? 發問於 科學及數學數學 · 1 十年前

amaths

http://i270.photobucket.com/albums/jj88/ucc041087/...

help me to prove the Q25 above .thz!~

4 個解答

評分
  • 1 十年前
    最愛解答

    cos(x+y+z) + cos(x+y-z) + cos(x-y+z) + cos(-x+y+z) = 4cosxcosycosz

    cos(x+y+z) + cos(x+y-z)

    = 2 cos[(x+y+z+x+y-z)/2] cos[(x+y+z-x-y+z)/2]

    = 2 cos(x+y) cosz

    cos(x-y+z) + cos(-x+y+z)

    = 2 cos[(x-y+z-x+y+z)/2] cos[(x-y+z+x-y-z)/2]

    = 2 cosz cos(x-y)

    L.H.S. = 2 cos(x+y) cosz + 2 cosz cos(x-y)

    = 2 cosz [cos(x+y) + cos(x-y)]

    = 2 cosz (2) cos[(x+y+x-y)/2] cos[(x+y-x+y)/2]

    = 2 cosz (2) cosx cosy

    = 4cosxcosycosz

    = R.H.S.

    只重複用左一條公式

    cos M cos N = 2 cos[(M+N)/2] cos[(M-N)/2]

    OK? 要補習嘛? 有我, 囉 A, 無難度

    2008-02-18 21:32:30 補充:

    oh, 高手好多, 搶答問題, 有難度...記得投我一票

  • 1 十年前

    都唔係靠實力攞最佳ge! 講咩「OK? 要補習嘛? 有我, 囉 A, 無難度」, 你估你真係補習天王咩?

  • 1 十年前

    cos(x+y+z)+cos(x+y-z)+cos(x-y+z)+cos(-x+y+z)=4cosxcosycosz

    L.H.S=cos(x+y+z)+cos(x+y-z)+cos(x-y+z)+cos(-x+y+z)

    =2cos(x+y+z+x+y-z/2)cos(x+y+z-x-y+z)+2cos(x-y+z-x+y+z)cos(x-y+z+x-y-z)

    =2cos(x+y)cos(z)+2cos(z)cos(x-y)

    =2cos(z)[cos(x+y)+cos(x-y)]

    =2cos(z)[2cos(x+y+x-y)cos(x+y-x+y)]

    =2cos(z)[2cos(x)cos(y)]

    =4cosxcosycosz

    2008-02-18 21:28:31 補充:

    sorry.....我所有部份公式都漏左除2...cos(x y z) cos(x y-z) cos(x-y z) cos(-x y z)=4cosxcosycoszL.H.S=cos(x y z) cos(x y-z) cos(x-y z) cos(-x y z)=2cos(x y z x y-z/2)cos(x y z-x-y z) 2cos(x-y z-x y z)cos(x-y z x-y-z)=2cos(x y)cos(z) 2cos(z)cos(x-y)=2cos(z)[cos(x y) cos(x-y)]

    2008-02-18 21:28:41 補充:

    =2cos(z)[2cos(x y x-y/2)cos(x y-x y/2)]=2cos(z)[2cos(x)cos(y)]=4cosxcosycosz=R.H.S

  • 1 十年前

    Prove the identity:

    cos(x+y+z)+cos(x+y-z )+cos(x-y+z)+cos(-x+ y+z)=4cosxcosycosz

    cosx +cosy = 2 cos((x +y)/2) cos((x –y)/2).

    cosx –cosy = -2 sin((x +y)/2) sin((x –y)/2).

    Consider

    cos(x+y+z)+cos(x+y-z )

    =2cos(x+y)cos(z )

    cos(x-y+z)+cos(-x+ y+z)

    =2cos(z)cos(x- y)

    So

    cos(x+y+z)+cos(x+y-z )+cos(x-y+z)+cos(-x+ y+z)

    =2cos(x+y)cos(z )+2cos(z)cos(x- y)

    =2[cos(x+y)cos(z )+cos(z)cos(x- y)]

    =2cos(z)[cos(x+y) +cos(x- y)]

    =2cos(z)[2cos(x)cos(y)]

    =4cosxcosycosz

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