Luonggiac-Chuong9 - C HNG IX: HE PHNG TRNH L N G GIA C I....

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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 1c o s x 2 = xk 2 k 3 π ⇔= ±+ π∈ Z Vôùi 3 2 π =+ π thay vaøo (2), ta ñöôïc 23 sin 2x sin k4 32 π ⎛⎞ =+ π = ⎜⎟ ⎝⎠ x 3 π =− + π k 2 sin 2x sin k4 π =− + π = 3 2 (loaïi) Do ñoù nghieäm c a heä laø: 2, 3 π = +π∈ ± k Baøi 174: sin x sin y 1 xy 3 + = π += Caùch 1: Heä ñaõ cho 2sin .cos 1 22 3 +− = π π− = = ⎪⎪ ⇔⇔ ⎨⎨ π π 2.sin 1 cos 1 62 2 3 3
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4 2 2 3 3 −= π ⎪⎪ ⇔⇔ π ⎨⎨ π += xy x yk k () 2 6 2 6 π =+ π ⇔∈ π =−π xk kZ Caùch 2: Heä ñaõ cho 3 3 31 sin sin 1 cos sin 1 3 22 3 3 sin 1 2 3 32 2 6 2 6 π π =− π ⎛⎞ +− = + = ⎜⎟ ⎝⎠ π π π ππ + π π ± yx xx x x x x k k Baøi 175 : Giaûi heä phöông trình: sin x sin y 2 (1) cos x cos y 2 (2) Caùch 1: xy xy 2sin cos 2 (1) 2cos 2 (2) = = Laáy (1) chia cho (2) ta ñöôïc: + = tg 1 ( do cos 0 = khoâng laø nghieäm cuûa (1) vaø (2) ) 24 ⇔= + π ⇔+=+ π⇔=−+ π k x k thay vaøo (1) ta ñöôïc: sin x sin x k2 2 2 π + π = sin x cos x 2 ⇔+=
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2 cos 2 4 2, 4 π ⎛⎞ ⇔− ⎜⎟ ⎝⎠ π ⇔− = π∈ = ± x xh h Do ñoù: heä ñaõ cho () 4 2,, 4 π =+π∈ π = +− π ± ± h yk h k h Caùch 2 : Ta coù A BA C B 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 2 2 sin sin 0 sin sin 0 sin 1 4 sin sin 2 sin 1 4 2 42 2 sin sin 0 xy x y xk yh π ⎛⎞⎛⎞ = ⎜⎟⎜⎟ π ⎝⎠⎝⎠ π ⇔⇔ + = ⎪⎪ π +=+π ⇔+ = + π π π π π 2 4 2 , h , k 4 Z Baøi 176: Giaûi heä phöông trình: −− = tgx tgy tgxtgy 1 (1) cos2y 3cos2x 1 (2)
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Ta coù: tgx tgy 1 tgxtgy =+ () 2 1t g x t g y 0 tg x y 1 tgx tgy 0 g x t g g x 0 ( V N ) += −= ⎪⎪ ⇔∨ = ⎨⎨ +≠ ( xy k kZ 4 π ⇔−=+π ) , vôùi x, y k 2 π k 4 π ⇔=++π , x, y k 2 π Thay vaøo (2) ta ñöôïc: cos2y 3 cos 2y k2 1 2 π ⎛⎞ + ++ π= ⎜⎟ ⎝⎠ cos 2 3 s 2 1 31 1 s2 c o s 2 s i n2 222 6 yi n y in y y y ⇔− = π = 1 2 = 5 22 2 2 66 6 6 y h hay y h h Z ππ π π = + π π ,, 62 (lo ïai) yh h h a y h ⇔=+π ∈ =+π ∈ ±± Do ñoù: Heä ñaõ cho 5 6 , 6 xk h hk Z 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 + = 3 sin x cos x tg x 1 tgx 1 ( k 4 ⇔= π −+π∈ Z ) Thay vaøo (1) ta ñöôïc: ( ) 32
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This note was uploaded on 11/26/2011 for the course MATH 1002 taught by Professor Chuck during the Spring '11 term at University of Western States.

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Luonggiac-Chuong9 - C HNG IX: HE PHNG TRNH L N G GIA C I....

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