Luonggiac-Chuong5 - CHNGV PHNG TRNH OI XNG THEO SINX, COSX...

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CHÖÔNGV PHÖÔNG TRÌNH ÑOÁI XÖÙNG THEO SINX, COSX () ( ) as inx c o sx bs inxc o sx c 1 ++ = Caùch giaûi Ñaët =+ t sin x cos x vôùi ñieàu kieän t 2 Thì t 2 sin x 2 cos x 44 ππ ⎛⎞ =− ⎜⎟ ⎝⎠ Ta coù : ( ) 2 t 1 2sin x cos x neân 1 thaønh 2 b at t 1 c 2 +− = 2 bt 2at b 2c 0 ⇔+− = Giaûi (2) tìm ñöôïc t, roài so vôùi ñieàu kieän t2 giaûi phöông trình π + = 2sin x t 4 ta tìm ñöôïc x Baøi 106 : Giaûi phöông trình ( ) 23 sin x sin x cos x 0 * ++= (*) ( ) 2 sin x 1 sin x cos x 1 sin x 0 ⇔+ +−= ( ) = + = 1s i n x 0h a y s i n xc o s x1s i n x 0 ( ) sin x 1 1 sin x cos x sin x cos x 0 2 = 2 1x k 2 k Z 2 Xeùt 2 : ñaët t sin x cos x 2 cos x 4 ñieàu kieän t 2 thì t 1 2sin x cos x π •⇔ = + π π •= + = ≤= + Vaäy (2) thaønh 2 t1 t0 2 −= 2 t 1 0 t1 2 l o a ï i ⇔−− = ⇔ ⎢ Do ñoù ( 2 ) 2cos x 1 2 4 π
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π ⎛⎞ ⇔− = = ϕ< ϕ < ⎜⎟ ⎝⎠ π ⇔−= ± ϕ + π∈ ϕ = − π ⇔=± ϕ ϕ ± ± 2 cos x 1 cos vôùi 0 2 42 2 xh 2 , h , v ô ù i c o s 2 2 , h , v ô ù i c o s π 1 1 Baøi 107 : Giaûi phöông trình () 33 3 1 sin x cos x sin 2x * 2 −+ + = () ( ) ( ) 3 * 1 sin x cos x 1 sin x cos x sin 2x 2 ⇔− + + = Ñaët ts i n xc o s x 2 s i nx 4 π =+= + Vôùi ñieàu kieän t2 Thì 2 t1 2 s i n x c o s =+ x Vaäy (*) thaønh : 2 2 3 1t1 t 1 22 = 32 2 2t3t 3 t3 t 1 0 t1t 4 t1 0 t1t 2 3t 2 3 l o a ï i = ⇔+ −− = ++= ⇔=∨= −+ ∨= vôùi t = 1 thì 1 sin x sin 44 2 ππ += = π π ⇔+= = π ∨+= + π∈ π ⇔= π ∨=+ π ∈ ± ± 3 xk 2 x k 2 , k 4 4 2 x k 2 , k 2 vôùi π− =− = 2 t h ì s i n x s i n 4 2 ϕ + = ϕ + = π ϕ = = ϕ ϕ = ϕ ± ± xm 2 x m 2 , m , v ô ù i s 2 2 x m 2 , m , v ô ù is i n 2 ϕ i n 2 Baøi 108 :Giaûi phöông trình ( ) 2s inx c o sx t g x c o tg x* + Ñieàu kieän sin x 0 sin 2x 0 cos x 0 ⇔≠ Luùc ñoù (*) sin x cos x o cos x sin x = +
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() 22 sin x cos x 1 2s inx c o sx sinxcosx + ⇔+ = = Ñaët ts i n xc o s x 2 s i nx 4 π ⎛⎞ =+= + ⎜⎟ ⎝⎠ Thì =+ t1 2 s i n x c o s x v ô ù i t 2 v a ø (*) thaønh 2 2 2t = 3 2t 2t 2 0 ⇔− = (Hieån nhieân t khoâng laø nghieäm) 1 2 2 t22 t2 t20 t 2t 1 0 voâ nghieäm ++ = = ⇔ ⎢ = Vaäy * 2sin x 2 4 π += π = ππ ⇔+=+ π∈ π ⇔=+ π∈ ± ± sin x 1 4 xk 2 , k 42 2 , k 4 Baøi 109 : Giaûi phöông trình ( ) ( ) 3co tgx cosx 5tgx s inx 2* −−−= Vôùi ñieàu kieän sin , nhaân 2 veá phöông trình cho sinxcosx thì : 2x 0 0 () ( ) ( ) −− = * 3 cos x 1 sin x 5 sin x 1 cos x 2 sin x cos x ( ) ( ) () () ( = + + ⎡⎤ ⎣⎦ + + +− = −= s x1 s 5s in x1 c o inxc o sx 3s inxco 3cos x cos x 1 sin x sin x 5sin x sin x 1 cos x cos x 0 3cos x cos x sin x cos x sin x 5sin x sin x sin x cos x cos x 0 sin x cos x sin x cos x 0 1 3cosx 5sinx 0 2 ) = = ( Ghi chuù: A.B + A.C = A.D A = 0 hay B + C = D ) Giaûi (1) Ñaët i n o s s i 4 π + vôùi ñieàu kieän : 2 2 s i n x c o s x t 2 vaø t 1 ≠± (1) thaønh : 2 2 t0 t 2 t 2 1 0 =⇔ − −= t1 2 l o a ï i d o t 2 t 1 2 nhaän so vôùi ñieàu kieän =−
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Vaäy () 12 sin x sin 0 2 42 π− ⎛⎞ += =α< α < π ⎜⎟
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Luonggiac-Chuong5 - CHNGV PHNG TRNH OI XNG THEO SINX, COSX...

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