Problem 132 - 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. \(r=14.2\) inches, \(s=23.8\) inches

Step-by-Step Solution

Answer

The central angle \( \theta \) measures 96 degrees, 2 minutes, and 41 seconds.

Step 1: Understand the Problem

We are given a circle with a radius \( r = 14.2 \) inches and an arc length \( s = 23.8 \) inches. We need to find the central angle \( \theta \) in degrees, minutes, and seconds.

Step 2: Use the Arc Length Formula

The formula for arc length is \( s = r \theta \), where \( \theta \) is in radians. To find \( \theta \), rearrange the formula as \( \theta = \frac{s}{r} \). Substitute the given values: \( \theta = \frac{23.8}{14.2} \).

Step 3: Calculate \( \theta \) in Radians

Compute \( \theta = \frac{23.8}{14.2} \approx 1.6761 \) radians.

Step 4: Convert Radians to Degrees

The conversion factor from radians to degrees is \( \frac{180}{\pi} \). Multiply to convert: \( \theta \approx 1.6761 \times \frac{180}{\pi} \approx 96.0446 \) degrees.

Step 5: Convert Degrees to Degrees, Minutes, and Seconds (DMS)

Use the fact that 1 degree has 60 minutes and 1 minute has 60 seconds. For \( 96.0446 \) degrees, the whole number is 96 degrees. Multiply the decimal part by 60: \( 0.0446 \times 60 = 2.676 \) minutes. Take the whole number 2 minutes. To get the seconds, multiply \( 0.676 \) by 60: \( 0.676 \times 60 \approx 40.6 \) seconds. Round to the nearest second, which is 41 seconds.

Key Concepts

Arc Length Formula

Understanding how to calculate the length of an arc is fundamental in geometry, especially when working with circles. The arc length formula is crucial for these calculations. Simply put, the arc length formula is given by:

• \( s = r\theta \)

where:

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

To find the central angle \( \theta \) when you know the radius and the arc length, rearrange the formula as:

• \( \theta = \frac{s}{r} \)

This simple rearrangement helps solve for the angle in radians. When given specific values for \( s \) and \( r \), substituting them into the formula gives the measurement of \( \theta \). It's a straightforward step, crucial for progressing with angle calculations.

Radians to Degrees Conversion

Angles can be expressed in different units, the most common being degrees and radians. Converting between these two is an essential part of solving geometrical problems. The conversion factor from radians to degrees is derived from the relationship that a full circle equals \( 360 \) degrees or \( 2\pi \) radians. Thus, converting from radians to degrees involves multiplying by:

• \( \frac{180}{\pi} \)

So, if you have an angle in radians and you need it in degrees, simply multiply:

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

This formula helps transition seamlessly from the mathematical realm of radians, often used in calculus and higher-level mathematics, to degrees, which are more intuitive and commonly used in everyday contexts.

Degrees, Minutes, Seconds (DMS)

Once you have calculated an angle in degrees, it might be necessary to convert that measurement into Degrees, Minutes, and Seconds (DMS). This format is particularly helpful for more precise navigational and geographical readings.

• 1 degree is equal to 60 minutes.
• 1 minute is equal to 60 seconds.

To break down a degree measurement:

• The whole number represents the degrees.
• Multiply the decimal portion by 60 to find the minutes.
• Take the remaining decimal and multiply by 60 again to determine the seconds, then round to the nearest second.

For example, if we have an angle of \( 96.0446 \) degrees, take \( 96 \) as the degrees, convert \( 0.0446 \) to minutes, and proceed to convert the next decimal until you achieve seconds. So, you will have 96 degrees, 2 minutes, and around 41 seconds.

TI Calculator Functions

Using a TI calculator can simplify calculations involving angles. These calculators often have built-in functions specifically designed for angle conversions.If you need to convert an angle from degrees into DMS format on a TI calculator, you can use the

• \([\text{ANGLE}]\)

feature which usually includes the

• \([\text{DMS}]\)

function. This function helps convert a decimal degree into the Degrees, Minutes, and Seconds format in a few simple steps. Typically, after calculating your angle in decimal form, you can:

• Access the angle menu using the angle command button.
• Select the DMS option.
• Enter your angle and let the calculator convert it directly into the DMS format.

These calculator functions not only save time but also ensure accuracy in measurements and conversions, helping avoid manual calculation errors.