How many degrees are equivalent to $\frac{4 \pi}{15}$ radians?
To convert an angle from radians to degrees, we use the conversion factor \(\frac{180^{\circ}}{\pi}\), which represents the fact that \(\pi\) radians is equivalent to \(180^{\circ}\). The general formula for this conversion is \(\text{Degrees} = \text{Radians} \times \frac{180^{\circ}}{\pi}\). We are given the angle \(\theta = \frac{4\pi}{15}\) radians.
We substitute the value of \(\theta\) into our formula:
\(\text{Degrees} = \left( \frac{4\pi}{15} \right) \times \left( \frac{180^{\circ}}{\pi} \right)\).
By canceling the common \(\pi\) in the numerator and denominator, the equation becomes:
\(\text{Degrees} = \frac{4}{15} \times 180^{\circ}\).
Now, perform the division and multiplication:
\(180^{\circ} / 15 = 12^{\circ}\)
\(\text{Degrees} = 4 \times 12^{\circ} = 48^{\circ}\).
Comparing our calculated value of \(48^{\circ}\) with the given options:
The result matches option (a). Thus, the correct equivalent is \(48^{\circ}\). The final answer is \(\boxed{48^{\circ}}\).
Understanding angle measurement is essential in various fields, such as mechanical engineering or navigation. For instance, if a machine part rotates by an angle of \(\frac{\pi}{6}\) radians, converting this into degrees helps technicians quickly understand the rotation as \(30^{\circ}\), making it easier to adjust or calibrate machinery components precisely using standard tools. This conversion is a fundamental skill for translating theoretical models into practical applications in design and construction.