CHÖÔNG IX: HEÄ PHÖÔNG TRÌNH LÖÔÏNG GIAÙC
I. GIAÛI HEÄ BAÈNG PHEÙP THEÁ
Baøi 173: Giaûi heä phöông trình:
(
)
()
2cosx 1 0 1
3
sin 2x 2
2
−=
⎧
⎪
⎨=
⎪
⎩
Ta coù:
()
1
1cosx
2
⇔
=
()
xk2k
3
π
⇔=±+ π ∈Z
Vôùi xk
32
π
=+ π
thay vaøo (2), ta ñöôïc
23
sin 2x sin k4
32
π
⎛⎞
=+π=
⎜⎟
⎝⎠
Vôùi x
3
π
=− + πk2
thay vaøo (2), ta ñöôïc
23
sin 2x sin k4
32
π
⎛⎞
=−+π=−≠
⎜⎟
⎝⎠
3
2
(loaïi)
Do ñoù nghieäm của heä laø: 2,
3
π
=
+π∈
฀
xkk
Baøi 174: Giaûi heä phöông trình:
sin x sin y 1
xy 3
+
=
⎧
⎪π
⎨+=
⎪
⎩
Caùch 1:
Heä ñaõ cho
xy xy
2sin .cos 1
22
xy 3
+−
⎧
=
⎪
⎪
⇔⎨π
⎪+=
⎪
⎩
π− −
⎧⎧
=
=
⎪⎪
⎪⎪
⇔⇔
⎨⎨
π
π
⎪⎪
+=
+= ⎪
⎪⎩
⎩
xy xy
2.sin .cos 1 cos 1
62 2
xy
xy 3
3
4
2
2
3
3
−
⎧−= π
=π ⎧
⎪
⎪⎪
⇔⇔
⎨⎨
π+=
⎪⎪
+= ⎩
⎪
⎩
xy
x
yk
k
xy
xy
()
2
6
2
6
π
⎧=+ π
⎪
⎪
⇔∈
⎨π
⎪=−π
⎪
⎩
xk
kZ
yk
Caùch 2:
Heä ñaõ cho
33
31
sin sin 1 cos sin 1
322
33
sin 1 2
332
2
6
2
6
ππ
⎧⎧
=− =−
⎪⎪
⎪⎪
⇔⇔
⎨⎨
π
⎛⎞
⎪⎪
+−=
+
=
⎜⎟
⎪⎪
⎝⎠ ⎩
⎩
π
⎧π
⎧
=− =−
⎪⎪
⎪⎪
⇔⇔
⎨⎨
πππ
⎛⎞
⎪⎪
+=
+
=+ π
⎜⎟ ⎪
⎪⎩
⎝⎠
⎩
π
⎧=+ π
⎪
⎪
⇔∈
⎨π
⎪=− π
⎪
⎩
฀
yx yx
xx
x
x
yx yx
x
x
k
xk
k
yk
Baøi 175: Giaûi heä phöông trình: sin x sin y 2 (1)
cos x cos y 2 (2)
⎧+=
⎪
⎨+=
⎪
⎩
Caùch 1:
Heä ñaõ cho
xy xy
2sin cos 2 (1)
22
xy xy
2cos cos 2 (2)
22
+−
⎧=
⎪
⎪
⇔⎨+−
⎪=
⎪
⎩
Laáy (1) chia cho (2) ta ñöôïc:
+
⎛⎞
=
⎜⎟
⎝⎠
xy xy
tg 1 ( do cos 0
22
−
=
khoâng laø nghieäm cuûa (1) vaø (2) )
24
22
22
+π
⇔=+π
ππ
⇔+=+ π⇔=−+ π
xy k
x
yk yxk
thay vaøo (1) ta ñöôïc: sin x sin x k2 2
2
π
⎛⎞
+−+π=
⎜⎟
⎝⎠
sin x cos x 2⇔+=
2 cos 2
4
2,
4
π
⎛⎞
⇔−
⎜⎟
⎝⎠
π
⇔− = π∈
=
฀
x
xhh
Do ñoù: heä ñaõ cho
()
2,
4
2,,
4
π
⎧=+ π∈
⎪
⎪
⇔⎨π
⎪
=
+− π ∈
⎪
⎩
฀
฀
xhh
ykhkh
Caùch 2: Ta coù
A
BACB
CD ACBD
=+=
⎧⎧
⇔
⎨⎨
=−=
⎩⎩
D+
−
Heä ñaõ cho
(
)
(
)
()()
⎧− + − =
⎪
⇔⎨++−=
⎪
⎩
⎧π π
⎛⎞ ⎛⎞
−+ −=
⎜⎟ ⎜⎟
⎪
⎪⎝⎠ ⎝⎠
⇔⎨ππ
⎛⎞ ⎛⎞
⎪++ +=
⎜⎟ ⎜⎟
⎪⎝⎠ ⎝⎠
⎩
sin x cos x sin y cos y 0
sin x cos x sin y cos y 2 2
2sin x 2sin y 0
44
2sin x 2sin y 2 2
44
sin sin 0
44
sin sin 0
44
sin 1
4
sin sin 2
44
sin 1
4
2
42
2
42
sin sin 0
44
xy
xy
x
xy
y
xk
yh
xy
⎧π π
⎛⎞⎛⎞
−
+−=
⎜⎟⎜⎟
⎪
⎧π π ⎝⎠⎝⎠
⎛⎞⎛⎞ ⎪
−+ −=
⎜⎟⎜⎟
⎪⎪π
⎪⎝ ⎠ ⎝ ⎠ ⎛⎞
