ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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For every $θ \geq 0$ , the unit-circle path $t \mapsto (\cos t, \sin t)$ on $[0, θ]$ has length $θ$

Example

For every real $θ \geq 0$ , let

$γ_{θ} : [0, θ] \to R^{2}, γ_{θ} (t) = (\cos t, \sin t) .$

Then $L (γ_{θ}) = θ$ . No geometric definition of angle or of $π$ is used: sine and cosine are the published power-series functions.

Facts & Assumptions

Given: A real $θ \geq 0$ and the displayed path.

[L1]

Vector differentiation is componentwise (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L2]

$(\sin t)^{'} = \cos t$ and $(\cos t)^{'} = - \sin t$ (The derivatives of sine and cosine are cosine and minus sine).

[L3]

$\sin^{2} t + \cos^{2} t = 1$ for every real $t$ (Parity and the Pythagorean identity for sine and cosine).

[L4]

A $C^{1}$ path has length equal to the integral of its speed (If $γ : [a, b] \to R^{n}$ is continuous, differentiable on $(a, b)$ , and $γ^{'}$ extends continuously to $[a, b]$ , then $L (γ) = \int_{a}^{b} ∥ γ^{'} (t) ∥_{2} d t$ ).

[L5]

The integral of the constant function $1$ on $[0, θ]$ is $θ$ (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

Verification

technique · computation

1.1

If $θ > 0$ , [L1]--[L2] give $γ_{θ}^{'} (t) = (- \sin t, \cos t)$ .

givenL1L2

2.1

By [L3], $∥ γ_{θ}^{'} (t) ∥_{2} = \sqrt{\sin^{2} t + \cos^{2} t} = 1$ .

step 1.1L3

3.1

Apply [L4] and [L5] to get $L (γ_{θ}) = \int_{0}^{θ} 1 d t = θ$ .

step 2.1L4L5

4.1

If $θ = 0$ , the domain is a singleton and the defined length is $0 = θ$ .

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 168 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. R. Shastri, Metric Spaces, Section 5 (standard reference, not scraped)

For every θ≥0θ≥0, the unit-circle path t↦(cos⁡t,sin⁡t)t↦(cost,sint) on [0,θ][0,θ] has length θθ