Of the infinitely many lines that are tangent to the curve

Question

y = −4 sin x
and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.

Answer 1

A quick look at the graph shows that the point we want will be somewhere in the interval [pi,2pi].

The line through (0,0) and (x,-4sinx) is

y = -4sinx/x

we also know that at any point x, the slope is

y' = -4cos(x)

So, we need

-4sin(x)/x = -4cos(x)
tanx = x

So, let f(x) = tanx-x

Pick x = 4.7 (near 3pi/2, which is an asymptote) as the initial guess, and apply Newton's method to that. I get x=4.493

Answer 2

That's not correct Steve that's not to six decimal places like the problem says