Convert -390 degree to radian measure. Write your answer as an exact value.
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 \frac{\pi}{180}radians=degrees×180π
Step 1: Apply the Formula
Given:−390∘-390^\circ−390∘
Using the formula:−390∘×π180=−390π180-390^\circ \times \frac{\pi}{180} = \frac{-390\pi}{180}−390∘×180π=180−390π
Now simplify the fraction:−390π180=−13π6\frac{-390\pi}{180} = \frac{-13\pi}{6}180−390π=6−13π
Final Answer:
−13π6 radians\boxed{-\frac{13\pi}{6}} \text{ radians}−613π radians
Explanation
The process of converting degrees to radians involves a fundamental understanding of how angle measurement units relate. Degrees and radians are two different ways to measure angles, with degrees being more common in everyday use, and radians being more natural in advanced mathematics, especially in calculus and trigonometry.
One complete circle measures 360 degrees, which is equal to 2π2\pi2π radians. From this, we derive the conversion factor:1∘=π180 radians1^\circ = \frac{\pi}{180} \text{ radians}1∘=180π radians
So, to convert any angle in degrees to radians, we multiply it by π180\frac{\pi}{180}180π. In this case, we were asked to convert −390∘-390^\circ−390∘ into radians. Note that the angle is negative, which means it is measured in the clockwise direction. This does not change the conversion process—just carry the negative sign throughout.
We start by plugging the value into the formula:−390×π180=−390π180-390 \times \frac{\pi}{180} = \frac{-390\pi}{180}−390×180π=180−390π
We then simplify the fraction. Both 390 and 180 are divisible by 30:−390π180=−13π6\frac{-390\pi}{180} = \frac{-13\pi}{6}180−390π=6−13π
This result is in terms of π\piπ, which keeps the value exact. This is important in mathematics because it avoids approximation errors that come from using decimal representations of π\piπ.
Understanding this result is also helpful in trigonometry. For instance, −13π6-\frac{13\pi}{6}−613π radians is equivalent to rotating clockwise past two full circles (since 2π=12\pi = 12π=1 full circle), then an additional −π6-\frac{\pi}{6}−6π. That’s useful for interpreting angles on the unit circle.
Thus, the exact radian measure of −390∘-390^\circ−390∘ is:−13π6\boxed{-\frac{13\pi}{6}}−613π
