Problem 67 - 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. $$ r=9 \mathrm{~m}, s=15 \mathrm{~m} $$

Step-by-Step Solution

Answer

The central angle \( \theta \) is \( \frac{5}{3} \) radians or approximately 95.49 degrees.

Step 1: Understand the Problem

We are given a circle with radius \( r = 9 \) meters and an arc length \( s = 15 \) meters. We need to find the central angle \( \theta \) that subtends this arc, expressed in both radians and degrees.

Step 2: Formula for Central Angle in Radians

The formula for a central angle \( \theta \) in radians when the radius \( r \) and arc length \( s \) are given is: \[ \theta = \frac{s}{r} \]

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

Substitute the given values into the formula: \[ \theta = \frac{15 \text{ m}}{9 \text{ m}} = \frac{5}{3} \text{ radians} \]

Step 4: Convert Radians to Degrees

To convert the angle from radians to degrees, use the conversion formula: \[ \theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180}{\pi} \]

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

Substitute the radians value into the conversion formula: \[ \theta_{\text{degrees}} = \frac{5}{3} \times \frac{180}{\pi} \approx 95.49 \text{ degrees} \]

Key Concepts

Understanding Radian Measure

Radian measure is a way to express angles using the radius of a circle. Instead of measuring angles in degrees, radian measure uses the arc length that the angle subtends at the center of the circle. This can be very handy in circle geometry. To understand it, remember:

• A full circle is equal to an angle of 360 degrees, which corresponds to an arc length of the entire circumference, \(2\pi r\).
• Since radian measure is based on the radius, a full circle is \(2\pi\) radians as the circumference divided by the radius (\(r\)) is \(2\pi\).

This shows each radian corresponds to \(r\) units of length along the circle's circumference, offering a straightforward way to relate angles directly to the geometry of a circle.

Degree Conversion

Converting between radians and degrees is essential as some problems may require different angle measurements. Degrees are perhaps more familiar since they are widely used in various fields. To convert radians to degrees, use the formula:

• \(1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees}\)

Degree conversion involves multiplying the radian measure by \(\frac{180}{\pi}\). This conversion factor comes from dividing the angle of a full circle in degrees (360 degrees) by the angle in radians (\(2\pi\) radians). This results in the equality \(\frac{180}{\pi}\), which you can use to easily switch from radians to degrees and vice versa.

Exploring Circle Geometry

Circle geometry deals significantly with angles, radii, and arcs. Understanding the relationships between these elements is crucial. Some key concepts include:

• The center of a circle is equidistant from every point on the circle, and this distance is called the radius (\(r\)).
• A central angle is an angle whose vertex is at the center, and it subtends an arc—a portion of the circle itself.
• The arc length (\(s\)) is the measure along the circle between two points and is often related to the central angle.

Grasping these relationships helps in finding unknown measurements in circle problems, such as determining the arc length, the radius, or the central angle, given the other two.

Calculating Arc Length

Arc length is the distance along the curved path of a part of the circle's circumference. It varies depending on the size of the central angle. The formula to find the arc length is directly derived from the definition of a radian:

• \(s = r \times \theta\)

Where \(\theta\) is the angle in radians and \(r\) is the circle's radius. This formula indicates that the arc length is a fraction of the total circumference of the circle, proportional to the central angle. By understanding how each part of the circle relates to the others, you can effectively use this formula to solve for missing measurements.