Question

Convert the following angles to degree/ minute/second format: (a) \(7.3614^{\circ}\) (b) \(10.0932^{\circ}\) (c) \(14.9610^{\circ}\)

Short answer

Question: Convert the following angles from decimal degrees to degree/minute/second format: (a) 7.3614°, (b) 10.0932°, and (c) 14.9610°. Answer: (a) 7°21'41.04'', (b) 10°5'35.52'', and (c) 14°57'39.6''.

Step-by-step solution

(a) Convert \(7.3614^\circ\) to degree/minute/second format

: Step 1: Extract the whole degree part: \(7^\circ\) Step 2: Multiply the decimal part by 60 to find the minutes: \(0.3614 \times 60 \approx 21.684\) Step 3: Extract the whole minute part and find the seconds: \(21'\) and \((0.684 \times 60) \approx 41.04\) Step 4: Round the result: The final result is \(7^\circ 21' 41.04''\)

(b) Convert \(10.0932^\circ\) to degree/minute/second format

: Step 1: Extract the whole degree part: \(10^\circ\) Step 2: Multiply the decimal part by 60 to find the minutes: \(0.0932 \times 60 \approx 5.592\) Step 3: Extract the whole minute part and find the seconds: \(5'\) and \((0.592 \times 60) \approx 35.52\) Step 4: Round the result: The final result is \(10^\circ 5' 35.52''\)

(c) Convert \(14.9610^\circ\) to degree/minute/second format

: Step 1: Extract the whole degree part: \(14^\circ\) Step 2: Multiply the decimal part by 60 to find the minutes: \(0.9610 \times 60 \approx 57.66\) Step 3: Extract the whole minute part and find the seconds: \(57'\) and \((0.66 \times 60) \approx 39.6\) Step 4: Round the result: The final result is \(14^\circ 57' 39.6''\)

Key concepts

Degree Minute Second Format

Understanding the degree, minute, second (DMS) format is essential to performing precise angle measurements. In the DMS system, one degree \( (^\circ) \) is divided into 60 minutes \( (') \) and each minute is further divided into 60 seconds \( ('') \)—it's akin to how we measure time. Just as a minute is a smaller unit of an hour, a minute of angle is a smaller unit of a degree, and similarly, a second of angle is an even smaller unit of the minute.

Visualizing the DMS Format

Think of a pie sliced into 360 degrees. If you zoom in on one of those slices, you can split it further into 60 tiny slivers—these are your minutes. Then, if you look even closer, each sliver can be divided again into 60 tinier fragments—these are your seconds. This level of detail allows for incredibly accurate angle measurements, such as in surveying, navigation, and astronomy.

Angle Measurement

Measuring angles correctly is a fundamental aspect of various fields, including mathematics, engineering, and geography. An angle is formed by two rays (or line segments) that meet at a common endpoint, known as the vertex. The amount of turn between the two rays is what we measure as an angle.

Angles can be measured in several units, but degrees are the most common. Degrees provide a convenient measure for many practical applications due to their intuitive representation; a full circle is 360 degrees, a straight angle is 180 degrees, a right angle is 90 degrees, and so forth. To increase precision, degrees are subdivided into minutes and seconds, much like hours into minutes and seconds of time.

Applications of Angle Measurement

Angle measurement is not only crucial for various tasks such as navigation by a compass or setting the angle of a cut in carpentry but also plays a significant role in geographical orientation and even in sports like golf, where the swing's angle can greatly affect a shot's trajectory.

Decimal to DMS Conversion

Converting angles from decimal degrees to the DMS format allows for precision and is widely used in fields like surveying and astronomy. The process is relatively straightforward but requires attention to detail to ensure accuracy. Here's how the conversion is done step by step:

Step-by-Step Guide

Let's take the decimal degree \(10.0932^\circ\) as an example. The whole number before the decimal represents the degrees. Multiply the decimal by 60 to get the minutes. In this case, \(0.0932 \times 60 \) gives us approximately 5.592 minutes. The whole number part of this calculation signifies the completed minutes. Lastly, take the decimal part of the minutes and multiply by 60 to find the seconds, like \(0.592 \times 60\) to get approximately 35.52 seconds.

Remember to round off the seconds to the nearest whole number if needed. This conversion is critical when representation of an angle with greater accuracy is required.