5 The first and second terms of an arithmetic progression are \(\tan \theta\) and \(\sin \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
- Given that \(\theta = \frac { 1 } { 4 } \pi\), find the exact sum of the first 40 terms of the progression.
The first and second terms of a geometric progression are \(\tan \theta\) and \(\sin \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\). - Find the sum to infinity of the progression in terms of \(\theta\).
- Given that \(\theta = \frac { 1 } { 3 } \pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures.