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Math Help - Find tan A/2 + tan C/2?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find tan A/2 + tan C/2?

    In a triangle ABC, if sides a,b and c are in A.P(Arithmetic Progression), angle B = pi/4, find tan A/2 + tan C/2?
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    Quote Originally Posted by fardeen_gen View Post
    In a triangle ABC, if sides a,b and c are in A.P(Arithmetic Progression), angle B = pi/4, find tan A/2 + tan C/2?
    This is how I would start off.
    \tan \frac{A}{2} + \tan \frac{C}{2} = \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}} + \frac{\sin \frac{C}{2}}{\cos \frac{C}{2}} = \frac{\sin \left( \frac{A}{2} + \frac{C}{2} \right)}{\cos \frac{A}{2}\cos \frac{C}{2}}

    But A+B+C =  \pi and B=\frac{\pi}{4} this means \sin \left( \frac{A}{2}+\frac{C}{2} \right) = \sin \frac{3\pi}{8}.

    Thus, the problem comes down to computing \cos \frac{A}{2}\cos \frac{C}{2}.
    Which looks easier.
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  3. #3
    Super Member fardeen_gen's Avatar
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    If A + C = 3pi/4, sin(A + C) = sin 3pi/4. But does this necessarily imply by some formula(which I seem to be forgetting) that sin ((A + C)/2) = sin 3pi/8? Please help.
    Last edited by fardeen_gen; September 2nd 2008 at 08:00 AM. Reason: Grammer typo!!
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    Quote Originally Posted by fardeen_gen View Post
    If A + C = 3pi/4, sin(A + C) = sin 3pi/4. But does this necessarily imply by some formula(which I seem to be forgetting) that sin ((A + C)/2) = sin 3pi/8? Please help.
    Hey you !

    A+C=3pi/4
    Then (A+C)/2=3pi/8
    Thus sin((A+C)/2)=sin(3pi/8)

    But for your arithmetic progression, no idea !!
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  5. #5
    Super Member fardeen_gen's Avatar
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    Ouch!!It seems that trying to delve into something deep always makes me forget the most basic and fundamental things concerned.
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  6. #6
    Super Member fardeen_gen's Avatar
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    I solved it! Use conditional identity tan A/2.tan B/2 + tan B/2.tan C/2 + tan C/2. tan A/2 = 1, for A + B + C = pi. That made it easy
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