Problem 63 - Precalculus : Building Concepts and Connections

Question

Convert each angle from radians to degrees. $$3 \pi$$

Step-by-Step Solution

Answer

The angle \(3 \pi\) radians is equivalent to 540 degrees when converted.

Step 1: Identify the given angle in radian

The angle provided in the problem is \(3 \pi\) radians.

Step 2: Apply the conversion factor

One radian is equivalent to \( \frac{180}{\pi} \) degrees. Therefore, to convert \(3 \pi\) radians to degrees, multiply the given radian value by this conversion factor: \(3 \pi\) x \( \frac{180}{\pi} \). The \( \pi \) in the numerator and denominator will cancel each other out.

Step 3: Perform the calculation

This leaves \(3 \times 180\), which equals 540.

Step 4: Write down the final answer

After converting it to degrees, the solution is 540 degrees.

Key Concepts

Radians to Degrees

When it comes to angle conversion, understanding how to switch between radians and degrees is essential. An angle can be expressed in two main units: radians and degrees. Radians are often used in mathematical and engineering contexts, while degrees are more common in everyday situations, like navigating maps or adjusting angles in construction.
To convert an angle from radians to degrees, you multiply the radians by a specific conversion factor. This factor is used to bridge the gap between the two different measurements of angle. Recognizing when and how to apply this conversion is key to mastering problems that involve angle measurement in these two units.
For example, if we have an angle of \(3\pi\) radians, converting this to degrees makes it more relatable to most people, as degrees are a familiar concept and largely intuitive.

Conversion Factor

The conversion factor between radians and degrees is central to changing one unit of angle measurement to another. This factor acts as a bridge between the two systems:

• One complete turn around a circle is \(360\) degrees or \(2\pi\) radians.
• This relationship gives us the conversion: \(1\) radian equals \(\frac{180}{\pi}\) degrees.

When you multiply the radian measure by this conversion factor, the \(\pi\) terms cancel out. Thus, converting \(3\pi\) radians means multiplying by \(\frac{180}{\pi}\), resulting in \(3\times 180=540\) degrees.
This process demonstrates how the conversion factor seamlessly transforms the unit from radians to degrees, making sense of the angle in a context that might be more convenient or appropriate for the task at hand.

Angle Measurement

Angle measurement is an important concept in mathematics and various fields of study that involves the measure of rotation or direction.
There are two common units for measuring angles:

• Radians: Used widely in mathematics, particularly in calculus and trigonometry. One radian is the angle created when the radius is wrapped around the circumference of a circle.
• Degrees: Commonly used in everyday applications, such as navigation, engineering, and architecture. One complete circle measure is 360 degrees.

Understanding both units allows you to work in different contexts, translating solutions back into familiar terms when needed, like from \(3\pi\) radians to more intuitive degrees.
Grasping these different angle measurements empowers you to analyze and dissect problems that involve rotational motion, geometric calculations, or even waves and oscillations in physics.