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 xaxis = \(\displaystyle \theta\) (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 xaxis 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?
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 xaxis = \(\displaystyle \theta\) (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 xaxis 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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