The degree of $\pi/6$ radians is 30 degrees.
Understanding Angular Measurement: Radians and Degrees
Radians and degrees are two fundamental units used to measure angles. While degrees (°) are perhaps more commonly encountered in everyday contexts, radians are widely used in advanced mathematics, physics, and engineering due to their natural relationship with the radius of a circle.
The key relationship between these two units is that a full circle, which is 360 degrees, is equivalent to $2\pi$ radians. From this, we can derive that $\pi$ radians is equal to 180 degrees.
Converting $\pi/6$ Radians to Degrees
To convert an angle from radians to degrees, you utilize the conversion factor derived from the relationship $\pi$ radians = 180 degrees. This factor is $\frac{180}{\pi}$ degrees per radian.
Conversion Formula
The general formula for converting radians to degrees is:
Degrees = Radians $\times \frac{180}{\pi}$
Calculation for $\pi/6$ Radians
Let's apply this formula to convert $\pi/6$ radians:
- Start with the radian measure: $\pi/6$ radians.
- Multiply by the conversion factor:
Degrees = $(\pi/6) \times \frac{180}{\pi}$ - Cancel out $\pi$ from the numerator and denominator:
Degrees = $\frac{180}{6}$ - Perform the division:
Degrees = 30
Therefore, $\pi/6$ radians is exactly 30 degrees.
Common Angle Conversions
Understanding the relationship between radians and degrees is crucial, and recognizing common angle conversions can be very helpful. Here is a table illustrating some frequently encountered angles:
| Turns | Radian Measure | Degree Measure |
|---|---|---|
| 1/12 | $\pi/6$ | 30° |
| 1/10 | $\pi/5$ | 36° |
| ⅛ | $\pi/4$ | 45° |
| 1/2$\pi$ | 1 rad | 57.3° |
Why Angle Conversion Matters
The ability to convert between radians and degrees is fundamental in various scientific and engineering disciplines. It allows for consistent calculations and understanding across different contexts, whether you're working with trigonometric functions, analyzing periodic phenomena, or designing mechanical systems. For instance, in calculus, angles are almost always expressed in radians because it simplifies many formulas and makes them more natural.
