Convert 0.71 radians to degree measure. Round your answer to the nearest hundredth.
The Correct Answer and Explanation is :
To convert radians to degrees, you can use the conversion factor ( \frac{180}{\pi} ) since there are ( \pi ) radians in 180 degrees. The formula to convert radians (r) to degrees (d) is:
[
d = r \times \frac{180}{\pi}
]
For the conversion of ( 0.71 ) radians to degrees:
[
d = 0.71 \times \frac{180}{\pi}
]
Calculating this step by step:
- First, calculate ( \frac{180}{\pi} ):
- Using ( \pi \approx 3.14159 ),
- ( \frac{180}{3.14159} \approx 57.2958 ).
- Now, multiply ( 0.71 ) by this conversion factor:
- ( 0.71 \times 57.2958 \approx 40.64 ).
Thus, ( 0.71 ) radians is approximately ( 40.64 ) degrees when rounded to the nearest hundredth.
Explanation
The need to convert between radians and degrees arises from their different applications in mathematics and science. Radians are often used in calculus and higher mathematics due to their natural relationship with circular functions. For instance, the length of an arc in a circle is directly related to the radius and the angle in radians. In contrast, degrees are more commonly used in everyday applications, such as navigation, geography, and basic trigonometry.
Understanding the conversion between these two units is essential for various fields, including physics, engineering, and computer graphics, where angles need to be manipulated or transformed between different units of measure. For example, in programming graphics, functions often require angles in degrees, whereas the underlying mathematics might be based on radians.
In practice, when converting angles, it’s important to maintain accuracy and attention to detail. Rounding can significantly affect outcomes in sensitive calculations, thus rounding to the nearest hundredth, as requested, ensures a balance between precision and simplicity. This rounding is particularly relevant in contexts where fine detail is less critical than overall accuracy, such as in basic trigonometric calculations or graphical representations.