Problem 24 - Precalculus Mathematics for Calculus

Question

Find the degree measure of the angle with the given radian measure. $$\frac{5 \pi}{18}$$

Step-by-Step Solution

Answer

The degree measure of the angle is 50 degrees.

Step 1: Understand the Relationship Between Radians and Degrees

One complete circle is 360 degrees or \(2\pi\) radians. Therefore, the conversion factor between degrees and radians is \(\frac{180}{\pi}\). This means that you can convert radians to degrees by multiplying the radian measure by \(\frac{180}{\pi}\).

Step 2: Apply the Conversion Formula

To convert \(\frac{5 \pi}{18}\) radians into degrees, multiply by \(\frac{180}{\pi}\):\[\frac{5 \pi}{18} \times \frac{180}{\pi}\]

Step 3: Simplify the Expression

The \(\pi\) terms in the numerator and denominator cancel each other, so you are left with:\[\frac{5}{18} \times 180\]

Step 4: Perform the Multiplication

Now, calculate the multiplication:\[\frac{5}{18} \times 180 = \frac{900}{18}\]

Step 5: Divide to Get the Degree Measure

Perform the division to find the angle in degrees:\[\frac{900}{18} = 50\]

Key Concepts

Radians and Degrees Relationship

Understanding the relationship between radians and degrees is fundamental in learning about angle measurements. A full circle covers an angle of 360 degrees. However, there is another way to measure angles, known as radians. Radians come into play as a unit based on the radius of a circle. One full circle is equal to \(2\pi\) radians. Thus, remembering the conversion between degrees and radians is essential: \(360\text{ degrees} = 2\pi\text{ radians}\). This helps store two key facts:

• 180 degrees is equivalent to \(\pi\) radians.
• 1 radian equals approximately 57.2958 degrees (since \(\pi\approx 3.1416\)).

If you want to switch from one unit to another, you can use the conversion factor \(\frac{180}{\pi}\) because \(\frac{360}{2\pi} = \frac{180}{\pi}\). This conversion factor allows for easy transformation between radians and degrees.

Angle Measurement

Angles measure the amount of rotation between two rays meeting at a common endpoint, called the vertex. Degrees and radians are two ways to measure these angles:

• Degrees: This is the more familiar unit, usually taught first in school. A full circle is 360 degrees. Common angles include 90 degrees for a right angle and 180 degrees for a straight line.
• Radians: This unit is often used in higher mathematics, such as calculus, because it simplifies many mathematical equations. A full circle is \(2\pi\) radians, and \(\pi\) radians is equivalent to 180 degrees.

For day-to-day use, degrees provide a more intuitive understanding of angles. Radians, however, are frequently used in mathematical calculations and specific fields like engineering and physics.

Circle in Degrees

A circle is a shape with constant curvature, and the measure of angles within a circle is foundational to geometry. When we talk about a circle in degrees, we split the circular path into 360 equally spaced parts. This division is purely conventional, dating back to ancient civilizations who wanted a system that divided nicely into parts like halves, quarters, and so forth.Here's why a circle is so conveniently measured in degrees:

• Each degree represents \(\frac{1}{360}\)th of a full circle.
• A quarter turn on a circle, like a corner, is 90 degrees.
• Half a circle measures 180 degrees, making calculations like these conversions straightforward.

Though degrees are not the only way to measure a circle, they provide a clear concept easily visualized and converted when necessary to other units like radians. Knowing how to interconvert helps solve many geometry problems efficiently.