(a)     Integrate \(\int {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta \) .

(b)     Given that \(\int_{\frac{\pi }{2}}^a {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta  = \frac{1}{2}\) and \(\frac{\pi }{2} < a < \pi \), find the value of \(a\) .

(a)     \(\int {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta  = \int {\frac{{\left( {1 - \cos \theta } \right)'}}{{1 - \cos \theta }}} {\text{d}}\theta  = \ln \left( {1 - \cos \theta } \right) + C\)     (M1)A1A1

Note: Award A1 for \(\ln \left( {1 - \cos \theta } \right)\) and A1 for C.

(b)     \(\int_{\frac{\pi }{2}}^a {\frac{{\sin \theta }}{{1 - \cos \theta }}} {\text{d}}\theta  = \frac{1}{2} \Rightarrow \left[ {\ln \left( {1 - \cos \theta } \right)} \right]_{\frac{\pi }{2}}^a = \frac{1}{2}\)     M1

\(1 - \cos a = {{\text{e}}^{\frac{1}{2}}} \Rightarrow a = \arccos \left( {1 - \sqrt {\text{e}} } \right)\)) or \(2.28\)     A1     N2

[5 marks]

Generally well answered, although many students did not include the constant of integration.