Use Newton's Method to approximate the smallest positive solution of sin(x) + cos(x)=0

Question

Use Newton's Method to approximate the smallest positive solution of sin(x) + cos(x)=0. Round x values to 5 decimal places.

I got x₁ = 1 + [sin(1)+cos(1)] / [cos(1) - sin(1)] = 2.7159

Is this the right way to use Newton's Method and is that the correct answer for the first try? I'm stuck between x₃ and x₄ (3rd and 4th continuation), as my x₃ is = 3.9976 but my x₄ somehow is = - 12.5831 and that doesn't seem right...

Answer 1

I think you missed a sign in the formula.

Then look at the graph...the root lies between 2pi/3 and 5pi/6.

I would take the first approximation as 2.36

The second iteration for me is 2.355284 and, of course, the exact root is 3pi/4 = 2.356194