Final answer:
Yes, it does matter whether you use radians or degrees when calculating arc length due to the difference in conversion between the two units. Arc length is computed by the formula s = rθ in radians, or s = (rπ/180)° in degrees, where ° represents the degree measure.
Step-by-step explanation:
When calculating the length of an arc, the unit of measurement for the central angle does indeed matter, and it is important to be consistent in the unit used, whether it is radians or degrees. The reason is that these are two different measures of angular displacement, and the formula used to determine the arc length from the central angle differs based on the unit used. If we have a central angle θ in radians and radius r, the arc length (s) is calculated using the formula s = rθ. Conversely, if the angle is in degrees, the formula usually incorporates a conversion factor to switch from degrees to radians before applying it to find the arc length.
Since there are 2π radians in one complete revolution (360 degrees), to convert degrees to radians, we multiply the degree measurement by π/180. Therefore, if the central angle is given in degrees, the arc length is found by the formula s = (rπ/180)°, where ° represents the degree measure. This shows that while the core concept of arc length remains the same, the calculations differ slightly based on the unit of the central angle. It's also important to note that radians are dimensionless as they are defined as the ratio of the arc length to the radius, simplifying some aspects of trigonometric calculations in physics and engineering.