Problem 66 - Precalculus with Calculus Previews

Question

In Problems \(65-68\), find the measure of a central angle \(\theta\) in a circle of radius \(r\) that subtends an arc length s. Give \(\theta\) in (a) radians and (b) degrees. $$ r=10 \mathrm{in}, s=36 \mathrm{in} $$

Step-by-Step Solution

Answer

\( \theta = 3.6 \) radians or approximately \( 206.26 \) degrees.

Step 1: Understand the Problem

We need to find the central angle \( \theta \) that subtends an arc length \( s = 36 \) inches in a circle of radius \( r = 10 \) inches. We'll find \( \theta \) in both radians and degrees.

Step 2: Use Formula for Radians

The formula for the central angle \( \theta \) in radians is given by \( \theta = \frac{s}{r} \). We'll plug in \( s = 36 \) inches and \( r = 10 \) inches to find \( \theta \) in radians.

Step 3: Calculate \( \theta \) in Radians

Using the formula \( \theta = \frac{s}{r} \), substitute \( s = 36 \) and \( r = 10 \): \( \theta = \frac{36}{10} = 3.6 \) radians.

Step 4: Convert Radians to Degrees

To convert radians to degrees, we use the conversion factor \( \frac{180}{\pi} \). The formula is \( \theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} \).

Step 5: Calculate \( \theta \) in Degrees

Apply the conversion: \( \theta_{degrees} = 3.6 \times \frac{180}{\pi} \). Calculating gives \( \theta_{degrees} \approx 206.26 \) degrees.

Key Concepts

Radians to Degrees Conversion

Converting an angle from radians to degrees is an essential skill in geometry and trigonometry. It lets you transition between these two units of measuring angles seamlessly. Radians are based on the radius of a circle, whereas degrees are based on dividing a circle into 360 equal parts. To convert radians to degrees, you multiply the radian measure by the conversion factor \( \frac{180}{\pi} \). This fraction stems from the fact that \( 180 \) degrees equals \( \pi \) radians. Consider an example where you have \( 3.6 \) radians. Using the conversion formula, the computation would look like:

• \( \theta_{degrees} = 3.6 \times \frac{180}{\pi} \)
• This simplifies to \( \theta_{degrees} \approx 206.26 \) degrees

This process scales similarly regardless of the radians involved, offering a simple yet powerful way to calculate angles in a different unit.

Arc Length

An arc is a segment of a circle, creating a curve that's part of the circle's circumference. Arc length, denoted as \( s \), quantifies the distance along this curve. In practical terms, the arc length is the part of the circle's perimeter that the angle subtends. To find an arc length, you can use the formula \( s = r \times \theta \). Here:

• \( r \) is the radius of the circle, and
• \( \theta \) is the central angle in radians.

For example, if you know the central angle is \( 3.6 \) radians and the circle's radius is \( 10 \) inches, calculating the arc length simply involves multiplying these two values: \( s = 10 \times 3.6 = 36 \) inches. This formula ties together the angle and radius in understanding the proportion of the circle covered by the arc.

Circle Radius

The circle radius \( r \) is the distance from the center point of the circle to any point on its edge. It's a fundamental property of the circle, serving as the backbone for many circular measurements. The radius directly influences calculations involving areas, circumferences, and arc lengths. For solving problems like determining the central angle, it acts as a scaling factor when combined with the arc length: \( \theta = \frac{s}{r} \). This formula shows:

• \( s \) is the arc length.
• \( r \) is the radius, dictating how large the circle is relative to its center.

By knowing the radius, you can easily find the measure of the angle subtending a particular arc. A larger radius means the same length of arc will subtend a smaller angle, as the extra circle size stretches out the arc. Thus, understanding the radius is critical for solving and appreciating structure within circular geometry.