Suppose we have a segment of length $a$:
Then we rotate this segment for some length $b$:
And we connect end points of starting segment and newly created segment to form segment of length $c$:
My question is this: knowing $a$ and $b$, how can one deduce $c$?
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The angle between the segments of length $a$ is $\theta=\frac b{a}$. Then if you bisect the angle between the $a$ segments you get two right triangles. $\frac c2=a \sin \frac \theta 2$ or $c=2a\sin \left(\frac b{2a}\right)$
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Having a good diagram goes a long way. By rotating a side you create an isosceles triangle with angle $\theta = \frac{b}{a}$
To find $c$ consider the right triangle created by the bisector (dashed line).
$$ \sin \left( \frac{\theta}{2} \right) = \frac{ \left( \frac{c}{2} \right) }{a} $$
or $$ \boxed{ c= 2 a \sin \left( \frac{b}{2 a} \right) }$$
Take a look at the Law of Cosines:
$$c^2 = a^2 + b^2 - 2ab \cos(x)$$
Where $x$ is the angle between sides $a$ and $b$. Applying this here, where both sides are of length $a$:
$$c^2 = a^2 + a^2 - 2a^2 \cos\left(\dfrac{b}{a}\right)$$
Notice that the cosine of $\frac{b}{a}$ is taken, because $b$ is the arc length, not the angle. Remember that $s = r \cdot \theta$ where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.