Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \sin{\left(x \right)}= \frac{169 x^{3}}{1000} - 6$ using $LaTeX: \displaystyle x_0=3$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{169 x_{n}^{3}}{1000} + \sin{\left(x_{n} \right)} + 6}{- \frac{507 x_{n}^{2}}{1000} + \cos{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 3$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (3.0000000000) - \frac{- \frac{169 (3.0000000000)^{3}}{1000} + \sin{\left((3.0000000000) \right)} + 6}{- \frac{507 (3.0000000000)^{2}}{1000} + \cos{\left((3.0000000000) \right)}} = 3.2841927139$ $LaTeX: x_{2} = (3.2841927139) - \frac{- \frac{169 (3.2841927139)^{3}}{1000} + \sin{\left((3.2841927139) \right)} + 6}{- \frac{507 (3.2841927139)^{2}}{1000} + \cos{\left((3.2841927139) \right)}} = 3.2642785341$ $LaTeX: x_{3} = (3.2642785341) - \frac{- \frac{169 (3.2642785341)^{3}}{1000} + \sin{\left((3.2642785341) \right)} + 6}{- \frac{507 (3.2642785341)^{2}}{1000} + \cos{\left((3.2642785341) \right)}} = 3.2641796856$ $LaTeX: x_{4} = (3.2641796856) - \frac{- \frac{169 (3.2641796856)^{3}}{1000} + \sin{\left((3.2641796856) \right)} + 6}{- \frac{507 (3.2641796856)^{2}}{1000} + \cos{\left((3.2641796856) \right)}} = 3.2641796831$ $LaTeX: x_{5} = (3.2641796831) - \frac{- \frac{169 (3.2641796831)^{3}}{1000} + \sin{\left((3.2641796831) \right)} + 6}{- \frac{507 (3.2641796831)^{2}}{1000} + \cos{\left((3.2641796831) \right)}} = 3.2641796831$