⇔⇔+=
⎨⎨
⎜⎟
ππ ⎝⎠
⎛⎞⎛⎞
⎪⎪
++ +=
⎜⎟⎜⎟
⎪⎪
π
⎝⎠⎝⎠ ⎛⎞
⎩+=
⎪⎜⎟
⎝⎠
⎩
⎧ππ
+=+π
⎪
⎪ππ
⎪
⇔+=+π
⎨
⎪
⎪ππ
⎛⎞⎛⎞
−+ −=
⎜⎟⎜⎟
⎪⎝⎠⎝⎠
⎩
π
⎧=+ π
⎪
⎪
⇔⎨π
⎪=+ π ∈
⎪
⎩
xk2
4
yh2,h,k
4Z
Baøi 176: Giaûi heä phöông trình: −− =
⎧
⎪
⎨+=−
⎪
⎩
tgx tgy tgxtgy 1 (1)
cos2y 3cos2x 1 (2)
Ta coù: tgx tgy 1 tgxtgy
−
=+
()
2
1tgxtgy 0
tg x y 1 tgx tgy 0
1tgxtgy 0 1tgx 0(VN)
⎧+=
−=⎧
⎪⎪
⇔∨−=
⎨⎨
+≠
⎪
⎩⎪+=
⎩
(
xy k kZ
4
π
⇔−=+π ∈
)
,
vôùi
x, y k
2
π
≠
+π
xy k
4
π
⇔=++π,
vôùi
x, y k
2
π
≠
+π
Thay vaøo (2) ta ñöôïc: cos2y 3 cos 2y k2 1
2
π
⎛⎞
+
++ π=−
⎜⎟
⎝⎠
cos 2 3 s 2 1
31 1
s2 cos2 sin2
222 6
yiny
in y y y
⇔− =−
π
⎛⎞
⇔−=⇔−
⎜⎟
⎝⎠
1
2
=
()
5
222 2
66 6 6
y h hay y h h Z
ππ π π
⇔−=+π −=+π ∈
,,
62
(loïai)yhhhayyhh
ππ
⇔=+π ∈ =+π ∈
฀฀
Do ñoù:
Heä ñaõ cho
() ()
5
6,
6
xkh
hk Z
yh
π
⎧=++π
⎪
⎪
⇔∈
⎨π
⎪=+π
⎪
⎩
Baøi 177: Giaûi heä phöông trình
3
3
cos x cos x sin y 0 (1)
sin x sin y cos x 0 (2)
⎧−+=
⎪
⎨−+=
⎪
⎩
Laáy (1) + (2) ta ñöôïc: 33
sin x cos x 0
+
=
33
3
sin x cos x
tg x 1
tgx 1
xk(k
4
⇔=−
⇔=−
⇔=−
π
⇔=−+π∈Z)
Thay vaøo (1) ta ñöôïc:
(
)
32
sin y cos x cos x cos x 1 cos x=− = −
==
21
cos x.sin x sin 2x sin x
2
ππ
⎛⎞⎛
=− −+
⎜⎟⎜
⎝⎠⎝
1sin sin k
22 4
⎞
π
⎟
⎠
π
⎛⎞
=− − + π
⎜⎟
⎝⎠
1sin k
24
⎧
⎪
⎪
=⎨
⎪−
⎪
⎩
2(neáu k chaün)
4
2(neáu k leû)
4
Ñaët 2
sin 4
α= (vôùi
02
<
α< π
)
Vaäy nghieäm heä
()
ππ
⎧⎧
=− + π ∈ =− + + π ∈
⎪⎪
⎪⎪
∨
⎨⎨
=α+ π ∈ =−α+ π ∈
⎡⎡
⎪⎪
⎢⎢
⎪⎪
=π−α+ π ∈ =π+α+ π ∈
⎣⎣
⎩⎩
฀
฀
฀฀
฀฀
x2m,m x 2m1,m
44
yh2,h y 2h,h
yh2,hyh2,h
II. GIAÛI HEÄ BAÈNG PHÖÔNG PHAÙP COÄNG
Baøi 178: Giaûi heä phöông trình:
()
()
1
sin x.cos y 1
2
tgx.cotgy 1 2
⎧=−
⎪
⎨
⎪=
⎩
Ñieàu kieän: cos x.sin y 0
Caùch 1: Heä ñaõ cho
() ()
11
sin x y sin x y
22
sin x.cos y 10
cos x.sin y
⎧
+
+−=
⎡⎤
⎣⎦
⎪
⎪
⇔⎨
⎪−=
⎪
⎩
−
()
(
)
() ()
()
+
+−=⎧
⎪
⇔⎨−=
⎪
⎩
−
+
+−=⎧
⎪
⇔⎨−=
⎪
⎩
sin x y sin x y 1
sin x cos y sin y cos x 0
sin x y sin x y 1
sin x y 0
−
(
)
()
+=−
⎧
⎪
⇔⎨−=
⎪
⎩
π
⎧+=−+ π ∈
⎪
⇔⎨
⎪−=π ∈
⎩
฀
฀
sin x y 1
sin x y 0
xy k2,k
2
xy h,h
()
()
ππ
⎧=− + + ∈
⎪
⎪
⇔⎨ππ
⎪=− + − ∈
⎪
⎩
≠
฀
฀
x2kh,k,h
42
y2kh,k,h
42
(nhaän do sin y cos x 0)