Problem 24 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{5 \pi}{6}\)

Step-by-Step Solution

Answer

\( \frac{5\pi}{6} \) is 150 degrees.

Step 1: Convert Radians to Degrees Formula

To convert an angle from radians to degrees, we use the formula: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]},

Step 2: Identify Given Values

In this exercise, we are given the radian measure \( \frac{5\pi}{6} \). We will convert it to degrees using the formula identified in the previous step.

Step 3: Plug Values into Formula

Insert \( \frac{5\pi}{6} \) into the formula: \[ \text{Degrees} = \frac{5\pi}{6} \times \frac{180}{\pi} \]

Step 4: Simplify Expression

First, cancel out \( \pi \) in the numerator and denominator:\[ \frac{5 \times 180}{6} \]This reduces to:\[ \frac{900}{6} \]

Step 5: Calculate Final Result

Divide 900 by 6:\[ \text{Degrees} = 150 \]

Step 6: Conclusion

The radian measure \( \frac{5\pi}{6} \) is equivalent to 150 degrees.

Key Concepts

Angle Measurement

When we talk about angles, we're dealing with the concept of rotation around a point. Angles can be measured in different units, mainly degrees and radians. Understanding angles is important in fields like geometry, trigonometry, and even in real-world applications such as engineering and navigation.

Let's break down how angle measurement works:

• Degrees: This is the most common unit for measuring angles. A full circular turn is 360 degrees, with each degree representing 1/360th of a circle.
• Radians: Another method of measuring angles, where a full circle is 2π radians. This is often used in higher mathematics because of its natural relation to circles and trigonometric functions.

By understanding these two systems of measuring angles, you can switch between them using conversion formulas, making it easier to solve various problems.

Radian Measure

Radians are a unit of angular measure that relates directly to the radius of a circle. One radian is the angle created when the arc length equals the radius of the circle. Because of this relationship, radians align perfectly with trigonometric functions and are thus heavily utilized in advanced mathematics.

To understand radian measure, remember:

• One complete revolution (full circle) is defined as 2π radians.
• A half-rotation, or straight angle, measures π radians, approximately 3.1416.
• Radians offer a simpler and more natural way to perform calculus operations, especially in trigonometry.

Knowing how to measure an angle in radians is a critical skill for anyone studying mathematics, as it bridges the gap between geometry and calculus.

Degree Measure

Degrees are the most familiar unit of measuring angles for everyday use. When describing angles in degrees, we usually think in terms of portions of a full rotation, which is 360 degrees.

Key concepts about degree measure include:

• One degree is 1/360th of a full circle. This unit of measurement is intuitive for many geometric problems.
• Common angles include 90 degrees (a right angle), 180 degrees (a straight angle), and 270 degrees.
• Degrees are easily divided into smaller units: the minute (1/60th of a degree) and the second (1/60th of a minute).

Using degrees simplifies working with geometric shapes and angles, and they are one of the first concepts introduced in learning geometry.

Conversion Formula

Converting between radians and degrees is crucial for applying the correct units in mathematical problems. The conversion formula allows for the translation of an angle given in one unit to its equivalent in another, ensuring mathematical calculations maintain their integrity.

Here's how the conversion formula works:

• To convert from radians to degrees, use: \[ ext{Degrees} = ext{Radians} \times \frac{180}{\pi} \]
• To convert from degrees to radians, use: \[ ext{Radians} = ext{Degrees} \times \frac{\pi}{180} \]

Understanding and applying these formulas lets you effortlessly switch between radian measure and degree measure. In our example, using the formula to convert \(\frac{5\pi}{6}\) radians gives us 150 degrees, a straightforward way to represent the same angle in different units.