Problem 131 - Precalculus

Question

Find the measure (in degrees, minutes, and nearest seconds) of a central angle \(\theta\) that intercepts an are on a circle with indicated radius \(r\) and are length \(s .\) With the TI calculator commands \([\text { ANGLE }]\) and \([\text { DMS }],\) change to degrees, minutes, and seconds. $$r=78.6 \mathrm{cm}, s=94.4 \mathrm{cm}$$

Step-by-Step Solution

Answer

Central angle \(\theta\) is approximately 68 degrees, 46 minutes, 48 seconds.

Step 1: Understand the Formula

The formula to find the measure of a central angle in radians is \( \theta = \frac{s}{r} \), where \(s\) is the arc length and \(r\) is the radius of the circle. We need to compute \( \theta \) using the given values of \(s\) and \(r\).

Step 2: Plug In the Values

Substitute the given values into the formula: \( \theta = \frac{94.4}{78.6} \).

Step 3: Calculate the Central Angle in Radians

Perform the division: \( \theta = \frac{94.4}{78.6} \approx 1.2005 \) radians.

Step 4: Convert Radians to Degrees

To convert radians to degrees, use the conversion factor \( \frac{180}{\pi} \). Multiply \(1.2005 \times \frac{180}{\pi} \approx 68.78\). Therefore, \( \theta \approx 68.78\) degrees.

Step 5: Convert to Degrees, Minutes and Seconds

Use a calculator or manual conversion to change the decimal degrees 68.78 into degrees, minutes, and seconds. \(0.78\) degrees can be converted to minutes by multiplying by 60, giving approximately 46.8. This means 46 minutes. Take the decimal \(0.8\), multiply by 60 to convert to seconds (\(0.8 \times 60 = 48\)). Thus, the angle is 68 degrees, 46 minutes, and 48 seconds.

Key Concepts

Arc Length

Arc length is the distance along the curved line that makes up the arc. It is a part of the circumference of a circle. You can think of it as a piece of string that wraps around the curve of a circle. Calculating the arc length is crucial in determining the measure of a central angle.

• Arc length depends on the radius of the circle and the angle of the arc itself.
• The formula for arc length is: \( s = r \cdot \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.
• An essential part of solving problems involving central angles is to accurately measure the arc length or have this information provided.

With this, you can calculate the central angle when the radius is known.

Radius

The radius is a straight line from the center of a circle to any point on its perimeter. It plays a significant role in the geometric properties of a circle, including the calculation of arc length and central angles.

• The radius is half of the diameter of the circle.
• In equations involving circles, such as those for calculating arcs and angles, the radius serves as a crucial component.

In our equation for finding the central angle, \( \theta = \frac{s}{r} \), it's evident that the radius is key in transforming arc length into an angular measurement.

Degrees, Minutes, and Seconds

Degrees, minutes, and seconds are units used to measure angles. A full circle is composed of 360 degrees, with each degree further divided into 60 minutes and each minute into 60 seconds.

• Degrees are the primary unit and are the most commonly used.
• For a more precise measurement, minutes and seconds help specify smaller subdivisions of angles.
• Conversions between these units are often necessary in mathematics and sciences.

In the context of central angle calculations, converting degree decimal measurements into this precise format allows for accurate and practical assessments, especially when using tools like calculators that offer degrees, minutes, and seconds conversion functionality.

Radian to Degree Conversion

Radians and degrees are two different units for measuring angles. While degrees divide one full circle into 360 parts, radians measure the angle through the arc length equivalent to the radius of the circle. One complete rotation around a circle is \( 2\pi \) radians.

• A vital conversion formula is: \( 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \).
• This conversion is crucial when working with formulas that output angles in radians, as we commonly express angles in degrees.

Thus, understanding and applying radian to degree conversion is essential when initial calculations give radians and the answer is needed in degrees, like in problems involving central angles.