How would u relate arc length to chord length?

Question

What is relationship of arc length of 2 points of a circle and if you connect a chord to the same points of arc.

In the book it says. Points A and B of an arc is 2x length So the chord length is 2 sin(x)

How come? What is relationship of arc with sin()

Answer 1

Let $l$ be the length of the arc and let $\alpha$ the angle on this arc. We have that: $$\frac{\alpha}{2\pi}=\frac{l}{2\pi r}\leftrightarrow\alpha=\frac{l}{r}$$ where $r$ is the radius of the circle. Also, we know that: $$L=2r\sin\left(\frac{\alpha}{2}\right)$$ by the trigonometric properties of $\sin(x)$. Substituting, we have: $$L=2r\sin\left(\frac{\alpha}{2}\right)=2r\sin\left(\frac{l}{2r}\right)$$ where $L$ is the length of the chord.

Note that the answer of your book is not correct because if you consider $-2\leq2\sin(\alpha)\leq2$ and so it can't represent the length of all chords of a circle with diameter $10$.

Answer 2

I don’t think what the book says is correct, for example if the arc length was pi, on a unit circle, then the chord length would apparently be Sin(pi) which is 0.

Answer 3

The arc length is $\dfrac{d\alpha}2$ and the corresponding chord $d\sin\dfrac\alpha2$. Hence

$$\begin{cases}x=\dfrac{d\alpha}2,\\\sin x=d\sin\dfrac\alpha2.\end{cases}$$

This is true when $x$ is the half of the aperture angle and the diameter is unit.