Use Newton's method to find the first 5 approximations of the solution to the equation $LaTeX: \displaystyle \cos{\left(x \right)}= \frac{26 x^{3}}{125} - 2$ using $LaTeX: \displaystyle x_0=1$ .

Using the formula for Newton's method gives $LaTeX: x_{n+1} = x_{n} - \frac{- \frac{26 x_{n}^{3}}{125} + \cos{\left(x_{n} \right)} + 2}{- \frac{78 x_{n}^{2}}{125} - \sin{\left(x_{n} \right)}}$ Using $LaTeX: \displaystyle x_0 = 1$ and $LaTeX: \displaystyle n = 0,1,2,3,$ and $LaTeX: \displaystyle 4$ gives: $LaTeX: x_{1} = (1.0000000000) - \frac{- \frac{26 (1.0000000000)^{3}}{125} + \cos{\left((1.0000000000) \right)} + 2}{- \frac{78 (1.0000000000)^{2}}{125} - \sin{\left((1.0000000000) \right)}} = 2.5915035712$ $LaTeX: x_{2} = (2.5915035712) - \frac{- \frac{26 (2.5915035712)^{3}}{125} + \cos{\left((2.5915035712) \right)} + 2}{- \frac{78 (2.5915035712)^{2}}{125} - \sin{\left((2.5915035712) \right)}} = 2.0669307514$ $LaTeX: x_{3} = (2.0669307514) - \frac{- \frac{26 (2.0669307514)^{3}}{125} + \cos{\left((2.0669307514) \right)} + 2}{- \frac{78 (2.0669307514)^{2}}{125} - \sin{\left((2.0669307514) \right)}} = 1.9787172705$ $LaTeX: x_{4} = (1.9787172705) - \frac{- \frac{26 (1.9787172705)^{3}}{125} + \cos{\left((1.9787172705) \right)} + 2}{- \frac{78 (1.9787172705)^{2}}{125} - \sin{\left((1.9787172705) \right)}} = 1.9762944652$ $LaTeX: x_{5} = (1.9762944652) - \frac{- \frac{26 (1.9762944652)^{3}}{125} + \cos{\left((1.9762944652) \right)} + 2}{- \frac{78 (1.9762944652)^{2}}{125} - \sin{\left((1.9762944652) \right)}} = 1.9762926528$