Problem 57 - Algebra and Trigonometry

Question

An arc of length 100 \(\mathrm{m}\) 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

\( \theta = 2 \, \text{radians} \) or approximately \( 114.59^{\circ} \).

Step 1: Understanding the Given Values

We are given the arc length \( s = 100 \, \mathrm{m} \) and the radius \( r = 50 \, \mathrm{m} \) of the circle. These values will be used in the arc length formula for a circle.

Step 2: Using the Arc Length Formula

The formula for arc length is \( s = r \theta \), where \( \theta \) is in radians. Plug in the given values: \( 100 = 50 \theta \).

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

To solve for \( \theta \), divide both sides of the equation by 50: \( \theta = \frac{100}{50} = 2 \, \text{radians} \).

Step 4: Converting Radians to Degrees

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

Key Concepts

Arc Length

Arc length is a fundamental concept in trigonometry that relates to the distance measured along the curve of a circle. To calculate the arc length (often denoted as \( s \)), we use the formula:

• \( s = r\theta \)

Here, \( r \) represents the radius of the circle, and \( \theta \) is the central angle in radians.
Understanding how to compute arc length is crucial because it allows us to find how far a point on the circumference has traveled.
In this specific example, the arc length is 100 meters, and the radius of the circle is 50 meters.
By substituting into the formula \( s = r\theta \), we can solve for \( \theta \).
This formula shows the direct proportionality between arc length, radius, and the subtended angle. Larger angles or larger circles will result in longer arcs.

Central Angle

The central angle is an angle formed between two radii in a circle, whose vertex is at the circle's center. This is denoted by \( \theta \) and is measured in radians or degrees.
In the given problem, we used the known values from the arc length and the radius to find the central angle. Since the arc length formula \( s = r\theta \) was employed, it implied that the angle was already in radians.
Solving for the central angle \( \theta \) when the arc length \( s \) is given involves rearranging the formula:

• Divide both sides by the radius: \( \theta = \frac{s}{r} \).

This formula highlights the idea that the central angle and the arc length are directly proportional, provided the radius is constant.
In our example, after performing the division, \( \theta \) was found to be 2 radians.

Radian to Degree Conversion

Radians and degrees are two units for measuring angles. Converting between them is essential in trigonometry and geometry.

• 1 radian is approximately 57.2958 degrees.
• The full circle is \( 2\pi \) radians or 360 degrees.

To convert from radians to degrees, use the formula:

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

In our scenario, where \( \theta = 2 \) radians, conversion involves multiplying by the factor \( \frac{180}{\pi} \).
This gives us an approximate central angle of 114.59 degrees.
Understanding this conversion process is important because it facilitates interpreting angles in different contexts and applications. When angles are given in radians, often computations become simpler, especially in calculus and higher mathematics.