Problem 55 - 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

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

Step 1: Understand the relationship between arc length, angle, and radius

The formula for arc length \( L \) in terms of the central angle \( \theta \) in radians and the radius \( r \) of the circle is given by \( L = r \theta \). Here, \( L = 100 \) m and \( r = 50 \) m.

Step 2: Solve for \(\theta\) in radians

Using the formula \( L = r \theta \), substitute the given values: \( 100 = 50 \theta \). Solve for \( \theta \) by dividing both sides by 50, giving \( \theta = \frac{100}{50} = 2 \) radians.

Step 3: Convert \(\theta\) from radians to degrees

To convert an angle from radians to degrees, use the conversion factor \( \frac{180}{\pi} \). Therefore, \( \theta \) in degrees is \( 2 \times \frac{180}{\pi} \approx 114.59 \) degrees.

Key Concepts

Arc Length

Arc length is the distance along the curved line forming the arc, which is a section of the circumference of a circle. It can be thought of as a string that fits snugly along the arc's curve.
In a circle, the arc length can be determined using the formula:

• \( L = r \theta \)

Here, \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians. This relationship shows that the arc length is directly proportional to both the circle's radius and the central angle.
If either the central angle increases or the radius becomes longer, the arc stretches further, producing a greater arc length. Knowing how to find the arc length is crucial for various applications in geometry and real-life scenarios, such as finding the actual path length on a circular track.

Central Angle

The central angle is the angle that is formed at the center of a circle by two radii extending to the circumference. It serves as a critical component in determining both the arc length and sector area of a circle.
This angle, denoted as \( \theta \), can be measured in either degrees or radians, though it is often expressed in radians in mathematical formulas. Using radians simplifies calculations since the arc length-formula becomes straightforward. If the central angle \( \theta \) is given, finding the arc length or the sector area becomes an easy task.
The central angle's measure is directly used in the formula \( L = r \theta \) to calculate the arc length, emphasising its role as a significant element in circle geometry. The larger the angle, the more significant portion of the circle's circumference or area it encloses.

Circle Radius

The radius of a circle is the distance from the center point to any point on the circumference. It's one of the fundamental properties of a circle and is typically represented by the symbol \( r \).
The radius is essential because it forms the base of most calculations involving circles such as determining the circle's circumference (\( 2\pi r \)) or its area (\( \pi r^2 \)).

• It also plays a critical role in the arc length formula: \( L = r \theta \).
• In the context of central angles, the radius helps define the segment of the circle you are looking at and calculates how long the arc corresponding to that segment will be.

Thus, knowing the radius allows you to explore various properties of the circle and is indispensable in tasks involving geometry, circular motion, and even designing circular objects.

Degree to Radian Conversion

Degrees and radians are two units of measuring angles. Degrees measure angles in terms of parts of a circle, with a full circle amounting to 360 degrees. On the other hand, radians are a way of relating angles directly to the radius of the circle.
Converting an angle from degrees to radians involves understanding that \( 360 \) degrees is equivalent to \( 2\pi \) radians. Therefore, each degree equals \( \frac{\pi}{180} \) radians, and the conversion formula is

• \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

For instance, an angle of \( 45\) degrees becomes \( \frac{\pi}{4} \) radians. Conversely, when converting from radians to degrees, use the factor \( \frac{180}{\pi} \).
Understanding how to convert between these units is crucial, as it lets you switch calculations and perspectives depending on the context of a problem. In practical scenarios, most mathematical calculations involving circles, such as those in trigonometry, favor using radians because of their intrinsic link to the circle's properties.