Problem 57 - Precalculus Mathematics for Calculus

Question

An arc of length 100 m subtends a central angle \(\theta\) in a circle of radius \(50 \mathrm{m}\). Find the measure of \(\theta\) in degrees and in radians.

Step-by-Step Solution

Answer

The measure of \( \theta \) is 2 radians or approximately 114.59 degrees.

Step 1: Understand the Problem

We are given an arc length and need to find the central angle it subtends. The circle has a specified radius. We need to find the angle both in degrees and radians.

Step 2: Use the Arc Length Formula in Radians

The formula for arc length in terms of radians is given by \( s = r \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians. Here, \( s = 100 \) meters and \( r = 50 \) meters.

Step 3: Solve for \( \theta \) in Radians

Rearrange the formula to find \( \theta \): \( \theta = \frac{s}{r} = \frac{100}{50} = 2 \) radians.

Step 4: Convert Radians to Degrees

To convert from radians to degrees, use the conversion factor \( 180^\circ = \pi \) radians. Therefore, \( \theta = 2 \times \frac{180^\circ}{\pi} \approx 114.59^\circ \).

Key Concepts

Arc Length

The concept of arc length is essential when dealing with circle geometry. Arc length is the distance measured along the curved line forming part of the circle's circumference. It represents a fraction of the entire circumference. You might think of it like the measure of a slice of pie. To find the arc length, you multiply the radius of the circle by the central angle in radians. The formula is:

\[ s = r \theta \]where:

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

In our exercise, if you know two of these values, you can find the third by rearranging the formula. It shows how several aspects of a circle are interconnected. Understanding arc length helps in calculating distances found along a circle's edge.

Central Angle

Central angle is another cornerstone of circle geometry. It is the angle subtended at the center of the circle by an arc. Essentially, it's how much of the circle's "pie" the arc covers.

The central angle is crucial because it directly determines the arc length and sector area. In radians, this angle is equivalent to the arc length divided by the radius. Understanding the central angle concept allows different parts of the circle to be related back to one another. In our example, it helped us find the missing value when we rearranged the formula.

Central angles are split into:

• Radian measure – directly tied to the arc length
• Degree measure – more common in everyday usage

You can always rely on the formula \( \theta = \frac{s}{r} \) to find the central angle in radians and use conversions for degrees.

Radian Conversion

Radians and degrees are two different ways of measuring angles, much like feet and meters measure distance. Knowing how to convert between them is vital for solving trigonometric problems. A full circle is \(2\pi\) radians, equivalent to 360 degrees.

Hence, the conversion factor is:

• 1 radian = \(\frac{180^\circ}{\pi}\)
• 1 degree = \(\frac{\pi}{180^\circ}\)

In our example, once we found the central angle in radians as 2, we simply multiplied by the conversion factor to switch to degrees. This is helpful in understanding the scale of the angle in a variety of contexts, whether mathematical or real-world applications, like construction or navigation.

Establishing comfort with these conversions will allow for greater flexibility and understanding as you delve deeper into trigonometry.

Circle Geometry

Circle geometry involves the study of shapes, sizes, and the positions of figures within a circle. It's a foundational element of trigonometry, often beginning with simpler properties and progressing to more complex concepts. Key components include radius, diameter, circumference, arc, and the central and inscribed angles.

Here are some terms frequently used in circle geometry:

• Radius (\( r \)) – distance from the circle's center to the edge
• Diameter – twice the radius, spanning the circle's widest point
• Circumference – the total distance around the circle, given by \( 2\pi r \)
• Arc – fraction of the circumference, with its length linking directly to central angles

Together, these concepts allow for the calculation of area, angle measures, and more. Understanding circle geometry is crucial for a wide array of higher-level math problems and practical applications. It consolidates your comprehension of how all elements of a circle are interrelated.