# 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.