Problem 16 - Algebra 2 and Trigonometry

Question

In \(13-22\) , find the degree measure of each angle whose radian measure is given. \(\frac{2 \pi}{5}\)

Step-by-Step Solution

Answer

The degree measure of \( \frac{2\pi}{5} \) radians is 72 degrees.

Step 1: Identify the Conversion Formula

To convert an angle from radians to degrees, we use the formula: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). This formula allows us to convert any given radian measure to degrees by multiplying by the factor \( \frac{180}{\pi} \).

Step 2: Substitute the Given Radian Measure into the Formula

Now, substitute the given radian measure \( \frac{2\pi}{5} \) into the conversion formula: \[ \text{Degrees} = \frac{2\pi}{5} \times \frac{180}{\pi} \].

Step 3: Simplify the Expression

First, simplify the expression by canceling \( \pi \) in the numerator and the denominator: \[ \text{Degrees} = \frac{2 \times 180}{5} \].

Step 4: Perform the Multiplication

Next, multiply \( 2 \times 180 \) to get \( 360 \): \[ \text{Degrees} = \frac{360}{5} \].

Step 5: Division to Get the Final Answer

Finally, divide \( 360 \) by \( 5 \) to get the degree measure: \[ \text{Degrees} = 72 \].

Key Concepts

Radian Measure

Radian measure is a way to express angles. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians are based on the radius of a circle. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. This might sound a bit complex at first, but it's a neat system that beautifully ties together geometry and mathematics. In terms of numerical value, there are approximately 6.283 radians in a full circle (equivalent to the circumference of the circle divided by its radius). Since the circumference is given by the formula \( 2\pi r \), a complete circle measures \( 2\pi \) radians. This means:

• Half a circle is \( \pi \) radians
• Quarter of a circle is \( \frac{\pi}{2} \) radians

Understanding radian measure not only helps in solving geometry problems but also in calculus and physics because it often simplifies calculus computations.

Degree Measure

The degree measure is likely more familiar to many people as it is commonly used in geometric problems and everyday life. When you talk about a 90-degree angle or a right angle, you're using degree measure. Degree measure divides a complete circle into 360 equal parts.
The origins of the 360-degree system are somewhat historical. The ancient Babylonians had a base-60 number system which led them to use 360 (close to 365 days of a year) for ease of calculations.
Knowing how to convert between degrees and radians is necessary since certain problems in physics and engineering require one or the other.

• 360 degrees \( = 2\pi \) radians
• 180 degrees \( = \pi \) radians

When dealing with degree measures, remember that angles can be acute (less than 90 degrees), right (exactly 90 degrees), or obtuse (greater than 90). These classifications can help when visualizing problems and solutions.

Mathematical Conversion Formulas

Mathematical conversion formulas are essential tools for quickly switching between radian and degree measures. Understanding how and why these formulas work makes it much easier to apply them.
The main conversion formula to remember is:

• Degrees \( = \text{Radians} \times \frac{180}{\pi} \)
• Radians \( = \text{Degrees} \times \frac{\pi}{180} \)

These formulas are derived from the fact that \( 360 \) degrees is equivalent to \( 2\pi \) radians. Thus, \( 1 \) radian equals \( \frac{180}{\pi} \) degrees. Conversely, \( 1 \) degree is equal to \( \frac{\pi}{180} \) radians.
To apply these formulas, ensure you multiply across and cancel out appropriately, like when you dealt with \( \frac{2\pi}{5} \) radians earlier. You substituted it into the degree conversion formula:\[ \text{Degrees} = \frac{2\pi}{5} \times \frac{180}{\pi} \]By canceling \( \pi \), you simplify to \( \frac{360}{5} \), which gives you the degree measure once divided.