Problem 81 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-2$$

Step-by-Step Solution

Answer

-114.592 degrees

Step 1: Identify the conversion factor

Recognize that there are \(180^\circ\) in \(\pi\) radians, thus the conversion factor from radians to degrees is \(\frac{180}{\pi}\).

Step 2: Multiply the given angle by the conversion factor

Multiply the given angle (-2 radians) by the conversion factor \(\frac{180}{\pi}\), to find the equivalent measure in degrees.

Step 3: Round to three decimal places

Once the multiplication is complete, round the result to three decimal places, since that is the precision requested in the exercise.

Key Concepts

Radians to Degrees

Understanding how to convert an angle from radians to degrees is fundamental in mathematics. Radians and degrees are two common units for measuring angles but they originate from different systems. A full circle is equal to two pi (\(2\pi\)) radians or 360 degrees. Therefore, one pi (\(\pi\)) radians is equivalent to 180 degrees. Knowing this equivalence helps us switch between these two systems efficiently.To convert an angle from radians to degrees:

• Use the fact that 1 radian equals \(\frac{180}{\pi}\) degrees.
• Multiply the radians by this conversion factor to find the equivalent in degrees.

For example, if we have -2 radians, we multiply by \(\frac{180}{\pi}\) to get its measure in degrees.

Conversion Factor

The conversion factor plays a crucial role in transforming angle measurements between units. When converting from radians to degrees, the conversion factor you use is \(\frac{180}{\pi}\).This factor comes from the relationship that \(\pi\) radians corresponds to 180 degrees:- \(\pi\) radians = 180 degrees- Thus, 1 radian = \(\frac{180}{\pi}\) degreesTo apply this factor to convert -2 radians:

• Multiply -2 by \(\frac{180}{\pi}\).
• The operation looks like this: \(-2 \times \frac{180}{\pi}\).
• This yields the angle measurement in degrees before rounding.

Hence, conversion factors like \(\frac{180}{\pi}\) ensure precise transformation between radians and degrees.

Rounding Decimals

Rounding decimals is often necessary to present a cleaner, more concise number, especially in mathematical and scientific contexts. When rounding to three decimal places, you ensure that the number is both accurate and manageable.Here's how you round to three decimal places:- Look at the number in the fourth decimal place.- If it is 5 or greater, increase the third decimal place by 1. If less than 5, keep the third decimal unchanged.In this exercise, after calculating \(-2 \times \frac{180}{\pi}\), you obtain an unrounded result. Applying the rules above, you round this result to three decimal places, providing a precise answer without excessive detail.This step is important for precision while also meeting the requirements set by the problem or context you are working within.