The arc length of one sine period is $4\sqrt{2}E(1\mathrm{/}\sqrt{2})$

Given: The graph path $\gamma (x)=(x,\sin x)$ on $[0,2\pi ]$.

If $f:[a,b]\to \mathbb{R}$ has a continuous derivative $q$, then its graph has length ${\int }_a^b\sqrt{1+q(t{)}^2}dt$ (If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f^{\mathrm{\prime }}$ extends continuously to $[a,b]$, then the graph of $f$ has length ${\int }_a^b\sqrt{1+f^{\mathrm{\prime }}(t{)}^2}dt$).

$(\sin x{)}^{\mathrm{\prime }}=\cos x$ on $\mathbb{R}$ (The derivatives of sine and cosine are cosine and minus sine).

The supplementary, reflection, and quarter-turn identities give $\cos (\pi -x)=-\cos x$, $\cos (\pi \mathrm{/}2-x)=\sin x$, and $\cos (-x)=\cos x$ (Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions).

Integrals are additive over subintervals, and affine substitutions obey the substitution formula with oriented endpoints (For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$, Substitution: if $\phi$ is differentiable on $[c,d]$ with ${\phi }^{\mathrm{\prime }}$ integrable and $f$ is continuous on an interval containing $\phi ([c,d])$, then ${\int }_{\phi (c)}^{\phi (d)}f={\int }_c^d(f\circ \phi ){\phi }^{\mathrm{\prime }}$, The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$).

For every real $x$, ${\sin }^{2}x+{\cos }^{2}x=1$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains).

Every positive real has a unique positive square root (Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$, case $n=2$).

Differentiable real functions are continuous (A function differentiable at $c$ is continuous at $c$).

The number $\pi =2\gamma$ is positive because the smallest positive zero of cosine satisfies $\gamma \in (0,2)$ (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two).

For $\alpha \in \mathbb{R}$, the function $u\mapsto u^{\alpha }$ is continuous on $(0,\mathrm{\infty })$; in particular the positive square-root function is continuous there (Continuity and derivatives of positive-base real powers).

Polynomial algebra and composites preserve continuity, and continuous functions on a compact interval are Riemann integrable (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, A composite of continuous functions is continuous, with no side hypothesis of the kind the composition of limits needs, A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion).