This is Numerical Analysis (Theory of Approximation)

Consider the function Fm y) = (992 + 23)\" y- The purpose of this project is to approximate the level curve F (:19, y) = C by a trigono metric polynomial. Here 0 is 10\""1 times the number formed by the last 4 digits of your student ID. (For example, if your student ID number is 12345678, then C = 104 X5678 = 0.5678.) In the polar coordinates (T, (,0) the equation F (39, y) = C takes the form 7'10 rsinr,0 = C, 1" 6 (0,00), (,0 E [0, 271'). (1) This equation implicitly denes r as a function of (,0, so that the polar graph of T(p) is the level curve of F. In the equation (1) replace 0 by its value (by the one you found based on your student ID) and then do the following steps. Step 1. Prove that for any xed (,0 E [0, 27r) the equation (1) has exactly one positive solution 7' = r(