A geometric sequence has first term 1 and common ratio \(\frac { 1 } { 2 }\)
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Find the sum to infinity of the sequence.
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(ii) Hence, or otherwise, evaluate
$$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$
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Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation
$$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$