Problem 133 - Precalculus

Question

In calculus we work with real numbers; thus, the measure of an angle must be in radians. What is the measure (in radians) of a central angle \(\theta\) that intercepts an arc of length \(2 \pi \mathrm{cm}\) on a circle of radius \(10 \mathrm{cm} ?\)

Step-by-Step Solution

Answer

The measure of the central angle \(\theta\) is \(\frac{\pi}{5}\) radians.

Step 1: Understanding the Relationship

In a circle, the arc length \(s\) can be calculated using the formula \(s = r \theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians. We are given that the arc length \(s\) is \(2\pi \) cm and the radius \(r\) is 10 cm. We need to find \(\theta\).

Step 2: Rearranging the Formula

Since we want to find \(\theta\), we can rearrange the formula \(s = r \theta\) to solve for \(\theta\). This gives us \(\theta = \frac{s}{r}\).

Step 3: Substituting Known Values

Now, substitute the given values into the formula: \(\theta = \frac{2\pi}{10}\).

Step 4: Simplifying the Expression

Simplify \(\frac{2\pi}{10}\) by dividing both the numerator and the denominator by 2. This results in \(\theta = \frac{\pi}{5}\).

Key Concepts

Angle Measurement in Radians

When we measure angles, especially in calculus and circle geometry, we often use radians instead of degrees. Radians offer a natural way of measuring angles, relating them directly to the properties of the circle.
Simply put, an angle measured in radians is based on the radius of the circle.
Here are some key points:

• One radian is the angle formed when the arc length is equal to the radius of the circle.
• The full circumference of a circle is given by the formula \(2\pi r\), where \(r\) is the radius. Since the circumference is also an arc, the total angle in radians around a circle is \(2\pi\) radians.
• This relationship makes working with radians mathematically convenient, especially when dealing with trigonometric functions and calculus.

By understanding radians, you can easily convert between degrees and radians, use them in formulas for simplifying expressions and solving problems involving circles.

Circle Geometry Basics

Circle geometry revolves around understanding the various properties and segments of a circle.
A circle is not just a simple rounded shape; it includes:

• The radius \(r\), which is a line from the center to the circumference.
• The diameter \(d\), which is twice the radius.
• The circumference, which is the distance around the circle, represented as \(2\pi r\).
• Other parts like chords, arcs, and tangents.

These elements help in finding the relationships and solving problems involving angles, lengths, and areas.
One of the most fundamental relationships in circle geometry is the formula for arc length, which is used to determine the length of the portion of a circle's edge.

Understanding Arc Length

Arc length is the distance measured along the curved line making up the arc. It's a part of the measurement of a circle and depends directly on the angle and the radius.
The formula for arc length is:\[s = r \theta\]Where:

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

This formula shows a direct proportion between arc length and the size of the angle when the radius is constant. If either the angle or the radius changes, the arc length will change accordingly.
Using this concept, it is possible to determine the measure of angles or distances in real-world applications involving circles.

Exploring the Central Angle

The central angle is a specific type of angle whose vertex is the center of the circle.
Its interception of the circle's circumference makes it unique:

• The size of a central angle is directly related to the arc it intercepts.
• Because of this direct relationship, the formula \(\theta = \frac{s}{r}\) can be derived to find the angle in radians.
• This relationship is crucial not just for solving problems but also in engineering and design, where precise angle measurements are required.

By understanding the properties of central angles, you can find solutions easily when dealing with arcs or determining the position of components in a circle. Knowing the central angle allows for accurate predictions of how two points on a circle relate to each other.