To apply Newton’s Method we take the function f(x)=sin(x)-cos(2x) and then we find its zeroes.

The first step is to find f'(x)=df/dx=cos(x)+2sin(2x). This is the gradient at (x,f(x)) and is used to calculate progressively more accurate values for the zero or zeroes.

We need an initial value for x, so since the interval is [0,π/2] we could choose an initial value x₀=0. Then we calculate x₁=x₀-f(x₀)/f'(x₀). We get x₂ by using x₁ in place of x₀ in the formula, and so on. x is in radians.

So we have:

x₀=0, x₁=0-f(0)/f'(0)=-(-1)/1=1.

x₁=1, x₂=1-f(1)/f'(1)=1-1.2576/2.3589=0.4669.

x₂=0.4669

x₃=0.5236

x₄=0.5236, so we have reached 3 decimal place accuracy: x=0.524.

The exact solution is:

sin(x)-1+2sin²(x)=0=(2sin(x)-1)(sin(x)+1).

Therefore sin(x)=0.5, x=π/6=0.5236.

So Newton’s Method gave us an accurate solution.