Convert the following angle from degrees to radians. Express your answer in simplest form. 555°
The Correct Answer and Explanation is:
Correct Answer:
To convert 555° to radians:
We use the formula:Radians=Degrees×(π180)\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)Radians=Degrees×(180π)
Step 1: Multiply 555 by π and divide by 180555∘×(π180)=555π180555^\circ \times \left( \frac{\pi}{180} \right) = \frac{555\pi}{180}555∘×(180π)=180555π
Step 2: Simplify the fraction
First, find the greatest common divisor (GCD) of 555 and 180. Both numbers are divisible by 15.
Divide numerator and denominator by 15:555÷15180÷15=37π12\frac{555 \div 15}{180 \div 15} = \frac{37\pi}{12}180÷15555÷15=1237π
Final Answer:555∘=37π12 radians555^\circ = \frac{37\pi}{12} \text{ radians}555∘=1237π radians
Explanation:
To convert an angle from degrees to radians, we use the standard conversion factor between these two units. One full revolution around a circle is 360 degrees, which is equal to 2π2\pi2π radians. This means that:1∘=π180 radians1^\circ = \frac{\pi}{180} \text{ radians}1∘=180π radians
This relationship allows us to convert any degree measure into radians by multiplying the degree measure by π180\frac{\pi}{180}180π.
In this problem, the angle given is 555 degrees. To convert this to radians, we multiply:555∘×(π180)=555π180555^\circ \times \left( \frac{\pi}{180} \right) = \frac{555\pi}{180}555∘×(180π)=180555π
Next, we simplify the fraction 555180\frac{555}{180}180555. To do this, we find the greatest common divisor (GCD) of 555 and 180. Using prime factorization:
- 555 = 5 × 111 = 5 × 3 × 37
- 180 = 2 × 2 × 3 × 3 × 5
The common factors are 3 and 5, so the GCD is 15.
We divide both numerator and denominator by 15:555÷15180÷15=3712\frac{555 \div 15}{180 \div 15} = \frac{37}{12}180÷15555÷15=1237
So, the simplified form is:37π12 radians\frac{37\pi}{12} \text{ radians}1237π radians
This is the simplest exact form. The angle in radians provides a more natural way to describe angles in terms of the circle’s geometry, especially in trigonometry and calculus.
