Convert the following angle from degrees to radians. Express your answer in simplest form. 555° Answer Attempt 1 out of 2 ?
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 \frac{\pi}{180}Radians=Degrees×180π
Substituting the given angle:555∘×π180=555π180555^\circ \times \frac{\pi}{180} = \frac{555\pi}{180}555∘×180π=180555π
Next, simplify the fraction:
First, find the greatest common divisor (GCD) of 555 and 180.
Prime factorization:
- 555 = 5 × 111 = 5 × 3 × 37
- 180 = 2² × 3² × 5
The common factors are 3 and 5, so GCD is 15.
Now divide numerator and denominator by 15:555π180=555÷15×π180÷15=37π12\frac{555\pi}{180} = \frac{555 ÷ 15 \times \pi}{180 ÷ 15} = \frac{37\pi}{12}180555π=180÷15555÷15×π=1237π
Thus, the angle in radians is:37π12\boxed{\frac{37\pi}{12}}1237π
Explanation:
Angles are commonly measured in two units: degrees and radians. Degrees divide a full circle into 360 equal parts, while radians use the arc length of a circle in relation to its radius. In mathematical and scientific applications, radians are often preferred because they simplify formulas involving trigonometry and calculus.
To convert degrees to radians, the conversion factor is used:π radians180∘\frac{\pi \text{ radians}}{180^\circ}180∘π radians
This relationship comes from the fact that a full circle is 360 degrees, which equals 2π2\pi2π radians.
In this problem, we are converting 555 degrees to radians. First, multiply 555 by π\piπ and divide by 180:555∘×π180=555π180555^\circ \times \frac{\pi}{180} = \frac{555\pi}{180}555∘×180π=180555π
To simplify, both 555 and 180 share common factors. Their greatest common divisor (GCD) is 15. Dividing both the numerator and denominator by 15:555÷15×π180÷15=37π12\frac{555 ÷ 15 \times \pi}{180 ÷ 15} = \frac{37\pi}{12}180÷15555÷15×π=1237π
Therefore, 555 degrees is equivalent to 37π12\frac{37\pi}{12}1237π radians in simplest form. This form is exact and preferable in mathematics to maintain precision. Using radians is particularly helpful when dealing with periodic functions like sine, cosine, and tangent, especially in higher-level math topics.
The final simplified answer is:37π12\boxed{\frac{37\pi}{12}}1237π
