Problem 36 - Basic Technical Mathematics with Calculus

Question

The measure of \(\widehat{A B}\) on a circle of radius \(r\) is \(45^{\circ} .\) What is the length of the arc in terms of \(r\) and \(\pi ?\)

Step-by-Step Solution

Answer

The arc length is \( \frac{\pi r}{4} \).

Step 1: Convert Angle to Radians

The problem measures the angle in degrees, but it is often useful to work in radians when dealing with circles. The conversion from degrees to radians is given by the formula: \[ \text{radians} = \frac{\text{degrees} \times \pi}{180} \]Substitute the given angle of \(45^{\circ}\):\[ \text{radians} = \frac{45 \times \pi}{180} = \frac{\pi}{4} \]

Step 2: Use Arc Length Formula

The length of an arc \(L\) on a circle is given by the formula:\[ L = r \times \theta \]where \(r\) is the radius of the circle and \(\theta\) is the angle in radians. Substituting the converted angle \(\theta = \frac{\pi}{4}\) from Step 1 into this formula gives:\[ L = r \times \frac{\pi}{4} \]

Step 3: Simplify the Expression

The expression for the arc length is already in a simplified form: \[ L = \frac{\pi r}{4} \]No further simplification is needed.

Key Concepts

Radian Conversion

When studying circle geometry, working in radians simplifies calculations involving angles. A radian is an alternative way to express angles, often preferred in mathematics due to its direct relation to the circle's radius.
To convert an angle from degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This is because a full circle is 360 degrees, which corresponds to \( 2\pi \) radians.
Thus, the formula to convert degrees into radians is:

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

For instance, if you have an angle of 45 degrees, this converts into:

• \( \frac{45 \times \pi}{180} = \frac{\pi}{4} \) radians.

Understanding radian conversion is crucial, especially when dealing with trigonometric functions, as they naturally operate in radians in advanced mathematics.

Circle Geometry

Circle geometry is a fascinating field that revolves around the properties and measurements of circles. A key aspect of this is understanding arcs, which are portions of the circle's circumference.
The essential formula for finding an arc's length is derived from the relationship between the radius and the angle in radians:

• Arc Length \( L = r \times \theta \)

Here, \( r \) is the circle's radius, and \( \theta \) is the angle in radians. This formula highlights the proportional relationship between the angle and the circle's size, revealing how they combine to determine the arc length.
If the angle is in degrees, converting it to radians is necessary to use this formula effectively. Circle geometry also involves the central angle, inscribed angles, and finding the circumference, which enriches our understanding of circular shapes.

Degrees to Radians Conversion

Understanding how to convert degrees to radians is fundamental for solving problems in trigonometry and circle-related mathematics. Degrees offer an intuitive measure of angles, common in everyday observations, but radians provide a direct mathematical connection to the properties of a circle.
To switch from degrees to radians, apply the formula:

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

For example, converting an angle of 90 degrees becomes:

• \( \frac{90 \times \pi}{180} = \frac{\pi}{2} \) radians.

This method ensures compatibility with various mathematical models, especially those involving circular motion and periodic functions.
Degrees to radians conversion solidifies core math skills, making complex problems simpler and more manageable.