If you want to know $\sin (x)$ within $0.5$ of its true value, then how accurately do you need to know $x$?

Question

I don't really understand how to think about this question.

score 5 · Accepted Answer · answered Dec 11 '13 at 18:43

5

Well, since $|\sin'(x)| \le 1$ for all $x$, you have (by the mean value theorem) $|\sin x - \sin y| \le |x -y|$.

So a sufficient condition is that if you know $x$ within $0.5$ of the correct value, then $\sin x$ will be within $0.5$ of the correct value.

answered Dec 11 '13 at 18:43

copper.hat

172,524

How do you know that $\left | \sin '(x) \right |\leq 1$ for all $x$? – Dec 11 '13 at 18:50
$\sin' = \cos$, and $|\cos(x)| \le 1 $ for all $x$. – copper.hat Dec 11 '13 at 18:52

score 4 · Answer 2 · answered Dec 11 '13 at 18:52

Note that $$|\sin y-\sin x|=2\left|\cos\frac{x+y}2\cdot\right|\cdot\left|\sin\frac{x-y}2\right|\le 2\cdot\left|\sin\frac{x-y}2\right|$$ so that $|x-y|<2\arcsin\frac 14\approx 0.50536$ is sufficient. This is slightly better than the $0.5$ obtained from $\sin'x$ alone.

score 0 · Answer 3 · answered Dec 11 '13 at 18:44

0

You should use Taylor's formula.

answered Dec 11 '13 at 18:44

user95884

1

If you want to know $\sin (x)$ within $0.5$ of its true value, then how accurately do you need to know $x$?

3 Answers3