Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$\frac{\pi}{7}$$

Step-by-Step Solution

Answer

The angle is approximately \(25.714\) degrees when rounded to three decimal places.

Step 1: Understand the conversion factor

First, it is important to understand the conversion factor between radians and degrees. By definition, \(180\) degrees is equal to \(\pi\) radians. We can use this information to create a conversion factor, which is \(\pi\) radians equals \(180\) degrees or \(1\) radian equals \(180/\pi\) degrees.

Step 2: Apply the conversion factor

We are given the angle \(\pi/7\) radians and we need to convert it into degrees. We can do this by multiplying the angle in radians by the conversion factor, \(180/\pi\), to get the angle in degrees. That is, \((\pi/7) * (180/\pi)\). The \(\pi\) in the numerator and denominator cancel out.

Step 3: Calculate the result

Perform the multiplication to get the answer. The result is \(180/7\) degrees.

Step 4: Round the result

Lastly, round the result to three decimal places as required by the exercise. The result is approximately \(25.714\) degrees when rounded to three decimal places.

Key Concepts

Understanding Angle Measure

Angles are measured in two primary units: radians and degrees. Each unit represents a way to describe the size of an angle.
Radians relate arc lengths to the radius of a circle, making it ideal for mathematical calculations. In one full circle (360 degrees), there are exactly \(2\pi\) radians. This means that \(\pi\) radians is half a circle, or 180 degrees.
Degrees, on the other hand, split a circle into 360 equal parts. This makes it intuitively easier to grasp for many everyday applications. Know that whether using radians or degrees, you're describing the same angle, just in different ways.
Key points include:

• 1 full circle = 360 degrees or \(2\pi\) radians.
• \(\pi\) radians = 180 degrees.
• Both radians and degrees measure angle size.

Using the Conversion Factor

Converting between radians and degrees involves using a crucial conversion factor. This factor stems from the relationship between the two: \(\pi\) radians equals 180 degrees.
To convert radians to degrees, use the conversion factor \(\frac{180}{\pi}\). This means you'll multiply the radian measure by \(\frac{180}{\pi}\) to find its degree equivalent.

• Identify the radian measure.
• Multiply by \(\frac{180}{\pi}\).
• This factor effectively cancels out the radian unit, leaving the angle in degrees.

Applying this to \(\frac{\pi}{7}\), multiply \(\frac{\pi}{7} \times \frac{180}{\pi}\). The \(\pi\)s cancel, giving \(\frac{180}{7}\) degrees.

Rounding Numbers Accurately

Once you've converted an angle to degrees, the next step is to round your result to the required precision. Rounding numbers ensures consistency and clarity in mathematical communication.
When rounding to three decimal places:

• Look at the fourth decimal place.
• If it's 5 or more, increase the third decimal place by 1.
• If it's less than 5, leave the third decimal place as is.

For example, in the conversion of \(\frac{180}{7}\), the decimal result is approximately 25.7142857. Since the fourth decimal is 2 (less than 5), \(25.714\) is our rounded result. Rounding helps present a clear and precise answer.