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Law of Cosines oddities

  1. Feb 24, 2007 #1
    1. The problem statement, all variables and given/known data
    Solve triangle having indicated sides and angles.
    a=6
    b=9
    C=45degrees


    2. Relevant equations
    Law of Cosines
    Law of Sines


    3. The attempt at a solution
    I did c^2=9^2+6^2-(2*9*6*Cos45 degrees)
    c=6.37
    ...
    Then I did 6.37/cos45 =9/SinB.. That proportion would give me 87.5 derees.. However, the back of my book and..
    http://www.trig.ionichost.com/
    Says that B is 93.3 degrees..

    However, if I were to approach the problem by solving for A (smallest angle) instead of immediately to B (mid sized).. I would get 41.7 degrees..

    Then I could do B=180-A-C..or 93.3 degrees..

    But still, if I set up a ratio of sin41.7degs/6=sinB/9, that's not 93.3, or 87.5, but 86.2

    I'd like to know why math is stupid. And next time, what I should do to know whether 93.3, 87.5, or 86.2 would be the right answer. :smile:
     
  2. jcsd
  3. Feb 25, 2007 #2

    AlephZero

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    Science Advisor
    Homework Helper

    Your problem is you are not working accurately enough.

    My calculator gives c = 6.37456

    If you use c = 6.37 you get sin B = (9 sin 45)/ 6.37 = 0.99905

    If you use c = 6.37456 you get sin B = 0.9983373

    If you look at the graph of sin x, it is almost "horizontal" when x is near to 90 degrees and sin x is nearly 1.0

    if sin B = 0.9983373 then B = 86.70 or 93.30

    if sin B = 0.99905 then B = 87.50 or 92.50

    The small change in the value of sin B causes a big change in the value of B

    When you found the smallest angle of the triangle by the sine rule, you avoided this problem for two reasons:

    1 the slope of the graph of sin x is steeper so there is less error in going from sin x to x
    2 you know the smallest angle of a triangle must be less than 90 degrees, so the other solution (A = 130.3) is not possible.
     
    Last edited: Feb 25, 2007
  4. Feb 25, 2007 #3
    Ok.. So when given SSA, always find the S of the given A.. then find the A of the other smallest S?
     
  5. Feb 25, 2007 #4

    AlephZero

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    Yes that would work.

    A more general message to take away from this is: If you are working with any function and you are close to the point where the graph is horizontal (zero slope), then be careful if you are using the inverse function. For example sin x near 90 degrees, or cos x near 0 degrees.
     
  6. Feb 25, 2007 #5
    Also!
    You can only have the possibility of having 2 solutions with SSA, right?
     
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