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Math Help - [SOLVED] Trig chord angle problem

  1. #1
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    [SOLVED] Trig chord angle problem

    This should be simple but its got me stumped so I could use some help.
    Code:
    :****                :
    :       ***          :
    :             **   + :          : is the tangent line
    :              +\*   :          + marks the line in question
    :           +    \ * :          \ marks the chord line
    :        +        \ *:          - marks the radius line
    :     +            \ *
    :  +                \* circle
    :---------r---------*

    I have a circle of radius r. A line at 0degrees is drawn from the center to the circle edge. At the point of intersection a tangent line can be drawn. There is a chord line of known length (ra) that bisects the line between the radius line and the tangent line. Since the radius of the circle is known and the length of the chord line is know how can I calculate either the angle theta (angle between the two lines) or the angle of the chord line or the x and y point where the chord line (re)crosses the circle. sorry the drawing is so bad... hard to do with ascii art

    A simple metaphor is this: You grab a slice of pizza. You have a tape measure and measure the length of the side of the piece and you measure across the widest point at the top of the piece - remember the second width is not the arc length as you just measured the straight distance across the piece. Given these two how do you determine the angle of the piece, or the angle of incident between one side and the straight line that connects the two end points of the piece.

    thx in advance
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  2. #2
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    partial solution

    ok I think I got it but would appreciate if someone could check my math.

    I'm using the law of cosines to solve it which states: a^2=b^2+c^2-2bccos(A)
    Code:
        /C\ 
       /   \
     b/     \a
     /       \
    /A_______B\
        c
    
    A is the angle theta so to speak and is what I need to know. a is the length of the chord which was given as ra. Since both b and c are from the midpoint to the circle they have the same length as the radius.

    so:
    a=ra
    b=r
    c=r

    substituting into law of cosine:
    ra^2=r^2+r^2-2r^2cos(A)
    ra^2=2r^2-2r^2cos(A)
    ra^2=2r^2(1-cos(A))
    ra^2/2r^2-1=-cos(A)
    cos(A)=1-ra^2/2r^2
    A=Acos(1-ra^2/2r^2)

    to solve for B
    use the law of sines: a/sin(A)=b/sin(B)=c/sin(C)

    substitute
    ra/sin(A)=r/sin(B)
    ra/r*sin(A)=1/sin(B)
    sin(B)=r*sin(A)/ra
    B=Asin(r*sin(A)/ra)
    or can be completely substituted
    B=Asin(r*sin(Acos(1-ra^2/2r^2))/ra)
    can this be further factored?

    thx
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  3. #3
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Quote Originally Posted by hyperkinetic View Post
    This should be simple but its got me stumped so I could use some help.
    Code:
    :****                :
    :       ***          :
    :             **   + :          : is the tangent line
    :              +\*   :          + marks the line in question
    :           +    \ * :          \ marks the chord line
    :        +        \ *:          - marks the radius line
    :     +            \ *
    :  +                \* circle
    :---------r---------*

    I have a circle of radius r. A line at 0degrees is drawn from the center to the circle edge. At the point of intersection a tangent line can be drawn. There is a chord line of known length (ra) that bisects the line between the radius line and the tangent line. Since the radius of the circle is known and the length of the chord line is know how can I calculate either the angle theta (angle between the two lines) or the angle of the chord line or the x and y point where the chord line (re)crosses the circle. sorry the drawing is so bad... hard to do with ascii art

    A simple metaphor is this: You grab a slice of pizza. You have a tape measure and measure the length of the side of the piece and you measure across the widest point at the top of the piece - remember the second width is not the arc length as you just measured the straight distance across the piece. Given these two how do you determine the angle of the piece, or the angle of incident between one side and the straight line that connects the two end points of the piece.

    thx in advance
    I really had a hard look on your question and the posted diagram. ra = re?
    I got lost on the question, and I also got lost on the diagram.

    So let us talk on the metaphor.
    "...how do you determine the angle of the piece, or the angle of incident between one side and the straight line that connects the two end points of the piece."

    The straight line that connects the two end points of piece?

    You want to know the angle between the two radii? Or the angle between one radius and the chord?

    I am still confused, but since you really showed you want help, then here is what I can give you.....according to my understanding so far on your descriptions.

    Let us call the two lines that forms the V of the pizza as r each. They are the radii of the circle from where the pizza is cut.
    And let us call the chord as x. I think you called it "ra".

    Let us draw another radius that will bisect, or divide equally, the x.
    Now there are two equal right triangles formed. Each one of them has:
    one leg is half of the x, so it is x/2.
    the other leg is a part of the radius that cut the x. Ignore this for we don't need it.
    hypotenuse is r.
    angle between the other leg and the hypotenuse is half of the angle you want to find.....let us call it theta/2.

    So,
    sin(theta/2) = (x/2) / r
    sin(theta/2) = x / (2r)
    theta/2 = arcsin(x / 2r)
    theta = 2*arcsin(x / 2r) ---------answer.

    You know the x, you know the r, so you should be able to get the theta.
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  4. #4
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    Sep 2008
    Posts
    3
    Occams Razor... some how I missed that very simple solution - thx
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