Problem 10 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{19 \pi}{3}\)

Step-by-Step Solution

Answer

The measure is 1140 degrees.

Step 1: Convert Radians to Degrees Formula

To convert a radian measure to degrees, you use the formula: \[ Degrees = Radians \times \frac{180}{\pi} \]

Step 2: Substitute Radian Value

Plug in the provided radian measure into the formula: \[ Degrees = \frac{19 \pi}{3} \times \frac{180}{\pi} \]

Step 3: Simplify the Equation

First, cancel out \( \pi \) from the numerator and denominator: \[ Degrees = \frac{19}{3} \times 180 \] Next, calculate the resulting multiplication: \[ Degrees = 19 \times 60 \]

Step 4: Calculate Final Result

Perform the multiplication to get the final measurement in degrees: \[ Degrees = 1140 \]

Key Concepts

Angle Measurement

In the world of mathematics, measuring angles is fundamental. We primarily use two units to measure angles: degrees and radians. Understanding these units is key to solving various mathematical problems efficiently.

• Degrees: This is perhaps the most familiar unit of angle measurement. A full circle contains 360 degrees. Each degree is further divided into 60 minutes, and each minute is divided into 60 seconds. When we speak about angles like 90° or 180°, we're talking in degrees.
• Radians: This unit is rooted in the concept of "radius." A radian measures the angle created when a circle's radius is wrapped around its circumference. A complete circle is made up of approximately 6.283 radians, or \(2\pi\).

Recognizing when to use each unit is important. Degrees are often used in everyday situations, like reading maps or measuring turn angles. Radians, however, are more common in higher mathematics, physics, and engineering due to their natural properties in equations.

Radian to Degree Formula

Converting between radians and degrees is a crucial skill in mathematics and science. The conversion involves a simple formula, thanks to the mathematical relationships between these two units.

The central formula to remember for converting radians to degrees is: \[ Degrees = Radians \times \frac{180}{\pi} \]This formula stems from the fact that a full circle is 360 degrees or \(2\pi\) radians. Simplified, this means \(\pi\) radians is equivalent to 180 degrees. Hence, the multiplication factor (\(\frac{180}{\pi}\)) is derived from this equivalence.

Using the formula enables you to swap radians for degrees, making mathematical operations with angle measures flexible. This flexibility is powerful, allowing scientists and engineers to switch between units as they model natural phenomena, analyze data, and solve equations in diverse fields.

Mathematical Conversion Steps

To convert radians to degrees, following the step-by-step approach ensures accuracy and understanding. Let's walk through the steps:

• Step 1: Start with the Radian to Degree formula: \[ Degrees = Radians \times \frac{180}{\pi} \]
• Step 2: Substitute the given radian value into the formula. For our example, this would be: \[ Degrees = \frac{19\pi}{3} \times \frac{180}{\pi} \]
• Step 3: Simplify the expression by cancelling out \(\pi\) from both the numerator and the denominator. This gives us: \[ Degrees = \frac{19}{3} \times 180 \]
• Step 4: Perform the multiplication. First, calculate \(\frac{180}{3}\) resulting in 60, then multiply 19 by 60 to arrive at our final measure: \[ Degrees = 1140 \]

These steps break down the conversion process into manageable parts. With practice, this exercise enhances your skill in transforming angles between radians and degrees, and builds confidence in handling related problems.