Radian angle by unit-circle arc length

Let

$\gamma (t)=(\cos t,\sin t),0\le t\le 2\pi \mathrm{.}$

For $0<t\le 2\pi$ the restriction $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}$ is a circular arc of the unit circle (Circular arcs, circumference as arc length, and diameter), and the counterclockwise angle swept from $\gamma (0)=(1,0)$ to $\gamma (t)$ is defined to have radian measure

$L\left(\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}\right),$

the length of that arc. At $t=0$ nothing is swept, and $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,0]}$ is not a circular arc, that definition admitting only parameter intervals $[\alpha ,\beta ]$ with $\alpha <\beta$; it is the one-point path at $(1,0)$, whose length is $0$ by the singleton convention for path length (Paths in ${\mathbb{R}}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability), and the degenerate angle at $t=0$ is defined to have radian measure $0$. In every case, then, the radian measure of the swept angle is $L(\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]})$.

That measure is $t$. Fix $t$ with $0<t\le 2\pi$, and write $v(u):=(-\sin u,\cos u)$ for $u\in [0,t]$.

Sine and cosine are differentiable on $\mathbb{R}$ with $(\sin u{)}^{\mathrm{\prime }}=\cos u$ and $(\cos u{)}^{\mathrm{\prime }}=-\sin u$ (The derivatives of sine and cosine are cosine and minus sine), hence continuous on $\mathbb{R}$ (A function differentiable at $c$ is continuous at $c$), and so is $-\sin$, a scalar multiple of a continuous function (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, clause 1). Continuity of a real function passes to a subset of its domain, the condition on the restriction quantifying over fewer points (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point), so $\cos$, $\sin$ and $-\sin$ restricted to $[0,t]$ are continuous at every point of $[0,t]$; and for a real function on a subset of $\mathbb{R}$ the $\mathbb{R}$-native and the metric-space notions of continuity are the same notion (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, clause 1). A function into ${\mathbb{R}}^m$ is continuous at a point of its domain if and only if each of its components is (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, clause 1; Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions). Hence $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}$ is continuous on $[0,t]$, and so is $v:[0,t]\to {\mathbb{R}}^2$.

Every point of a nondegenerate interval is a limit point of it, and if a real function is differentiable at a point of its domain, so is its restriction to any subset still having that point as a limit point, with the same derivative (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). So $\cos$ and $\sin$ restricted to $[0,t]$ are differentiable at every $u\in (0,t)$, with derivatives $-\sin u$ and $\cos u$; and a vector-valued function is differentiable at a limit point of its domain exactly when each component is, its derivative there being the vector of the component derivatives (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). Hence $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}$ is differentiable at every $u\in (0,t)$ with derivative $(-\sin u,\cos u)=v(u)$, and $v$ is a continuous extension of that derivative to $[0,t]$ — the hypotheses of If $\gamma :[a,b]\to {\mathbb{R}}^n$ is continuous, differentiable on $(a,b)$, and ${\gamma }^{\mathrm{\prime }}$ extends continuously to $[a,b]$, then $L(\gamma )={\int }_a^b\parallel {\gamma }^{\mathrm{\prime }}(t){\parallel }_{2}dt$. Since $\parallel w{\parallel }_2=(\mathrm{\mid }{w}_0{\mathrm{\mid }}^2+\mathrm{\mid }{w}_1{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$) and ${\sin }^{2}u+{\cos }^{2}u=1$ (Parity and the Pythagorean identity for sine and cosine), we get $\parallel v(u){\parallel }_2=(\mathrm{\mid }-\sin u{\mathrm{\mid }}^2+\mathrm{\mid }\cos u{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}=({\sin }^{2}u+{\cos }^{2}u{)}^{1\mathrm{/}2}=1$ for every $u\in [0,t]$; and the integral of a constant over $[0,t]$ is that constant times $t$ (If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$). Therefore

$L\left(\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}\right)={\int }_0^t\parallel v(u){\parallel }_{2}du={\int }_0^{t}1du=t$

(If $\gamma :[a,b]\to {\mathbb{R}}^n$ is continuous, differentiable on $(a,b)$, and ${\gamma }^{\mathrm{\prime }}$ extends continuously to $[a,b]$, then $L(\gamma )={\int }_a^b\parallel {\gamma }^{\mathrm{\prime }}(t){\parallel }_{2}dt$), while at $t=0$ both the length and the parameter are $0$.

Thus the analytic parameter $t$ is the geometric radian measure of the swept angle. At $t=2\pi$ the path makes one full turn, so a full turn has radian measure $2\pi$, agreeing with the circumference of the unit circle (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).