If in two circles, arcs of the same length subtend angles $60^{\circ}$ and $75^{\circ}$ at the centre, find the ratio of their radii.

Let the radii of the two circles be $r_{1}$ and $r_{2} .$ Let an arc of length $l$ subtend an angle of  $60^{\circ}$ at the centre of the circle of radius $r_{1},$ while let an arc of length/subtend an angle of  $75^{\circ}$ at the centre of the circle of radius $r_{2}$

Now, $60^{\circ}=\frac{\pi}{3}$ radian and $75^{\circ}=\frac{5 \pi}{12}$ radian

We know that in a circle of radius $r$ unit, if an arc of length $l$ unit subtends an angle $\theta$ radian at the centre then

$\theta=\frac{l}{r}$ or $l=r \theta$

$\therefore l=\frac{r_{1} \pi}{3}$ and $l=\frac{r_{2} 5 \pi}{12}$

$\Rightarrow \frac{r_{1} \pi}{3}=\frac{r_{2} 5 \pi}{12}$

$\Rightarrow r_{1}=\frac{r_{2} 5}{4}$

$\Rightarrow \frac{r_{1}}{r_{2}}=\frac{5}{4}$

Thus, the ratio of the radii is $5: 4 $