Equivalence Class/Examples/Equal Sine of pi x over 6 on Integers

Example of Equivalence Class

Let $\Z$ denote the set of integers.

Let $\RR$ denote the relation on $\Z$ defined as:

$\forall x, y \in \Z: x \mathrel \RR y \iff \sin \dfrac {\pi x} 6 = \sin \dfrac {\pi y} 6$

Then the equivalence class of $1$ under $\RR$ is:

$\eqclass 1 \RR = \set {1 + 12 k: k \in \Z} \cup \set {5 + 12 k: k \in \Z}$

Proof

From Equivalence Relation Examples: Equal $\sin \dfrac {\pi x} 6$ on Integers, $\RR$ is an equivalence relation.

We have that:

$\eqclass 1 \RR = \set {x \in \Z: \sin \dfrac {\pi x} 6 = \sin \dfrac \pi 6}$

That is:

$\eqclass 1 \RR = \set {x \in \Z: \sin \dfrac {\pi x} 6 = \dfrac 1 2}$

We have that:

$\sin \dfrac \pi 6 = \dfrac 1 2$

and:

$\sin \dfrac {5 \pi} 6 = \dfrac 1 2$

The result follows from Sine and Cosine are Periodic on Reals.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.1 \ \text{(b)}$