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Math Help - trig applications # 3

  1. #1
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    trig applications # 3

    Points A and B lie on a circle, centre O, radius 5 cm. Find the value of angle AOB that produces a maximum area for the triangle AOB.

    This is my work and attempt:

    So I will describe my diagram. It is a circle, with a triangle inside (not right angled) in the middle of the circle is a dot labelled O. From here I randomly drew point A and point B, on the edge of the circle and then connected points AOB to form a triangle in the circle. Side OB will be labelled 5 cm as well as AO for radius.

    Then in the middle of AB i chopped it in half and drew a line down the middle to point O to form two right angled triangles.

    Now I can begin:
    cosx=a/h
    cosx=a/5
    5cosx=a

    sinx=o/h
    sinx=o/5
    5sinx=o

    Max area of triangle
    A=0.5 b h
    =(0.5) 5cosx (5sinx)
    =12.5(cosxsinx)

    A'=-12.5(sin^(2)x-cos^(2)x)

    A'=0 when x=pi/4
    But this is only half of theta or angle O, so times two.
    pi/4 x 2
    =pi/2

    Now this answer is correct with the back of the book BUT It doesn't work so well because when i plug it back into my area formula, it turns out the angle is 0.. which cannot be max?

    A(0)=0
    A(pi/2)=0
    So my area formula must be wrong.

    What is wrong from my work and how do i fix it? demonstrate.
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  2. #2
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    A = \frac{1}{2}r^2\sin{\theta}
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  3. #3
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    Quote Originally Posted by skeeter View Post
    A = \frac{1}{2}r^2\sin{\theta}
    how did you get the base and height to be that?
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  4. #4
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    Quote Originally Posted by skeske1234 View Post
    how did you get the base and height to be that?
    Math B - Area of Triangle Using Trigonometry
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  5. #5
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    shouldn't it be
    A=1/2 (r^2) (5sinx)?
    How did you just get sinx using the SAS rule?
    like even if you use the SAS rule, that means that you know two sides of a triangle and an angle inbetween.. i know that, but my question is how is the formula for that derived (sinx) to represent the height of the triangle?
    and another question, r^2 represents the diamater correct?
    Last edited by skeske1234; August 19th 2009 at 05:37 AM.
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  6. #6
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    Quote Originally Posted by skeske1234 View Post
    shouldn't it be
    A=1/2 (r^2) (5sinx)?
    How did you just get sinx using the SAS rule?
    like even if you use the SAS rule, that means that you know two sides of a triangle and an angle inbetween.. i know that, but my question is how is the formula for that derived (sinx) to represent the height of the triangle?
    and another question, r^2 represents the diamater correct?
    r represents the radius.
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