Problem 25 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. $$ -\frac{2 \pi}{15} $$

Step-by-Step Solution

Answer

The angle is \(-24\) degrees.

Step 1: Understand Radian to Degree Conversion Formula

To convert an angle from radians to degrees, use the formula: \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\). This means if you have an angle in radians, you multiply it by \(\frac{180}{\pi}\) to find the equivalent in degrees.

Step 2: Write the Radian Measure

The angle given is \(-\frac{2 \pi}{15}\). In this step, ensure that you understand that \(-\frac{2 \pi}{15}\) is the angle currently in radians that you will convert to degrees.

Step 3: Apply the Conversion Formula

Convert \(-\frac{2 \pi}{15}\) to degrees by multiplying it with \(\frac{180}{\pi}\):\[\text{Degrees} = -\frac{2 \pi}{15} \times \frac{180}{\pi}\]

Step 4: Simplify the Expression

Cancel the \(\pi\) in the numerator with the \(\pi\) in the denominator and calculate the result:\[= -\frac{2 \times 180}{15}\]

Step 5: Perform the Multiplication and Division

Calculate the multiplication and division:\(= -\frac{360}{15}= -24 \) Thus, \(-\frac{2 \pi}{15}\) radians is equivalent to \(-24\) degrees.

Key Concepts

Understanding Angle Conversion

Angle conversion is the process of changing the unit of an angle from one measurement to another. The most common conversions involve switching between degrees and radians. This concept is essential in fields such as mathematics, physics, and engineering.

Both degrees and radians serve as ways to measure angles, but they use different scales:

• Degrees divide a circle into 360 equal parts. For instance, a right angle equals 90 degrees.
• Radians use the arc length and radius, where the full circle equals approximately 6.283 radians (or twice \\(\pi\), because the circumference of a circle divided by its radius equals \\(2\pi\)).

By understanding the basic conversion formula, which is related to the circumference of a circle, converting between radians and degrees becomes much clearer.

Converting Radians to Degrees

To change an angle from radians to degrees, the fundamental formula is:

• \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)

This means if you have an angle in radians, you multiply it by \(\frac{180}{\pi}\) to find its degree measure.

Let's see how it works with the example of \(-\frac{2 \pi}{15}\). First, we apply the conversion formula:\[\text{Degrees} = -\frac{2 \pi}{15} \times \frac{180}{\pi}\]
Cancel out \(\pi\) in both the numerator and denominator to simplify the expression:\[-\frac{2 \times 180}{15}\]Then complete the multiplication and division to get:\[-\frac{360}{15} = -24\]Thus, \(-\frac{2 \pi}{15}\) radians equals \(-24\) degrees.

Mathematics Education and Angle Conversion

In mathematics education, understanding how to convert angles is crucial. It supports a deeper comprehension of concepts like trigonometry and geometry. Here are some educational tips:

• Always start by learning the basic conversion formulas. Memorizing \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\) is a useful step.
• Practice makes perfect: Regularly using the conversion process strengthens your skills.
• Visual aids, such as unit circles, can help illustrate how radians and degrees relate.
• Work through real-world examples, which can be valuable in seeing the applications of conversions.

Students who understand angle conversions often find it easier to tackle related math problems and apply their knowledge in practical situations, such as engineering or navigation.