Parameterization of Unit Circle is Simple Loop

Theorem
Let $\mathbb S^1$ denote the unit circle whose center is at the origin of the Euclidean space $\R^2$.

Let $p: \closedint 0 1 \to \R^2$ be defined by:


 * $\forall t \in \closedint 0 1 : \map p t = \tuple {\map \cos {2 \pi t}, \map \sin {2 \pi t} }$

Then $p$ is a simple loop with image equal to $\mathbb S^1$.

Proof
Parametric Equation of Circle shows that for all points $\tuple {x, y}$ on the unit circle $\mathbb S^1$, the point can be expressed as $\tuple {x, y} = \tuple {\map \cos {r}, \map \sin {r} }$ for some $r \in \R$.

Sine and Cosine are Periodic on Reals shows that sine and cosine functions are periodic functions with period equal to $2 \pi$.

It follows that the image of $p$ is equal to $\mathbb S^1$.

By definition of period, it follows that the only values of $t_1, t_2 \in \closedint 0 1$ with $t_1 \ne t_2$ such that $\map p {t_1} = \map p {t_2}$ is for $t_1, t_2 \in \set {0, 1}$, where:


 * $\tuple {\map \cos {0}, \map \sin {0} } = \tuple {\map \cos {2 \pi}, \map \sin {2 \pi} } = \tuple{ 1, 0 }$

by Cosine of Zero is One and Sine of Zero is Zero.

We have that Real Sine Function is Continuous and Cosine Function is Continuous.

Multiple Rule for Continuous Real Functions and Composite of Continuous Mappings between Metric Spaces is Continuous shows that the two coordinate projections:


 * $\map \cos {2 \pi t}, \map \sin {2 \pi t}$

are continuous.

Hence from Continuity of Composite with Inclusion: Inclusion on Mapping and Continuous Mapping to Product Space, it follows that $p$ is continuous.

By definition of simple loop, it follows that $p$ is a simple loop.