標題:
急求助!!F.4 A.maths
發問:
Solve the following equations for 0 ≦ x ≦ 360(Q1-Q4)1. ( 1 - tanx)(1 + sin2x)=1+tanx2.sin^4x + cos ^4x = 1/2sin2x3.cos4x + sin2x = 04.Suppose y =(sec^2x csc^2x) - 2a. Show that y =( 4/sin^2 2x ) -25 Show that sin3x/sinx ＋ cos3x/cosx = 4cos2x6 Let f(x) = sin^6x + cos^6 x Espressf(x) in the form A + Bcos4x... 顯示更多 Solve the following equations for 0 ≦ x ≦ 360(Q1-Q4) 1. ( 1 - tanx)(1 + sin2x)=1+tanx 2.sin^4x + cos ^4x = 1/2sin2x 3.cos4x + sin2x = 0 4.Suppose y =(sec^2x csc^2x) - 2 a. Show that y =( 4/sin^2 2x ) -2 5 Show that sin3x/sinx ＋ cos3x/cosx = 4cos2x 6 Let f(x) = sin^6x + cos^6 x Espressf(x) in the form A + Bcos4x where A and B are constants Hence find the maximun and minimun values of f (x) solve the equation sec^2x = (2 - cosx - sinx)/1 - sinx for 0 ≦ x ≦ 360
最佳解答:
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1) (1 - tanx)(1 + sin2x)=1+ tanx 1+ 2sinx cosx - tanx - 2sinx cosx tanx = 1+ tanx 2sinx cosx-2(sinx)^2=2 tanx sinx(cosx - sinx)=sinx/cosx sinx=0 or cosx - sinx=1/cosx sinx=0 or (cosx)^2 - sinxcosx=1 sinx=0 or 1 - (cosx)^2 + sinxcosx = 0 sinx=0 or (sinx)^2 + sinxcosx = 0 sinx=0 or sinx(sinx+cosx) = 0 sinx=0 or tanx = -1 x=0, 180 or -135, -315 2)(sinx)^4 + (cosx)^4 = 1/2sin2x ((sinx)^2 + (cosx)^2)^2 - 2(sinxcosx)^2 = 1/2sin2x (1)^2 - 0.5(sin2x)^2 = 1/2sin2x sin2x - 0.5(sin2x)^3 - 0.5 = 0 (sin2x)^3 -2 sin2x + 1 = 0 Let y = sin2x y^3 -2y + 1 = 0 (y-1)(y^2+y-1)=0 y=1 or (-1+(root5))/2 or (-1-(root5))/2 sin2x = 1 or (-1+(root5))/2 or (-1-(root5))/2 2x = 90,450 or 38.173, 141.827, 398.173, 501.827 or no solution x = 45 / 225 / 19.086 / 70.914 / 199.086 / 250.914 3) cos4x + sin2x = 0 1-2(sin2x)^2+sin2x=0 2(sin2x)^2 - sin2x - 1=0 (sin2x - 1)(2sin2x + 1)=0 sin2x=1 or -0.5 2x=90, 450 or 210, 330, 570, 690 x=45, 225 or 105, 165, 285, 345 4)Suppose y =(secx)^2 (cscx)^2 - 2 (a) y =(secx)^2 (cscx)^2 - 2 = 1/(cosxsinx)^2-2 =1/(sin2x/2)^2-2 = 4/(sin2x)^2-2 5) sin3x/sinx + cos3x/cosx = (sinxcos2x+cosxsin2x)/sinx + (cosxcos2x-sinxsin2x)/cosx =cos2x+2(cosx)^2 + cos2x - 2(sinx)^2 =2cos2x+2((cosx)^2-(sinx)^2) = 2cos2x+2cos2x = 4cos2x 6) f(x) = (sinx)^6 + (cosx)^6 =((sinx)^2 + (cosx)^2)^3 - 3(sinx)^2(cosx)^2((sinx)^2+(cosx)^2) =(1)^3 - 3(sinx)^2(1-(sinx)^2)(1) =1 - 3(sinx)^2 + 3(sinx)^4 =cos2x - (sinx)^2 + 3(sinx)^4 =cos2x - 0.5 + 0.5cos2x + 3/4 * (1 - 2cos2x+(cos2x)^2) =cos2x - 0.5 + 0.5cos2x + 3/4 - 1.5cos2x + 3/4 * (cos2x)^2 =0.25 + 0.75* (0.5+0.5cos4x) =0.25 + 0.385 + 0.385cos4x = 5/8 + 3/8 * cos4x max value=5/8+3/8*(1)=1, min value=5/8+3/8*(-1)=1/4 7)(secx)^2 = (2 - cosx - sinx)/(1 - sinx) 1+(tanx)^2 = (1-cosx)/(1-sinx)+1 (tanx)^2 = (1-cosx)/(1-sinx) (1-sinx)(tanx)^2 = (1-cosx) (sinx)^2-(sinx)^3 = (cosx)^2 - (cosx)^3 (sinx)^2-(cosx)^2 = (sinx)^3 - (cosx)^3 (sinx+cosx)(sinx-cosx) = (sinx-cosx)((sinx)^2+sinxcosx+(cosx)^2) sinx-cosx=0 or sinx+cosx=1+sinxcosx tanx=1 or 1+sinxcosx-sinx-cosx = 0 tanx=1 or 1-sinx+cosx(sinx-1) = 0 tanx=1 or (1-cosx)(1-sinx)= 0 tanx=1 or sinx=1 or cosx=1 x=0,45,90,225 sec 90 is undefined, thus x is not 90 Thus, x=0, 45, 225 2007-02-07 17:59:09 補充： 1)x=0, 180, 360 or -135, -3157)x=0, 45, 225, 360
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