Problem 56 - Precalculus Mathematics for Calculus

Question

A central angle \(\theta\) in a circle of radius \(5 \mathrm{m}\) is subtended by an arc of length \(6 \mathrm{m}\). Find the measure of \(\theta\) in degrees and in radians.

Step-by-Step Solution

Answer

\(\theta = 1.2\) radians or \(68.75^\circ\).

Step 1: Identify the Relationship

The relationship between the arc length \(s\), radius \(r\), and central angle \(\theta\) in radians is given by the formula: \(s = r \theta\). Here, \(s = 6 \mathrm{m}\) and \(r = 5 \mathrm{m}\).

Step 2: Calculate \(\theta\) in Radians

Rearrange the formula \(s = r \theta\) to solve for \(\theta\): \(\theta = \frac{s}{r}\). Substituting the given values, \(\theta = \frac{6}{5} = 1.2\) radians.

Step 3: Convert Radians to Degrees

To convert an angle from radians to degrees, use the conversion factor \(180^\circ/\pi\). Thus, \(\theta \text{ in degrees} = 1.2 \times \frac{180^\circ}{\pi} \approx 68.75^\circ\).

Key Concepts

Arc Length

Arc length is an essential concept in circle geometry. It refers to the length of a segment of the circumference of a circle. To calculate the arc length, we need to understand the relationship between the arc length, the circle’s radius, and the central angle. This relationship is represented by the formula:

• \( s = r \theta \)

where \( s \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. If an arc is subtended by a central angle \( \theta \) and you know the radius, you can easily determine the arc length by multiplying the radius and the angle in radians.
For example, when a circle has a radius of \( 5 \mathrm{m} \) and an arc length of \( 6 \mathrm{m} \), you can use the formula to find the measure of the central angle. Plug the values into the formula, and you will solve for \( \theta \):

• \( 6 = 5 \theta \)
• \( \theta = \frac{6}{5} = 1.2 \text{ radians} \)

This simple relationship allows you to find unknown measures as long as you have two pieces of related data.

Radians to Degrees Conversion

When dealing with angles, it’s often necessary to convert between different units of measurement, namely radians and degrees. Degrees measure how much of a circle an angle covers, with a full circle being \( 360\text{°} \). Radians offer a more natural measure, especially in higher mathematics, being the ratio of the arc length to the radius.
To convert an angle from radians to degrees, we use the formula:

• \( \text{Degrees} = \text{Radians} \times \frac{180^\circ}{\pi} \)

This conversion factor arises because there are \( 2\pi \) radians in a complete circle, which equals \( 360^\circ \).
So, if you have an angle of \( 1.2 \) radians and want to express it in degrees, multiply by \( \frac{180^\circ}{\pi} \):

• \( 1.2 \times \frac{180^\circ}{\pi} \approx 68.75^\circ \)

Understanding how to convert between these two units is crucial for working with angles efficiently.

Circle Geometry

Circle geometry is an entire branch of mathematics dedicated to understanding the properties and measurements associated with circles. A central concept here is the central angle, which is an angle whose vertex is at the center of the circle. The central angle subtends an arc, and its size can be measured in either radians or degrees.
The relationship between an angle, a radius, and an arc is foundational to circle geometry formulas. The central angle is especially significant because it can determine the measure of both the arc length and the sector area. Central angle formulas, such as:

• \( s = r \theta \)

for arc length, are applicable only when \( \theta \) is measured in radians.
The comprehensive understanding of these relationships allows students to solve various real-world and theoretical problems involving circles. Whether calculating the length of a circular track or the slice of a pie, the foundational principles of circle geometry, including the concept of central angles, are key to unlocking these solutions.