Answer
\[\boxed{\begin{array}{*{20}{c}}
n&{{x_n}} \\
0&{1.500000} \\
1&{0.101436} \\
2&{0.501114} \\
3&{ - 1.465956} \\
4&{0.510961} \\
5&{0.510973} \\
6&{0.510973} \\
7&{0.510973} \\
8&{0.510973} \\
9&{0.510973} \\
{10}&{0.510973}
\end{array}}\]
Work Step by Step
\[\begin{gathered}
{\text{Let }}f\left( x \right) = \sin x + x - 1,{\text{ and }}{x_0} = 1.5 \hfill \\
{\text{Using the Newton's Method }}{x_{n + 1}} = {x_n} - \frac{{f\left( {{x_n}} \right)}}{{f'\left( {{x_n}} \right)}} \hfill \\
f'\left( x \right) = \cos x + 1 \hfill \\
{\text{Then evaluating }}f\left( {{x_n}} \right){\text{ and }}f'\left( {{x_n}} \right) \hfill \\
{x_{n + 1}} = {x_n} - \frac{{\sin {x_n} + {x_n} - 1}}{{\cos {x_n} + 1}} \hfill \\
\hfill \\
{\text{Taking }}{x_0} = 1.5 \hfill \\
{x_0} = 1.5 \hfill \\
{x_{0 + 1}} = {x_1} = 1.5 - \frac{{\sin \left( {1.5} \right) + \left( {1.5} \right) - 1}}{{\cos \left( {1.5} \right) + 1}} \approx 0.101436 \hfill \\
{x_{1 + 1}} = {x_2} = 0.101436 - \frac{{\sin \left( {0.101436} \right) + \left( {0.101436} \right) - 1}}{{\cos \left( {0.101436} \right) + 1}} \approx 0.501114 \hfill \\
{x_{1 + 2}} = {x_3} = 0.501114 - \frac{{\sin \left( {0.501114} \right) + \left( {0.501114} \right) - 1}}{{\cos \left( {0.501114} \right) + 1}} \approx 0.510961 \hfill \\
{x_{1 + 3}} = {x_4} = 0.510961 - \frac{{\sin \left( {0.510961} \right) + \left( {0.510961} \right) - 1}}{{\cos \left( {0.510961} \right) + 1}} \approx 0.510973 \hfill \\
{x_{1 + 4}} = {x_5} = 0.510973 - \frac{{\sin \left( {0.510973} \right) + \left( {0.510973} \right) - 1}}{{\cos \left( {0.510973} \right) + 1}} \approx 0.510973 \hfill \\
{x_{1 + 5}} = {x_6} = 0.510973 - \frac{{\sin \left( {0.510973} \right) + \left( {0.510973} \right) - 1}}{{\cos \left( {0.510973} \right) + 1}} \approx 0.510973 \hfill \\
{x_{1 + 6}} = {x_7} = 0.510973 - \frac{{\sin \left( {0.510973} \right) + \left( {0.510973} \right) - 1}}{{\cos \left( {0.510973} \right) + 1}} \approx 0.510973 \hfill \\
{x_{1 + 7}} = {x_8} = 0.510973 - \frac{{\sin \left( {0.510973} \right) + \left( {0.510973} \right) - 1}}{{\cos \left( {0.510973} \right) + 1}} \approx 0.510973 \hfill \\
{x_{1 + 8}} = {x_9} = 0.510973 - \frac{{\sin \left( {0.510973} \right) + \left( {0.510973} \right) - 1}}{{\cos \left( {0.510973} \right) + 1}} \approx 0.510973 \hfill \\
{x_{1 + 9}} = {x_{10}} = 0.510973 - \frac{{\sin \left( {0.510973} \right) + \left( {0.510973} \right) - 1}}{{\cos \left( {0.510973} \right) + 1}} \approx 0.510973 \hfill \\
\hfill \\
{\text{Thus}}{\text{, we obtain}} \hfill \\
\boxed{\begin{array}{*{20}{c}}
n&{{x_n}} \\
0&{1.500000} \\
1&{0.101436} \\
2&{0.501114} \\
3&{ - 1.465956} \\
4&{0.510961} \\
5&{0.510973} \\
6&{0.510973} \\
7&{0.510973} \\
8&{0.510973} \\
9&{0.510973} \\
{10}&{0.510973}
\end{array}} \hfill \\
\end{gathered} \]
