Simple question but stumping me - on ratio of areas

Apr 2009
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 = \(\displaystyle \theta\) (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 \(\displaystyle 2 \theta / \pi\)

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

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?


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