What is the approximate value of 120 degrees in radians?
The correct answer and explanation is:
The approximate value of 120 degrees in radians is 2π3\frac{2\pi}{3} radians, which is about 2.094 radians.
To convert degrees to radians, the formula is: radians=degrees×π180\text{radians} = \text{degrees} \times \frac{\pi}{180}
Using this formula for 120 degrees: 120×π180=120π180=2π3120 \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3}
In decimal form, since π\pi is approximately 3.1416, the calculation becomes: 2×3.14163=6.28323≈2.094\frac{2 \times 3.1416}{3} = \frac{6.2832}{3} \approx 2.094
Radians and degrees are both units used to measure angles. The radian is the standard unit of angular measure used in many areas of mathematics. One radian is defined as the angle created when the radius of a circle is wrapped along its circumference. Since the circumference of a circle is 2π2\pi times the radius, a full circle contains 2π2\pi radians. In contrast, a full circle is divided into 360 degrees.
This means that 360 degrees is equivalent to 2π2\pi radians. Therefore, 1 degree equals π180\frac{\pi}{180} radians. When converting, multiplying the degree measure by this factor gives the radian measure.
For practical use, angles like 90 degrees, 180 degrees, and 360 degrees convert nicely into radians as π2\frac{\pi}{2}, π\pi, and 2π2\pi respectively. For 120 degrees, it lies between 90 and 180 degrees, so its radian value 2π3\frac{2\pi}{3} is between π2\frac{\pi}{2} (about 1.57) and π\pi (about 3.14).
Understanding radians is important because many mathematical functions, especially trigonometric functions like sine and cosine, use radians as their input. Calculations involving angular velocity, oscillations, and waves also rely on radians for consistency and simplicity.