August 6th, 2012, 09:09 AM  #1 
Member Joined: Sep 2007 Posts: 77 Thanks: 0  Area of a Circle problem
The area of a circle i. Let r be a positive real number and let O = (0, 0) and P = (r, 0) be points in the coordinate plane. Let ?? be the (half)line that emanates from O and makes an angle ? with respect to the positive xaxis (see figure in attachment). Let Q be the point on ?? whose distance to O is r and let Q' be the point on ?? whose perpendicular projection onto the positive xaxis is P (see figure in attachment). Find the x and y coordinates of Q and Q'. ii. Partition the angle 2? into N equal parts of size ? and let Cr = area of a circle with radius r, A = area of the triangle ?OPQ, A' = area of the triangle ?OPQ', where O,P,Q,Q' are as in the above figure with angle ? = ?. (a) Show that A = r^2 sin(?) / 2 and A' = r^2 tan(?) / 2 (b) Explain why the inequalities Nr2 sin(?) / 2 ? Cr ? Nr2 tan(?)/ 2 hold for any r > 0 and any natural number N ? 3. (c) Find an expression for N in terms of ? and take the limit as ? > 0 in (7) to find Cr. Can anyone help get me started on this? Thank you! 
August 6th, 2012, 11:15 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Area of a Circle problem
It is straightforoward that the x coordinate of Q' is simply r. Now, So, the y coordinate of Q' is Can you proceed for Q now? 
August 6th, 2012, 11:59 AM  #3 
Member Joined: Sep 2007 Posts: 77 Thanks: 0  Re: Area of a Circle problem
From there, I know that the coordinates of QP are (rcos x, rsinx). To find the area, I need to take the antiderivative of the the function. However, even though we have the two coordinates, how do I know which formula to take the antiderivative of?

August 6th, 2012, 01:42 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Area of a Circle problem
10.) i) We may use the definition of sine and cosine to find: Since we then find ii) a) Using the formula for the area of a triangle we then find: b) For any natural number and the geometrical interpretation of the given inequality is that for a circle of radius r, its area is greater than an inscribed Ngon and less than that of a circumscribed Ngon. c) and hence: 
August 6th, 2012, 01:47 PM  #5 
Member Joined: Sep 2007 Posts: 77 Thanks: 0  Re: Area of a Circle problem
Thank you for the help! As you can see from my prior post, I was really off from what I needed to do, so I appreciate the help in preparation for my final. Thank you thank you thank you!

August 6th, 2012, 02:02 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Area of a Circle problem
To be more formal about the last part, we could state: Apply L'Hpital's rule to the limits (although this isn't necessary, just quicker): 

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