Radians to Degrees Conversion Basics

The problem asks us to find the approximate degree measure for an angle given as 4 radians. This involves converting units of angular measurement.

The Angle Conversion Formula

To convert an angle from radians to degrees, we use the fundamental relationship that $\pi$ radians is equal to $180^\circ$. The formula for conversion is:

Degree Measure = Radians $\times \frac{180^{\circ}}{\pi}$

Calculating 4 Radians in Degrees

We are given the angle in radians as 4. Substituting this value into the conversion formula:

Degree Measure = $4 \times \frac{180^{\circ}}{\pi}$

Degree Measure = $\frac{720^{\circ}}{\pi}$

Approximating Pi for Calculation

To find a numerical approximation in degrees, we need to substitute a value for $\pi$. A common approximation used in calculations is $\pi \approx \frac{22}{7}$. Using this approximation:

Degree Measure $\approx 4 \times \frac{180^{\circ}}{22/7}$

Let's simplify this expression:

Degree Measure $\approx 4 \times \frac{180^{\circ} \times 7}{22}$

We can simplify $\frac{180}{22}$ by dividing both numerator and denominator by 2:

Degree Measure $\approx 4 \times \frac{90^{\circ} \times 7}{11}$

Degree Measure $\approx \frac{4 \times 630^{\circ}}{11}$

Degree Measure $\approx \frac{2520^{\circ}}{11}$

Converting Decimal Degrees

Now, we convert the improper fraction $\frac{2520}{11}$ degrees into the degrees, minutes, and seconds format:

1. Convert to Degrees and Fraction: Divide 2520 by 11.

   $2520 \div 11 = 229$ with a remainder of $1$. So, $\frac{2520^{\circ}}{11} = 229^{\circ} + \frac{1}{11}^{\circ}$.

2. Convert Fractional Degree to Minutes: Multiply the fractional part of the degree by 60 (since $1^{\circ} = 60'$).

   Minutes = $\frac{1}{11} \times 60' = \frac{60}{11}'$.

3. Convert to Minutes and Fraction: Divide 60 by 11.

   $\frac{60}{11} = 5$ with a remainder of $5$. So, $\frac{60}{11}' = 5' + \frac{5}{11}'$.

4. Convert Fractional Minute to Seconds: Multiply the fractional part of the minute by 60 (since $1' = 60''$).

   Seconds = $\frac{5}{11} \times 60'' = \frac{300}{11}''$.

5. Calculate Seconds: Divide 300 by 11.

   $\frac{300}{11}'' \approx 27.27''$. Rounding to the nearest second gives $27''$.

Combining these parts, we get the angle measure as $229^{\circ} 5' 27''$.

Matching the Best Degree Approximation

The calculated value is approximately $229^\circ 5' 27''$. Let's compare this with the given options:

• 1. 231°5′27″
• 2. 220°5'27"
• 3. 229°5'27"
• 4. 230°5'
• 5.

Our calculated value, $229^\circ 5' 27''$, matches option 3 exactly.