For $A=\{\sin (2n\pi/7) \mid n \in \mathbb{N}\},$ how do I find $\sup(A)$ and $\min(A)$?

Question

I'm kinda new at this and I know what $\sup$ and $\min$ mean, but the problem is when calculating them like the example above. Can you enlighten me please?

Think about it for a while! Start plotting! Or listing elements! What do you know about suprema/infima of finite sets? — Jyrki Lahtonen, Oct 10 '14 at 19:35

score 3 · Answer 1 · answered Oct 10 '14 at 19:41

3

Call $a_n = \sin(2\pi n/7)$. For every $n,k \in \mathbb{N}$, you have $$a_{n + 7k} = \sin(2\pi n /7 + 2 k \pi) = \sin(2\pi n /7) = a_n.$$ Thus, you just need to check the minimum and the maximum among $\{a_0,...,a_6\}$.

answered Oct 10 '14 at 19:41

Paglia

1,274

Ri-Li · Accepted Answer · 2014-10-10T19:59:58.107

1

Note that i) $−1≤\sin(x)≤1$. ii) Now here the values of the set depends only on $\{\sin (0), \sin (\frac {2\pi}7), \sin (\frac {4\pi}7), \sin (\frac {6\pi}7), \sin (\frac {8\pi}7), \sin (\frac {10\pi}7), \sin (\frac {12\pi}7)\}$. So it is finite. Now just check the minimum & maximum value. because after this the values are repeating.

edited Oct 10 '14 at 19:59

answered Oct 10 '14 at 19:41

Ri-Li

9,038

you made it so easy thank you so much. – Tébina1 Oct 10 '14 at 22:22

For $A=\{\sin (2n\pi/7) \mid n \in \mathbb{N}\},$ how do I find $\sup(A)$ and $\min(A)$?

2 Answers2