Jade 發問於 科學及數學數學 · 2 月前

a triangle ABC, how to prove sin^2(A)-sin^2(B)=sin(A-B)sinC?

更新:

I tried rewriting LHS into (sinA+sinB)(sinA-sinB), and used A=180-(B+C) and B=180-(A+C), and use differences identities sin(180-(B+C)) ->sin180cos(B+C)-cos180sin(B+C) and so on, and ended up horribly long and complicated process and never gets to RHS...

4 個解答

評分
  • GONG
    Lv 6
    2 月前
    最愛解答

    sin(A-B)sinC=sin(A-B)sin(A+B)=(sinAcosB-cosAsinB)(sinAcosB+cosAsinB)

    =sin^2Acos^2B-cos^2Asin^2B=sin^2Acos^2B-(1-sin^2A)(1-cos^2B)

    =sin^2Acos^2B-(1-sin^2A-cos^2B+sin^2Acos^2B)

    = -1+sin^2A+cos^2B= -1+sin^2A+(1-sin^2B)=sin^2A-sin^2B

    硬要從左到右需要不少技巧,需加上一些臨時輔佐數字,例如:加1再減1..等等,不是一時想得到的,若是規定如此只好請你將我寫得從右寫到左,至於其他方法目前還沒想到

  • 1 月前

    LHS--->RHS:-

    We are learning -0 to +0 and +0 to -0,properly,

    For the -0 to +0 refers to +sinC

    For the +0 to -0 refers to +sinC

    ;

    Both improperly !?

    Yet properly to sinC.

    Attachment image
  • 1 月前

    由要證明的等式 sin^2(A)-sin^2(B)=sin(A-B)sinC

    來看, 顯然是從右式變到左式比較自然. 也沒有規

    定一定要從左式變到右式才正規. 若一定要, 那麼

    把右式變成左式的過程反著寫就可以了.

  • 2 月前

    Gong, can you rewrite LHS instead of RHS?

還有問題嗎?立即提問即可得到解答。