# Epimorphism from Real Numbers to Circle Group

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## Theorem

Let $\struct {K, \times}$ be the circle group, that is:

- $K = \set {z \in \C: \cmod z = 1}$

under complex multiplication.

Let $f: \R \to K$ be the mapping from the real numbers to $K$ defined as:

- $\forall x \in \R: \map f x = \cos x + i \sin x$

Then $f: \struct {\R, +} \to \struct {K, \times}$ is a group epimorphism.

Its kernel is:

- $\map \ker f = \set {2 \pi n: n \in \Z}$

## Proof

$f$ is a surjection from ...

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$\Box$

Then:

\(\ds \map f x \times \map f y\) | \(=\) | \(\ds \paren {\cos x + i \sin x} \paren {\cos y + i \sin y}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \cos x \cos y + i \sin x \cos y + \cos x i \sin y + i \sin x i \sin y\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {\cos x \cos y - \sin x \sin y} + i \paren {\sin x \cos y + \cos x \sin y}\) | as $i^2 = -1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \cos {x + y} + i \map \sin {x + y}\) | Cosine of Sum and Sine of Sum | |||||||||||

\(\ds \) | \(=\) | \(\ds \map f {x + y}\) |

So $f$ is a (group) homomorphism.

$\Box$

Thus $f$ is seen to be a surjective homomorphism.

Hence, by definition, it is a (group) epimorphism.

$\Box$

From Cosine of Multiple of Pi:

- $\forall n \in \Z: \cos n \pi = \paren {-1}^n$

and from Sine of Multiple of Pi:

- $\forall n \in \Z: \sin n \pi = 0$

From Sine and Cosine are Periodic on Reals, it follows that these are the only values of $\Z$ for which this holds.

For $\cos x + i \sin x = 1 + 0 i$ it is necessary that:

- $\cos x = 1$
- $\sin x = 0$

and it can be seen that the only values of $x$ for this to happen is:

- $x \in \set {2 \pi n: n \in \Z}$

Hence, by definition of kernel:

- $\map \ker f = \set {2 \pi n: n \in \Z}$

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Example $12.1$