The Correct Answer and Explanation is:
Here are the solutions to the problems presented in the image.
Fraction: 285° = 19/12 of a semicircle
Angle measure: 285° = 19π/12 radians
Explanation
1. Finding the Fraction of a Semicircle
To express 285 degrees as a fraction of a semicircle, you first need to know how many degrees are in a semicircle. A full circle contains 360 degrees, so a semicircle, being half of a circle, contains 360 / 2 = 180 degrees.
Next, you create a fraction by placing the given angle (285°) over the angle of a semicircle (180°):
Fraction = 285 / 180
To simplify this fraction, you find the greatest common divisor (GCD) for both the numerator and the denominator. Both numbers are divisible by 5:
285 ÷ 5 = 57
180 ÷ 5 = 36
This gives the fraction 57/36.
Now, we can simplify this fraction further. Both 57 and 36 are divisible by 3:
57 ÷ 3 = 19
36 ÷ 3 = 12
This simplifies the fraction to 19/12. Since 19 is a prime number, the fraction is in its simplest form. Thus, 285° is 19/12 of a semicircle.
2. Converting Degrees to Radians
To convert an angle from degrees to radians, you multiply the degree measure by the conversion factor π/180°.
The formula is: Radians = Degrees × (π / 180°)
Applying this to 285 degrees:
Radians = 285 × (π / 180)
Radians = (285π) / 180
As we did in the first problem, we simplify the fraction 285/180. We already determined that this simplifies to 19/12.
Therefore, the angle measure of 285° in radians is 19π/12.
