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        riddles >> putnam exam (pure math) >> Polar Coordinates
        
(Message started by: mikedagr8 on Feb 6th, 2008, 12:54am)
Title: Polar Coordinates
Post by mikedagr8 on Feb 6th, 2008, 12:54am
Could I get an explanation as to why the answer I get is different to what the intended answer is.

y2=8x

My working is as follows.
y2-8x=0
r2sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif-8rcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif=0
r (rsin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif- 8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)=0
r=0 or rsin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif-8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif=0

The answer intended is (8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif.
Am I missing something or is it something I have not been taught. If it is somethign I have not been taught like cot and sec (I heard these terms) then just say so and I will be fine.
Thanks guys.
Title: Re: Polar Coordinates
Post by pex on Feb 6th, 2008, 1:01am

on 02/06/08 at 00:54:01, mikedagr8 wrote:
Could I get an explanation as to why the answer I get is different to what the intended answer is.

y2=8x

My working is as follows.
y2-8x=0
r2sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif-8rcoshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif=0
r (rsin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif- 8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)=0
r=0 or rsin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif-8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif=0

The answer intended is (8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif.
Am I missing something or is it something I have not been taught. If it is somethign I have not been taught like cot and sec (I heard these terms) then just say so and I will be fine.
Thanks guys.

As long as http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif is not a multiple of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif, your result r sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif - 8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 0 is equivalent with r = (8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/(sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif). And the r=0 case is covered by, for instance, http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2.
Title: Re: Polar Coordinates
Post by pex on Feb 6th, 2008, 1:04am
And just to clarify, cot and sec are nothing special: cothttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 1/tanhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, sechttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 1/coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif, and cschttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 1/sinhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif.
Title: Re: Polar Coordinates
Post by mikedagr8 on Feb 6th, 2008, 1:17am
Thanks for that. Is this for all cases that are not multiples of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifthat I can write my second term over my first if they are being subtracted? Would it be multiplied if it were added?
Thanks again.
Title: Re: Polar Coordinates
Post by pex on Feb 6th, 2008, 1:27am

on 02/06/08 at 01:17:06, mikedagr8 wrote:
Thanks for that. Is this for all cases that are not multiples of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifthat I can write my second term over my first if they are being subtracted? Would it be multiplied if it were added?
Thanks again.

???

It is a simple rearranging of terms:
r sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif - 8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 0
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif r sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif r = (8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/(sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif).

The condition that http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif is not a multiple of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif ensures that we are not dividing by zero, that's all.

So addition would not lead to multiplication, but to another division:

r sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif + 8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = 0
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif r sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif = -8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif
http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bigto.gif r = (-8coshttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif)/(sin2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif).
Title: Re: Polar Coordinates
Post by mikedagr8 on Feb 6th, 2008, 1:31am
Of course!
Sorry about that. Mental blank.
I still appreciate the help.
Thanks.


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