In a circle, two arcs of unequal lengths are in the ratio 5:2. If the central angle corresponding to the second arc is 30°, what is the radian measure of the central angle corresponding to the first arc?

Q.In a circle, two arcs of unequal lengths are in the ratio 5:2. If the central angle corresponding to the second arc is 30°, what is the radian measure of the central angle corresponding to the first arc?

Let the degree measure of the first angle be \(θ^\circ\). According to the condition, \[ \cfrac{θ^\circ}{30^\circ} = \cfrac{5}{2} \Rightarrow θ^\circ = 30^\circ \times \cfrac{5}{2} = 75^\circ \] Since \(180^\circ = \pi\) radians, \[ 75^\circ = \cfrac{75}{180} \times \pi = \cfrac{5}{12} \pi \] ∴ The radian measure of the first angle is \( \cfrac{5}{12} \pi \).