What is the equivalent of pi over 6 radians in degrees?
The Correct Answer and Explanation is:
To convert (\frac{\pi}{6}) radians to degrees, we use the conversion formula between radians and degrees:
[
\text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi}
]
- Substitute (\frac{\pi}{6}) for the radians: [
\text{Degrees} = \frac{\pi}{6} \times \frac{180^\circ}{\pi}
] - Simplify by canceling (\pi) from the numerator and denominator: [
\text{Degrees} = \frac{180^\circ}{6}
] - Divide 180 by 6 to get: [
\text{Degrees} = 30^\circ
]
Explanation
Radians and degrees are two units for measuring angles. One full circle is (360^\circ) in degrees or (2\pi) radians. The relationship between radians and degrees is defined by this full rotation equivalence: (360^\circ = 2\pi) radians. From this, we derive that (180^\circ = \pi) radians, which is the foundation for converting between the two units.
To convert from radians to degrees, we multiply the radian value by (\frac{180^\circ}{\pi}). In our case, (\frac{\pi}{6}) radians represents an angle that is one-sixth of (\pi) radians. Therefore, when we multiply (\frac{\pi}{6}) by (\frac{180^\circ}{\pi}), the (\pi) terms cancel out, simplifying the calculation to (\frac{180^\circ}{6} = 30^\circ).
This means an angle of (\frac{\pi}{6}) radians is equivalent to (30^\circ). Understanding this conversion is essential in trigonometry, physics, and engineering, as radians are commonly used in mathematical functions and formulas, especially when dealing with circular motion, oscillations, or wave patterns. In practical applications, converting radians to degrees helps visualize angles in terms of the more familiar degree unit.