3 The equation \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x ) - x + 0.1 = 0\) has a root \(\alpha\) such that \(- 1 < \alpha < 0\).
Alex uses an iterative method to find a sequence of approximations to \(\alpha\). Some of the associated spreadsheet output is shown in the table.
| C | D | E |
| 4 | \(r\) | \(\mathrm { x } _ { \mathrm { r } }\) | \(\mathrm { f } \left( \mathrm { x } _ { \mathrm { r } } \right)\) |
| 5 | 0 | - 1 | - 0.4707963 |
| 6 | 1 | - 0.8 | - 0.0272952 |
| 7 | 2 | - 0.787691 | - 0.0193610 |
| 8 | 3 | - 0.7576546 | - 0.0020574 |
| 9 | 4 | - 0.7540834 | - 0.0001740 |
| 10 | 5 | | |
| 11 | 6 | | |
The formula in cell D7 is
$$= ( \mathrm { D } 5 * \mathrm { E } 6 - \mathrm { D } 6 * \mathrm { E } 5 ) / ( \mathrm { E } 6 - \mathrm { E } 5 )$$
and equivalent formulae are in cells D8 and D9.
- State the method being used.
- Use the values in the spreadsheet to calculate \(x _ { 5 }\) and \(x _ { 6 }\), giving your answers correct to 7 decimal places.
- State the value of \(\alpha\) as accurately as you can, justifying the precision quoted.
Alex uses a calculator to check the value in cell D9, his result is - 0.7540832686 .
- Explain why this is different to the value displayed in cell D9.
The value displayed in cell E11 in Alex's spreadsheet is \(- 1.4629 \mathrm { E } - 09\).
- Write this value in standard mathematical notation.