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Math Help - Simple question but stumping me - on ratio of areas

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    Simple question but stumping me - on ratio of areas

    I thought this would be a simple question but when I thought more I got more confused.

    Here is the question -

    1. Consider first quadrant of cartesian plane.
    2. Draw a ray - starting from (0,0) and extending to infinty. Angle this ray makes with +ve x-axis = \theta (in radians)
    3. This ray divides the area first quadrant in two parts
    4. Question - What is the ratio of area between the ray and x-axis to the area of first quadrant?

    I have tried to solve this problem. First I just drew a circle with center at (0,0) and radius 'r' and found the ratio. The answer to the question asked should be the limit of this ratio when r tends to infinite. It came to 2 \theta / \pi

    Then I though why circle? Why can't I draw a square? When I did that I found answer is Tan(\theta)/2

    Please see the attached image for my working - it is pretty straight forward.

    Now this rang a bell. What is going wrong? Which answer is correct? Infact depending on what initial figure I choose to draw (circle/sqaure/rectangle) the answer varies? Is there a limit or limit is not defined? What is the correct / rogoruos mathematical argument?

    Any help please?
    Attached Thumbnails Attached Thumbnails Simple question but stumping me - on ratio of areas-untitled.pdf  
    Last edited by aman_cc; August 6th 2010 at 03:40 AM.
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