Problem 21 - Precalculus with Calculus Previews

Question

In Problems \(21-24\), express the given angle in terms of degrees, minutes, and seconds. $$ 210.78^{\circ} $$

Step-by-Step Solution

Answer

210° 46' 48"

Step 1: Separate Whole Degrees

The whole number part of the angle is already given in degrees. For the angle \(210.78^{\circ}\), the whole number degrees are \(210^{\circ}\).

Step 2: Convert Decimal Degrees to Minutes

Take the decimal part \(0.78\) and convert it to minutes. Multiply by 60 because there are 60 minutes in a degree: \[0.78 \times 60 = 46.8\] So, there are 46 full minutes.

Step 3: Convert Remainder to Seconds

Take the decimal part \(0.8\) from Step 2, which represents minutes, and convert it to seconds. Multiply by 60, because there are 60 seconds in a minute: \[0.8 \times 60 = 48\] So, there are 48 seconds.

Step 4: Combine the Components

Combine all the components to express the angle in degrees, minutes, and seconds: \[210^{\circ} 46' 48"\].

Key Concepts

Degrees to Minutes

Converting degrees to minutes is an important skill in understanding angle measurements. To convert from degrees to minutes, it's essential to know that one degree is equivalent to 60 minutes.

When given an angle that includes decimal degrees, like 210.78°, you start by isolating the fractional part, which is 0.78 in this case. To transform this decimal into minutes, you multiply by 60. For example, 0.78 multiplied by 60 gives 46.8 minutes.

To simplify our answer, we focus on whole numbers, so in this conversion, we consider the 46 minutes here. This step is crucial in breaking down angle measurements for practical use and geometric calculations.

Minutes to Seconds

The next step in angle measurement involves converting the remaining fraction of a minute into seconds. Remember, each minute contains 60 seconds.

In our ongoing example, we had 46.8 minutes after converting degrees to minutes. We will focus on the decimal part, which is 0.8 minutes. This number needs to be converted into seconds by multiplying by 60. Here, 0.8 times 60 results in 48 seconds.

This step completes the process by converting all smaller fractional components into comprehensible units like seconds, often more cler than using decimal numbers directly in measurements.

Decimal Degrees

Understanding decimal degrees can make navigating calculations involving angles simpler. Decimal degrees are a form of expressing angles using a decimal rather than minutes or seconds, simplifying certain types of mathematical operations.

The format breaks down degrees into one single decimal figure, placing it as a shortcut method without losing precision by converting directly into minutes and seconds. Real-life applications, such as GPS technology, frequently use decimal degrees for their ease of integration into computational systems.

In calculations, when you need further precision or diplay changes, converting back into traditional degree-minute-second (DMS) format ensures that all details remain accurate and practical for different fields.

Angle Measurement

Angle measurement is a fundamental concept in geometry and science that describes the rotation needed to superimpose one of two intersecting lines onto another.

It's measured in degrees (°), and each degree can be further divided into minutes (') and seconds (''), which allows for precise expression of angle size.

This is crucial in fields from astronomy to engineering, illustrating everything from the Earth's rotation to architectural designs. Learning to convert and express angles correctly is essential in these fields, ensuring calculations' precision and accuracy.