Trig identities

  1. Trig identities plz help

    In triangle ABC if sin (A/2) sin (B/2) sin (C/2) = 1/8
    prove that the triangle is equilateral plz show steps
     
  2. jcsd
  3. James R

    James R 562
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    Perhaps you can use the sine rule (?)
     
  4. Sure .. I tried but had no success .. If you find an answer plz post your steps
     
  5. one method suggested:

    (an absurd reasonning)

    if it is an equilateral triangle then:
    A = B = C = pi/3 rad

    implies ---> A/2 = B/2 = C/2 = pi/6 rad

    implies ---> sin(A/2) = sin(B/2) = sin(C/2) = 1/2

    implies ---> sin(A/2)sin(B/2)sin(C/2) = 1/2*1/2*1/2 = 1/8

    thus it is indeed an equilateral triangle

    if i come with another one i will post it :)
    hope it will help
     
  6. VietDao29

    VietDao29 1,422
    Homework Helper

    Try expand the equation
    [tex]\sin{\frac{A}{2}}\sin{\frac{B}{2}}\sin{\frac{C}{2}} = \frac{1}{8}[/tex]
    to
    [tex]4\sin{\frac{C}{2}}^{2} - 4\sin{\frac{C}{2}}\cos{\frac{A - B}{2}} + 1 = 0[/tex]
    Then to
    [tex](2\sin{\frac{C}{2}} - \cos{\frac{A - B}{2}})^{2} + (\sin{\frac{A - B}{2}})^{2} = 0[/tex]
    Now you have something like [itex]A^{2} + B^{2} = 0[/itex] so
    [tex]\left\{ \begin{array}{c} A = B \\ \sin{\frac{C}{2}} = \frac{1}{2}\cos{\frac{A - B}{2}} \end{array}\right[/tex]
    So you will have A = B = C = 60 degrees, which implies the triangle ABC is equilateral.
    Hope it help.
    Viet Dao,
     
  7. For AI thank you but this won't do
     
  8. For VietDao29 If A^2 + B^2 = 0 then we're stuck because no two +ve numbers addto zero ...right ? Then it should be A^2 = - B^2
    How did you expand 1st step
    How did you get last step
    plz go in more details
     
  9. dextercioby

    dextercioby 12,314
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    Both A and B are real.So their square is larger or equal to zero.In order for the sum of the squares to be 0,each if the squares must be 0.

    Daniel.
     
  10. Well that's a good point. How did I miss that :) Now for the first step plz how did we expand Sin (A/2) Sin (B/2) Sin (C/2) to next step
    ie. How to start .... the rest is ok
     
  11. dextercioby

    dextercioby 12,314
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    Use this IDENTITY:

    [tex] \sin x\sin y\equiv \frac{1}{2}[\cos(x-y)-\cos(x+y)] [/tex]

    The result is immediate.

    Daniel.
     
  12. Tom Mattson

    Tom Mattson 5,538
    Staff Emeritus
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    Try starting by eliminating a variable. Since you know that A, B, and C are all in the same triangle, you have:

    A+B+C=180
    C=180-A-B

    See where that gets you.
     
  13. Thank you all ... I can do it now following your steps
    The rule supplied by Dextercioby did not look familiar (but it's correct I checked) ..well memory is not what it used to be :) isn't that a bit complicated though ... I thought the answer should be more straight forward .. any way thank you all again
     
  14. dextercioby

    dextercioby 12,314
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    That's interesting.The checking part.I've said IDENTITY. :wink: There may have been a chance i didn't invent it,but either picked it from a book or deduced starting other identities (which i have actually done).

    Daniel.
     
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