If in two circles, arcs of the same length subtend angles \[{60^0}\] and \[{75^0}\] at the Centre, find the ratio of their radii.

As we know that arc length \[l = r\theta \]
Converting the given angles into radian as,
For \[{60^0}\] ,
\[\theta = 60 \times \dfrac{\pi }{{180}} = \dfrac{\pi }{3}\]
Similarly, for \[{75^0}\] we get,
\[\theta = 75 \times \dfrac{\pi }{{180}} = \dfrac{{5\pi }}{{12}}\]
Now as arc lengths are equal so we get,
\[{l_1} = {l_2}\]
On substituting the above equation we get,
\[{r_1}{\theta _1} = {r_2}{\theta _2}\]
Now, as we need the ratio of the radii so we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{{\theta _2}}}{{{\theta _1}}}\]
On putting the value of above obtained angle in radians we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{\dfrac{{5\pi }}{{12}}}}{{\dfrac{\pi }{3}}}\]
Hence, on simplifying above equation we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{{\dfrac{5}{{12}}}}{{\dfrac{1}{3}}}\]
On calculating above ratio we get,
\[\dfrac{{{r_1}}}{{{r_2}}} = \dfrac{5}{4}\]
Hence, above is our required answer.

Note:
Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. Substitute the value and solve the above equation carefully so that without any mistake the correct answer can be calculated.