Problem 54 - Algebra and Trigonometry

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

The angle \(\theta\) is 1.2 radians or approximately 68.75 degrees.

Step 1: Understanding the Arc Length Formula

To find the central angle \(\theta\), we use the arc length formula: \( L = r\theta \), where \(L\) is the arc length and \(r\) is the radius of the circle. We're given that \(L = 6\,\mathrm{m}\) and \(r = 5\,\mathrm{m}\).

Step 2: Solving for the Angle in Radians

By substituting the given values into the arc length formula, we get \(6 = 5\theta\). To solve for \(\theta\), divide both sides by 5: \(\theta = \frac{6}{5} = 1.2\) radians.

Step 3: Converting Radians to Degrees

To convert the measure of \(\theta\) from radians to degrees, use the conversion \(1\text{ radian} = \frac{180}{\pi}\) degrees. Thus, \(1.2\) radians is equivalent to \(1.2 \times \frac{180}{\pi}\) degrees. Calculating this gives approximately \(68.75\) degrees.

Key Concepts

Arc Length Formula

When working with circles, one of the useful formulas is the arc length formula, which helps determine the central angle subtended by an arc. The formula is expressed as \( L = r\theta \), where:

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

To solve for a central angle, you can rearrange the formula to \( \theta = \frac{L}{r} \). In the exercise, the circle has a radius \( r = 5 \, \mathrm{m} \) and the arc length \( L = 6 \, \mathrm{m} \). By substituting these values into the formula, the central angle \( \theta \) is determined as:\[ \theta = \frac{6}{5} = 1.2 \, \text{radians} \]This straightforward calculation is foundational in connecting distances along a curve with angle measurements within a circle.

Radian Measure

Radian measure is an alternative to degree measure for determining angles. Instead of dividing a circle into 360 degrees, the radian measure considers the length of the arc subtended by the angle as a fraction of the circle's circumference. A complete circle is \( 2\pi \) radians.Radian measure is based directly on the radius of the circle. This makes mathematical calculations involving angles more natural and simpler compared to degrees. For example:

• A full circle is \( 2\pi \) radians.
• A half circle is \( \pi \) radians.
• A quarter circle is \( \frac{\pi}{2} \) radians.

For the given problem, the measure of the angle in radians is \( \theta = 1.2 \) radians. Using this measure, you can easily relate the angle to physical properties of the circle in real-world applications.

Degree Conversion

Converting between radians and degrees is essential because both units are commonly used in different contexts. The conversion between these units is straightforward:

• \( 1 \) radian equals \( \frac{180}{\pi} \) degrees.
• \( 360 \) degrees equals \( 2\pi \) radians.

To convert radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \).For instance, converting \( 1.2 \) radians to degrees involves:\[ 1.2 \times \frac{180}{\pi} \approx 68.75 \text{ degrees} \]Understanding this conversion helps bridge the conceptual gap between the often-used degree system and the radian measure, which provides elegance in certain mathematical and engineering calculations.