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How many solutions? (Posted on 2007-01-31) Difficulty: 4 of 5
a) Find the number of solutions of the equation sin(x)=x/573204.

b) Find the approximate difference between the two largest solutions.

No Solution Yet Submitted by Art M    
Rating: 3.7500 (4 votes)

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Solution solution (spoiler) | Comment 2 of 3 |

Part a)

Let's start with the solutions where x>=0.

573,204/(2*pi) ~= 91,228.25000004677, so when x/573,204 reaches a value of 1, the sine function has gone through just over 91,228.25 cycles. Through each cycle, including that just-bigger-than-a-quarter-cycle, there are two points where the sine curve intersects the straight line, so this accounts for 91,229 * 2 = 182,458 solutions.

Both functions are odd, so there are just as many solutions with x<=0, but one of the solutions is in fact at x=0, common to both sets, so there are 182,458 + 182,457 = 364,915 solutions altogether

Part b)

The following program first brackets 91228*2*pi and 91228.25*2*pi to find the next-to-last solution, by doing a binary search, and then does the same between 91228.25*2*pi and 91228.251*2*pi to find the last solution. One solution is then subtracted from the other.

   5   point 10
  10   Number=573204/(2*#pi)
  15   print Number
  20   High=91228.25*2*#pi:Low=91228*2*#pi
  25   Diff=10
  30   while High>Low and abs(Diff)>0
  40     Avg=(High+Low)/2
  50     Diff=sin(Avg)-Avg/573204
  60     if Diff>0 then High=Avg
  70     if Diff<0 then Low=Avg
  80   wend
  90   Sol1=Avg
 120   High=91228.251*2*#pi:Low=91228.25*2*#pi
 125   Diff=10
 130   while High>Low and abs(Diff)>0
 140     Avg=(High+Low)/2
 150     Diff=sin(Avg)-Avg/573204
 160     if Diff<0 then High=Avg
 170     if Diff>0 then Low=Avg
 180   wend
 190   Sol2=Avg
 200   print Sol1:print Sol2:print Diff:print Sol2-Sol1
 
 

This finds:
 
  91228.250000046774044067248700178722375600406777015828
  573203.999995944357426571559091389974452083351136109094
  573203.999999978703418122669322463268847448704168081184
  0
  0.00000403434599155111023107329439536535303197209
 
 where the first line indicates how many cycles it takes for the linear function to reach 1, the second line is the penultimate solution, the third line is the last solution, the zero confirms a solution really was found, and the last line shows the difference between the two largest solutions.
 
 I would not trust the last few digits, due to rounding of the narrow-gapped functions, so I've bolded some trustworthy parts above.  So the part b answer would be 0.000004034345991551110231073294395365353.


  Posted by Charlie on 2007-01-31 11:26:34
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