Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{159 x^{3}}{200} - 4$ using $LaTeX: \displaystyle x_0=1$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{159 x_{n}^{3}}{200} + \cos{\left(x_{n} \right)} + 4}{- \frac{477 x_{n}^{2}}{200} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 1$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (1.0000000000) - \frac{- \frac{159 (1.0000000000)^{3}}{200} + \cos{\left((1.0000000000) \right)} + 4}{- \frac{477 (1.0000000000)^{2}}{200} - \sin{\left((1.0000000000) \right)}} = 2.1608045829$ $LaTeX: x_{2} = (2.1608045829) - \frac{- \frac{159 (2.1608045829)^{3}}{200} + \cos{\left((2.1608045829) \right)} + 4}{- \frac{477 (2.1608045829)^{2}}{200} - \sin{\left((2.1608045829) \right)}} = 1.7783182796$ $LaTeX: x_{3} = (1.7783182796) - \frac{- \frac{159 (1.7783182796)^{3}}{200} + \cos{\left((1.7783182796) \right)} + 4}{- \frac{477 (1.7783182796)^{2}}{200} - \sin{\left((1.7783182796) \right)}} = 1.6988734037$ $LaTeX: x_{4} = (1.6988734037) - \frac{- \frac{159 (1.6988734037)^{3}}{200} + \cos{\left((1.6988734037) \right)} + 4}{- \frac{477 (1.6988734037)^{2}}{200} - \sin{\left((1.6988734037) \right)}} = 1.6955970803$ $LaTeX: x_{5} = (1.6955970803) - \frac{- \frac{159 (1.6955970803)^{3}}{200} + \cos{\left((1.6955970803) \right)} + 4}{- \frac{477 (1.6955970803)^{2}}{200} - \sin{\left((1.6955970803) \right)}} = 1.6955916293$