QQuestion
Two unequal arcs of ratio 5 : 2 of a circle subtend angle at the centre. If the second angle is 30°, find the first angle in circular measure.
✓Answer
Since the angles subtended at the center are proportional to the lengths of the arcs, we set up the proportion:
\frac{\text{First Angle}}{\text{Second Angle}} = \frac{5}{2}
Let the first angle be x, then:
\frac{x}{30} = \frac{5}{2}
Cross multiplying:
x = \frac{5}{2} \times 30
x = \frac{150}{2}
x = 75°
Now, convert degrees to radians:
Radians = \frac{75\pi}{180}
Simplify:
Radians = \frac{5\pi}{12}
The first angle is 75° or \frac{5\pi}{12} radians.
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