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Math Help - Trigonometry

  1. #1
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    Smile Trigonometry

    This is a question involving Non-right angled triangle trigonometry...
    I hope you can help me solving this question.
    Thanks,
    Amine

    A boat sailing from A to B travels in a straight line until the captain realises he is off course. The boat is turned through an angle of 60 degrees, and then travels another 10 km to B. The trip would have been 4 km shorter if the boat had gone straight from A to B. How far did the boat travel?
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  2. #2
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    Non-right angled triangle

    Hi-

    A boat sailing from A to B travels in a straight line until the captain realises he is off course. The boat is turned through an angle of 60 degrees, and then travels another 10 km to B. The trip would have been 4 km shorter if the boat had gone straight from A to B. How far did the boat travel?
    Suppose that the boat has sailed x km to a point C before the captain changes course. So you can draw a triangle ABC with the following information:

    • AC = x km
    • CB = 10 km
    • Angle ACB = 180 - 60 = 120 degrees
    • AB is 4 km less than (x + 10). So?

    You can now use the Cosine Rule on this triangle: AB^2=AC^2 +  BC^2 -2.AC.BC.cos C

    Now remove brackets and simplify. (Be careful with the cosine of 120 degrees - it's negative.) If you do it right, it turns out to be a very easy equation for x.

    If yours doesn't work out to be easy, let me know, and I'll give you a bit more help.

    Grandad
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  3. #3
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    Hi, thanks for the help!
    My equation turned out to be:

    AB^2 = X^2 + 10^2 - 2 * X * 10 * cos(120)
    Is this correct, or have I already done something wrong?
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  4. #4
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    Hello, Aminekhadir!

    My equation turned out to be:
    . . AB^2 \:= \:x^2 + 10^2 - 2(x)(10)\cos120^o

    Is this correct? . . . . well yes, but . . .
    Code:
                                      *
                              C   *
                              *  60
                     x    * 120*
                      *           * 10
                  *                 *
              *                       *
          * - - - - - - - - - - - - - - *
          A           x + 6             B

    Distance AB is 4 km less than x + 10.
    . . Hence: . AB \:=\:x+6

    Law of Cosines: . (x+6)^2 \;=\;x^2 + 10^2 - 2(x)(10)\cos120^o

    This comes out to: . x \:=\:32


    Don't forget to answer the question.

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  5. #5
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    Hi,
    Thanks a lot for helping ,
    but are you completely sure that the answer is 32? Isn't it 42? :S
    Or am I doing something wrong again?
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  6. #6
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    Non-right angled triangle

    Hello Aminekhadir

    Yes, the answer is 42, but x is 32, as Soroban said. Can you see why?

    Grandad
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  7. #7
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    yes, I can see it now.
    Thanks.
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