Problem 44 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{29 \pi}{4}\)

Step-by-Step Solution

Answer

The angle \(\frac{29\pi}{4}\) radians converts to 1305 degrees.

Step 1: Understand the Conversion

To convert an angle given in radians to degrees, we use the conversion factor, where 180 degrees is equivalent to \(\pi\) radians. Therefore, 1 radian equals \(\frac{180}{\pi}\) degrees.

Step 2: Apply the Conversion Factor

Multiply the given radian measure by \(\frac{180}{\pi}\) to convert it to degrees. So, for \(\frac{29\pi}{4}\) radians, the conversion to degrees is computed as such:\[\frac{29\pi}{4} \times \frac{180}{\pi} \]

Step 3: Simplify the Expression

Remove \(\pi\) from the numerator and denominator, and simplify the expression:\[\frac{29 \times 180}{4}\]

Step 4: Perform the Multiplication and Division

Calculate the simplified expression:1. Calculate \(29 \times 180 = 5220\).2. Divide by \(4\): \(\frac{5220}{4} = 1305\) degrees.

Key Concepts

Radians to Degrees

Converting angles from radians to degrees is a common task in mathematics. Radians and degrees are both units for measuring angles, and they are essential in various fields like geometry and trigonometry. The conversion between these units is important because certain formulas or problems require angles to be expressed in a specific measure. In this context, converting \(\frac{29\pi}{4}\)radians to degrees involves multiplying by a special number known as the conversion factor.
Understanding this conversion allows you to tackle mathematical problems flexibly, as you can easily switch between these two systems depending on what your problem requires.

Conversion Factor

The conversion factor is a crucial element when converting an angle from radians to degrees. This factor is derived from the fundamental equivalence that \(180\)degrees equals \(\pi\)radians.

• Therefore, \(1\)radian is equivalent to \(\frac{180}{\pi}\)degrees.
• To convert radians to degrees, simply multiply the radian measure by this factor.
• This conversion is handy because many mathematical and real-world problems might give angle measurements in radians but require interpretation or action in degrees.

In the example of converting \(\frac{29\pi}{4}\)radians into degrees, multiplying by \(\frac{180}{\pi}\)provides the required degree measure.

Degree Measure

When translating radians to degrees, the goal is to express the angle measure in a more intuitive unit for most people. Degrees are often preferred in educational and practical settings because they split the circle into 360 parts.
In the solution, multiplying \(\frac{29\pi}{4}\)by \(\frac{180}{\pi}\)simplifies the \(\pi\)terms and lets you compute:

• The numerator calculation: \(29 \times 180 = 5220\)
• The division by \(4\)leads to \(1305\) degrees.

This calculation leads to the degree measure, \(1305\)degrees, which represents the same rotation or angle size as \(\frac{29\pi}{4}\)radians but in a format that's typically easier to visualize and use. Understanding degree measure simplifies the representation and comprehension of angles.

Mathematics Education

In the realm of mathematics education, converting between radians and degrees is a fundamental skill. It's critical for students to understand both units and how they relate to each other.

• Radial measurements are often used in calculus and higher mathematics.
• Degrees are more visual and are used in fields like engineering and navigation.
• This conversion shows the application of algebraic manipulation and simplification, supporting a deeper understanding of trigonometric concepts.

Utilizing practical examples like converting \(\frac{29\pi}{4}\)radians into \(1305\)degrees helps learners see the real-world application of these conversions. As students frequently encounter both units in different math problems, mastering conversion fosters mathematical flexibility and problem-solving skills.