dvhtlbz的部落格

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A Maths問題

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Slove the following equations for 0≦x≦360 1.cos x +cos 3x=0 2.sin3x+cos3x=sin2x+cps2x 3.cos (x+15)cos(x-15)-sinx sin(x-30)=cos165 4.cos x+cos2x+cos3x+cos4x=0 5.tan(30+x)tan(30-x)=1-2cos2x

最佳解答:

1.cos x +cos 3x=0 2cos2xcox=0 cos2x=0 or cosx=0 2x=90?,270?,450?,630? or x=90?,270? x=45?,90?,135?,225?,270? or 315? 2.sin3x+cos3x=sin2x+cos2x sin3x-sin2x=cos2x-cos3x 2cos(5x/2)sin(x/2)=-2sin(5x/2)sin(-x/2) cos(5x/2)sin(x/2)=sin(5x/2)sin(x/2) cos(5x/2)sin(x/2)-sin(5x/2)sin(x/2)=0 sin(x/2)[cos(5x/2)-sin(5x/2)]=0 sin(x/2)=0 or tan(5x/2)=1 x/2=0?,180? or 5x/2=45?,225?,405?,585?,765? x=0?,18?,90?,162?,234?,306? or 360? 3.cos (x+15?)cos(x-15?)-sinx sin(x-30?)=cos165? 1/2[cos2x+cos30?]+1/2[cos(2x-30?)-cos30?]=cos165? cos2x+cos30?+cos(2x-30?)-cos30?=2cos165? cos2x+cos(2x-30?)=2cos165? 2cos(2x-15?)cos15?=2cos(180?-15?) cos(2x-15?)cos15?=-cos15? cos(2x-15?)=-1 2x-15?=270? or 630? 2x=285? or 645? x=142.5? or 322.5? 4.cos x+cos2x+cos3x+cos4x=0 cosx+cos3x+cos2x+cos4x=0 2cos2xcosx+2cos3xcosx=0 cosx(cos2x+cos3x)=0 cosx[2cos(5x/2)cos(x/2)]=0 cosx=0 or cos(5x/2)=0 or cos(x/2)=0 x=90?,270? or 5x/2=90?,270?,450?,630?,810? or x/2=90? x=36?,90?,180?,252?,270?,324? 5. tan(30?+x) tan(30?-x) = 1 - 2cos2x sin(30?+x)sin(30?-x)/cos(30?+x)cos(30?-x) = 1 - 2cos2x (1/2)(cos2x-cos60?)/(1/2)(cos60*+cos2x) = 1 - 2cos2x cos2x- 1/2 = (1-2cos2x)( 1/2 + cos2x ) cos2x-1/2 = 1/2+cos2x-cos2x-2cos22x 2cos22x+cos2x-1=0 (2cos2x-1)(cos2x+1)=0 cos2x=1/2 or cos2x=-1 2x = 60?, 300?,420?, 660? or 2x = 180?, 540? x = 30?, 150?, 210?, 330? or 90?, 270?

其他解答:

Q.5 tan(30 +x)tan(30-x)= sin(30+x)sin(30-x)/cos(30+x)cos(30-x)=(cos2x-cos60)/2 divided by (cos60 + cos2x)/2 = (cos2x-1/2)/(1/2 +cos2x) = (2cos2x - 1)/(1+2cos2x), which is equal to 1-2cos2x according to the equation. That is (2cos2x - 1) = (1-2cos2x)(1 + 2cos2x) (2cos2x - 1) -(1-2cos2x)(1+2cos2x) = 0 (2cos2x - 1) + (2cos2x - 1)(1 + 2cos2x) = 0 (2cos2x - 1)(1 + 1 + 2cos2x) = 0 Therefore, cos2x = 1/2...........(1) and cos2x = -1.............(2) For (1) x=30, 150. For (2) x=90. Therefore, x = 30, 90 and 150. 2008-05-25 13:53:02 補充： x can also = 210,270 and 330.7638E748CCC65837