Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \sin{\left(x \right)}= \frac{43 x^{3}}{200} - 8$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{43 x_{n}^{3}}{200} + \sin{\left(x_{n} \right)} + 8}{- \frac{129 x_{n}^{2}}{200} + \cos{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{43 (3.0000000000)^{3}}{200} + \sin{\left((3.0000000000) \right)} + 8}{- \frac{129 (3.0000000000)^{2}}{200} + \cos{\left((3.0000000000) \right)}} = 3.3438002337$ $LaTeX: x_{2} = (3.3438002337) - \frac{- \frac{43 (3.3438002337)^{3}}{200} + \sin{\left((3.3438002337) \right)} + 8}{- \frac{129 (3.3438002337)^{2}}{200} + \cos{\left((3.3438002337) \right)}} = 3.3146178206$ $LaTeX: x_{3} = (3.3146178206) - \frac{- \frac{43 (3.3146178206)^{3}}{200} + \sin{\left((3.3146178206) \right)} + 8}{- \frac{129 (3.3146178206)^{2}}{200} + \cos{\left((3.3146178206) \right)}} = 3.3144010175$ $LaTeX: x_{4} = (3.3144010175) - \frac{- \frac{43 (3.3144010175)^{3}}{200} + \sin{\left((3.3144010175) \right)} + 8}{- \frac{129 (3.3144010175)^{2}}{200} + \cos{\left((3.3144010175) \right)}} = 3.3144010056$ $LaTeX: x_{5} = (3.3144010056) - \frac{- \frac{43 (3.3144010056)^{3}}{200} + \sin{\left((3.3144010056) \right)} + 8}{- \frac{129 (3.3144010056)^{2}}{200} + \cos{\left((3.3144010056) \right)}} = 3.3144010056$