4.
$$f ( x ) = \sin \left( \frac { 3 } { 2 } x \right)$$
- Find the Taylor series expansion for \(\mathrm { f } ( x )\) about \(\frac { \pi } { 3 }\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 4 }\)
- Hence obtain an estimate of \(\sin \frac { 1 } { 2 }\), giving your answer to 4 decimal places.