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Exercise 1.15

The sine procedure is applied \(5\) times because that’s the minimum number of times we need to divide \(12.15\) by \(3\) to get a number smaller than \(0.1\). The number of times the sine procedure is applied on an angle \(a\) is the smallest positive integer \(n\) such that \(\left|a \times (1/3)^n \right|< 0.1\), solving this equation we get \(n > \log_{3}\left(\frac{|a|}{0.1}\right)\). We can conclude that the sine procedure requires approximately \(\log_{3}\left(\frac{|a|}{0.1}\right)\) steps, the order of growth in time is logarithmic.

Similarly we can see that the order of growth in space is logarithmic.