Given a number d, find a fraction of the form p/q with q at most equal to d which is closest to π.
For example 22/7 is the closest with d=10.
Here, we define π as the smallest positive number such that sin(π)=0. Moreover, we can assume that we already know that π lies between 3 and 4.
Hint: Near π the sine function decreases linearly. Hence, the value of |sin(p/q)| can be taken to be a measure of how close p/q is to π when p/q is near π.
Note that it is good to think of a method that does not use any other aspect of 
. This will allow you to adapt your method to find the nearest rational number to a positive root of (say) 
, by a small modification of your program.
Using Sin(pi)=0 as our definition of pi , and solving the taylor approximation of sin(x), using the Newton-Raphson algorithm, will probably give us the answer.
Suroj, just carefully look at the hint given by Professor Paranjape. He has essentially given you the algorithm. Nothing like Newton-Raphson and Taylor series should be needed for it. Essence of the hint lies in understanding the behaviour of sin in the vicinity of 
.
Why doesn't someone write a Python code for it?
import math
d = 10000
err = abs(math.sin(3))
pgood = 3
qgood = 1
for q in range(1,d+1):
for p in range(3*q,4*q+1):
newerr = abs(math.sin(float(p)/float(q)))
if newerr < err:
err = newerr
pgood = p
qgood = q
print pgood,"/",qgood
Questions
as the smallest real number 
for which 
? as the smallest
real number for which 
? would you trust most? The one using sine, tangent or cosine function?Keep 
-thoning!
Isn't this better:
import math
d=input("Enter the maximum value of d:")
min=math.pi-3
pc=3
qc=1
for q in range(1,d+1):
for p in range(3*q,4*q):
l=abs(math.pi-float(p)/q)
if l<min:
min=l
pc=p
qc=q
print str(pc)+"/"+str(qc)
Is the answer cos(x/2)?
I think because the slope is smaller in magnitude (i.e. 1/2) around x = pi for y = cos(x/2), but it is 1 for y = sin(x) or y = tan(x), so the function changes slowly in the first case and there is less scope for error.
The greater is the slope, the lesser is the error based only on comparing the values of the function!