
2
~ ^
bo'ladi. Agar
a=—
bo'lsa, 1—3sin
2
xcos
2
x= — -
3
-3sin
2
xcos2x hosil bo'ladi, bundan 1 = —
tenglik hosil bo'ladi, buning
bo'lishi mumkin emas.
1
Agar -
bo'lsa, * ~~ ±
2
arcsin
a
- 1
2 a - 3
k n
Agar
a< —
va
a>
1 bo'lsa, yechim yo'q.
3x
.
. x
6
-misol. sin — + sin
x - m
sin —
tenglamani yeching.
Y e c h is h . s m |x + | = s in x c o s ~ + c o s x s i n ^ , bunga ko'ra
X
.
X
.
X
„
sin x + s m x c o s — + c o sx sm — - m s m — =
0
,
2
2
2
X
\
X
s in x |
1
+ c o s— + sin — (cos x -
m) =
0
,
. x
sin —
2
2 cos — + 4 cos
2
— - 1
- m
2
2
=
0
.
1
) s i n ^ =
0
, x =
2
kn, k e Z ;
>
2
x
»
x n
.
n
x
- l ± V 5 + m -4
2
) 4cosz — + 2 cos — - (1 +
m)
= 0,
cos — ---------------------
2
2
2
4
a)
c o s ^
m - 4 ,
bunda quyidagi shartlar bajarilishi kerak:
221
5 +
Am >
О,
-1 ± V5 + 4
m
< 1 .
Bu tengsizliklarni yechsak,
~ —^ m < 5
hosil bo'ladi. Bu shartga ko'ra
x
= ±2 arccos —— V
+
4kn, k e Z;
x
-1 -
yj5 + 4m
b) C0S- = -----------------
^
—
1
± -n/5 + 4
m
„
Bu yechilsa,
jc = ± 2
arccos
------------
-
------------+
4 k n , к e Z , - ~ < m <
1 .
4
«v
4
Javobi:
Agar
- —< m < \.
bo'lsa,
x
=2I
ck
, keZ.
, »
- l± > /5 +
4m
,
„
.
,
, ,
х1Ъ
= ± 2 arccos
---------
+4kn, k e Z ,
Agar l , „
- I
+ yj5 + 4m
..
,
„
xy=2kn, k e Z x2 =
±2 arccos-----------------+ 4
kn, k e Z ,
A gar«i> 5,
m<- - -
bo'lsa,
x=2kn, keZ.
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