Convert 600∘ to radian measure in terms of π. n radians
The Correct Answer and Explanation is:
To convert degrees to radians, we use the following relationship:
[
\text{Radians} = \text{Degrees} \times \left(\frac{\pi \text{ radians}}{180^\circ}\right)
]
Given the degree measure of (600^\circ), we can substitute this value into the formula:
[
\text{Radians} = 600^\circ \times \left(\frac{\pi \text{ radians}}{180^\circ}\right)
]
Next, we simplify the expression. First, we can reduce (600) and (180) by dividing both by (60):
[
600 \div 60 = 10 \quad \text{and} \quad 180 \div 60 = 3
]
Now substituting these reduced values back into the equation, we have:
[
\text{Radians} = 10 \times \left(\frac{\pi \text{ radians}}{3}\right)
]
This simplifies to:
[
\text{Radians} = \frac{10\pi}{3} \text{ radians}
]
Thus, the conversion of (600^\circ) to radians is
[
\frac{10\pi}{3} \text{ radians}.
]
Explanation
Understanding the conversion between degrees and radians is crucial in various fields such as mathematics, physics, and engineering. Degrees and radians are both units used to measure angles, but they represent the same concepts in different ways. Degrees are based on a circle being divided into (360) equal parts, while radians are based on the radius of a circle. In fact, one radian is the angle formed when the length of the arc is equal to the radius of the circle.
The formula for conversion derives from the relationship between the circumference of a circle and its radius. The circumference of a circle is given by (C = 2\pi r), and if we consider a full circle (which is (360^\circ)), this circumference also represents (2\pi) radians. Therefore, by using the conversion factor (\frac{\pi}{180}), we can efficiently translate any degree measure into radians.
In this case, converting (600^\circ) to radians allows us to understand the same angle in a format that may be more useful in mathematical equations, particularly those involving trigonometric functions, where radians are typically the preferred unit of measure. Thus, the conversion is not only a numerical process but also a fundamental aspect of working with angular measures in various applications.