Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \sin{\left(x \right)}= \frac{977 x^{3}}{1000} - 9$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{977 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 9}{- \frac{2931 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{977 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 9}{- \frac{2931 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 2.3701675356$ $LaTeX: x_{2} = (2.3701675356) - \frac{- \frac{977 (2.3701675356)^{3}}{1000} + \sin{\left((2.3701675356) \right)} + 9}{- \frac{2931 (2.3701675356)^{2}}{1000} + \cos{\left((2.3701675356) \right)}} = 2.1774423944$ $LaTeX: x_{3} = (2.1774423944) - \frac{- \frac{977 (2.1774423944)^{3}}{1000} + \sin{\left((2.1774423944) \right)} + 9}{- \frac{2931 (2.1774423944)^{2}}{1000} + \cos{\left((2.1774423944) \right)}} = 2.1591384668$ $LaTeX: x_{4} = (2.1591384668) - \frac{- \frac{977 (2.1591384668)^{3}}{1000} + \sin{\left((2.1591384668) \right)} + 9}{- \frac{2931 (2.1591384668)^{2}}{1000} + \cos{\left((2.1591384668) \right)}} = 2.1589787906$ $LaTeX: x_{5} = (2.1589787906) - \frac{- \frac{977 (2.1589787906)^{3}}{1000} + \sin{\left((2.1589787906) \right)} + 9}{- \frac{2931 (2.1589787906)^{2}}{1000} + \cos{\left((2.1589787906) \right)}} = 2.1589787785$