Consider the roots of the equation

(a) Write Newton’s method for computing x^(k+1) from the previous value x^(k) for this problem.

(b) Simplify your result for large x^(k).

(c) Assume you have computed root x^∗_n of this equation for some large value of x using Newton’s method. Use the result from part (b) to explain why x⁽⁰⁾_n+1 = x^∗_n + π is a good guess for the next root when using Newton’s method. You may (or may not) find the following trigonometric identities useful.

sin(a + b) = sin(a) cos(b) + cos(a) sin(b)

sin(a − b) = sin(a) cos(b) − cos(a) sin(b)

cos(a + b) = cos(a) cos(b) − sin(a) sin(b)

cos(a − b) = cos(a) cos(b) + sin(a) sin(b)

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