Converting Angle Measure: Radians to Degrees

Understanding how to convert angle measures between radians and degrees is a fundamental concept in trigonometry and geometry. Radians and degrees are two different units used to measure angles.

Understanding Radian-Degree Conversion

There are $2\pi$ radians in a full circle ($360^\circ$). This relationship provides the basis for converting between the two units.

The key conversion factor is derived from the fact that $\pi$ radians is equal to $180^\circ$. Therefore:

• To convert radians to degrees, multiply by $\frac{180}{\pi}$.
• To convert degrees to radians, multiply by $\frac{\pi}{180}$.

Calculating the Angle in Degrees

The question asks for the measure of the angle $-5$ radians in degrees. We will use the conversion formula:

Angle in degrees = Angle in radians $\times \frac{180^\circ}{\pi}$

Substitute the given angle measure ($-5$ radians):

Approximating the Value

To find the value in degrees, minutes, and seconds, we first calculate the approximate decimal value of $\frac{-900}{\pi}$. Using a value for $\pi \approx 3.14159265...$:

Converting Decimal Degrees to Degrees, Minutes, and Seconds

The angle is $-286.479087...^\circ$. The whole number part represents the degrees.

Degrees: $-286^\circ$

Now, we need to convert the decimal part ($0.479087...$) into minutes and seconds. Note that we work with the absolute value for this conversion and keep the negative sign for the final result.

Calculating Minutes:

Multiply the decimal part by 60 (since there are 60 minutes in a degree):

The whole number part is the number of minutes: $28'$.

Calculating Seconds:

Now, take the decimal part of the minutes ($0.74524...$) and multiply by 60 (since there are 60 seconds in a minute):

Rounding to the nearest second gives approximately $45''$. However, if we truncate or consider the value before rounding, it's closer to $44''$. Let's consider the value $44.7''$ which is between $44''$ and $45''$.

Final Angle Measure

Combining the degrees, minutes, and seconds, the angle is approximately:

Comparing this result with the given options, the closest representation is often chosen.