Question

The degree measure of \(\frac{\pi}{32}\) is equal to

• A. \(5^\circ 30' 20''\)
• B. \(5^\circ 37' 30''\)
• C. \(5^\circ 37' 20''\)
• D. \(4^\circ 30' 30''\)

Answer 1

The Correct Option is B

To convert radians to degrees, use the formula:
\[ \text{Degrees} = \text{Radian} \times \frac{180^\circ}{\pi} \]

Given: \(\frac{\pi}{32} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{32} = 5.625^\circ\)

Now convert decimal degrees into degrees, minutes, and seconds:

\[ 0.625^\circ \times 60 = 37.5' \Rightarrow 37' \text{ and } 0.5' \times 60 = 30'' \]

So: \[ \frac{\pi}{32} = 5^\circ 37' 30'' \]

Correct Answer: \(5^\circ 37' 30''\)

Answer 2

we know that π radians is equivalent to 180 degrees.
So,\(\frac{\pi}{32}\) radians can be converted to degrees as follows:
\(\left(\frac{\pi}{32}\right) \times \left(\frac{180}{\pi}\right) = \frac{180}{32} \text{ degrees} = 5.625 \text{ degrees}\)
Since 0.625 degrees is equal to \(0.625 \times 60 = 37.5 \text{ minutes}\), we have: 
5.625 degrees = 5 degrees 37.5 minutes. 
To express the minutes in terms of minutes and seconds, we can calculate \(0.5 \times 60 = 30 \text{ seconds}\)
Therefore, the degree measure of \(\frac{\pi}{32}\) is equal to 5 degrees 37 minutes 30 seconds. 
Hence, the correct option is (B) \(5^\circ \ 37' \ 30''\).

Answer 3

We want to convert the angle from radians to degrees, minutes, and seconds.

The given angle is \(\frac{\pi}{32}\) radians. 

We use the conversion factor \(\pi \text{ radians} = 180^\circ\).

To convert radians to degrees, we multiply by \(\frac{180^\circ}{\pi}\):

\[ \text{Angle in degrees} = \frac, we use the conversion factor \(\pi \text{ radians} = 180^\circ\).

Multiply the radian measure by \(\frac{180^\circ}{\pi}\):

\[ \text{Degrees} = \frac{\pi}{32} \times \frac{180^\circ}{\pi} = \frac{180}{32}^\circ \]

Simplify the fraction \(\frac{180}{32}\) by dividing both the numerator and denominator by their greatest common divisor, which is 4:

\[ \frac{180 \div 4}{32 \div 4} = \frac{45}{8} \]

So the angle is \(\frac{45}{8}\) degrees.

Now, convert this improper fraction into degrees, minutes, and seconds.

First, convert the fraction to a mixed number:

\[ \frac{45}{8} = 5 \frac{5}{8} \]

This means the angle is 5 degrees plus \(\frac{5}{8}\) of a degree.

Next, convert the fractional part (\(\frac{5}{8}\) degrees) into minutes. Since \(1^\circ = 60'\) (60 minutes), multiply the fraction by 60:

\[ \text{Minutes} = \frac{5}{8} \times 60' = \frac{300}{8}' \]

Simplify the fraction \(\frac{300}{8}\) by dividing by 4:

\[ \frac{300 \div 4}{8 \div 4} = \frac{75}{2}' \]

Convert this improper fraction to a mixed number:

\[ \frac{75}{2} = 37 \frac{1}{2} \]

This means we have 37 minutes plus \(\frac{1}{2}\) of a minute.

Finally, convert the fractional part (\(\frac{1}{2}\) minutes) into seconds. Since \(1' = 60''\) (60 seconds), multiply the fraction by 60:

\[ \text{Seconds} = \frac{1}{2} \times 60'' = 30'' \]

So we have 30 seconds.

Combining the parts, the angle \(\frac{\pi}{32}\) radians is equal to \(5^\circ 37' 30''\).

Comparing this with the given options, the correct option is:

\(5^\circ 37' 30''\)