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Trigonometry Preparing for math contest; Please help!

  1. Feb 4, 2005 #1
    Prove that the triangle ABC is:

    a) acute, if and only if tan(alfa)*tan(beta)>1
    b) right, if and only if tan(alfa)*tan(beta)=1
    c) obtuse, if and only if tan(alfa)*tan(beta)<1
     
  2. jcsd
  3. Feb 4, 2005 #2
    start with the following...
    k=alpha+beta
    90<k<180 in an acute triangle
    k=90 in a right triangle
    k<90 in an obtuse triangle

    (sinx/cosx)(sin(k-x)/cos(k-x))
    =(sinxsinkcosx-cosksin^2x)/(cos^2xcosk+sinksinxcosx)
    sinxsinkcosx=A and cosk=B
    (A-Bsin^2x)/(A+Bcos^2x)= (A-Bsin^2x)/(A+B-Bsin^2x)
    we then want B equal to zero. cosk=0 when k=90. when k is greater B becomes negative.
     
    Last edited: Feb 4, 2005
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