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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.
The derivative of f is f′(x)=cosx. Now use Newton's Method:
xn+1=xn−f(xn)f′(xn) ⟶
xn+1=xn−sinxncosxn ⟶
xn+1=xn−tanxn
I will choose to let x0=2. Using Newton's Method formula for 7 iterations in a spreadsheet results in :
Thus π to ten decimal places is π≈3.1415926536.
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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