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SOLUTION 4: We know from trigonometry that  sinπ=0, so I will choose to solve the equation  sinx=0  for x. Let's define function f(x)=sinx, whose graph is given below.

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The derivative of f is f(x)=cosx. Now use Newton's Method: xn+1=xnf(xn)f(xn)     xn+1=xnsinxncosxn     xn+1=xntanxn I will choose to let x0=2. Using Newton's Method formula for 7 iterations in a spreadsheet results in :

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Thus  π  to ten decimal places is  π3.1415926536.

IMPORTANT NOTE: Choosing a different initial guess can lead to other solutions to the equation  sinx=0 . For example, if I choose to let x0=1, then Newton's Method leads to the solution  x=0 . Using Newton's Method formula for iterations in a spreadsheet results in :

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