Convert the following angle in degrees to radians. 345o
The Correct Answer and Explanation is:
To convert an angle from degrees to radians, we use the following formula: Radians=Degrees×(π180)\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)
Given the angle is 345°, we can apply the formula: 345∘×(π180)=345π180345^\circ \times \left( \frac{\pi}{180} \right) = \frac{345\pi}{180}
Now simplify the fraction: 345π180=23π12\frac{345\pi}{180} = \frac{23\pi}{12}
So, 345 degrees is equal to 23π12\frac{23\pi}{12} radians, which is approximately 6.021 radians when using π≈3.1416\pi \approx 3.1416.
Explanation
Understanding how to convert degrees into radians is essential in mathematics, especially in trigonometry and calculus. Angles can be measured in two main units: degrees and radians. A full circle has 360 degrees, which corresponds to 2π2\pi radians. Therefore, 1 degree is equal to π180\frac{\pi}{180} radians. This conversion factor allows us to convert any angle from degrees to radians by multiplying the degree measure by π180\frac{\pi}{180}.
Let’s take the angle of 345°. This is close to a full circle and lies in the fourth quadrant of the unit circle. To convert this angle to radians, we multiply: 345×π180=345π180345 \times \frac{\pi}{180} = \frac{345\pi}{180}
To simplify this, we divide both the numerator and the denominator by 15: 345π180=23π12\frac{345\pi}{180} = \frac{23\pi}{12}
Thus, 345 degrees is equivalent to 23π12\frac{23\pi}{12} radians. This form is exact and is typically preferred in higher-level mathematics. If a decimal approximation is needed, you can substitute π\pi with its approximate value (3.1416), which gives: 23×3.141612≈6.021\frac{23 \times 3.1416}{12} \approx 6.021
This means 345° is about 6.021 radians.
Knowing how to convert between degrees and radians is particularly useful when working with trigonometric functions, as many calculators and software packages use radians by default. It’s also important for understanding angular motion in physics and engineering.
