Problem 115 - Algebra and Trigonometry

Question

In Exercises \(115-116,\) convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. $$ 30.42^{\circ} $$

Step-by-Step Solution

Answer

Therefore, in degrees-minutes-seconds form (rounded to the nearest second), \(30.42^\circ\) equals \(30^\circ 25' 12''\).

Step 1: Decimal Degrees to Degrees

The given angle is already in degrees. Therefore, no conversion is needed. The degrees portion is \(30^\circ\).

Step 2: Convert Decimal Degrees to Minutes

To obtain the 'minutes' portion, take the decimal part of the degree measure, multiply by 60 (since there are 60 minutes in a degree). The decimal part of 30.42 is 0.42, so multiplying 0.42 by 60 gives 25.2. Therefore, the 'minutes' part is 25.

Step 3: Convert Decimal Minutes to Seconds

To obtain the 'seconds' portion, again take the decimal part of the 'minutes' measure from Step 2, and multiply it by 60 (as there are 60 seconds in a minute). The decimal part of 25.2 is 0.2, so multiplying 0.2 by 60 gives 12. Therefore, the 'seconds' part is 12.

Key Concepts

Degrees to Minutes

When converting angles, it's common to express them using degrees, minutes, and seconds (DMS). Each degree can be divided into smaller units.
To convert degrees to minutes, you need to know that one degree is equivalent to 60 minutes. This relationship allows us to convert the fractional part of a degree into minutes.
For example, if you have an angle of

• 30.42°

identify the decimal portion, which is 0.42. To find out how many minutes this represents, you multiply 0.42 by 60. This calculation results in 25.2 minutes.
This means the angle is made up of 30 degrees and 25 minutes, and a little extra, which we'll address in the next section.

Minutes to Seconds

Once you have converted the fractional part of the degrees into minutes, you may notice there can still be some leftover decimal — this is where seconds come in.
Remember, each minute is made up of 60 seconds. So, to convert the remaining decimal portion of the minutes into seconds, multiply it by 60.
Using the previous example, you start with 25.2 minutes. The decimal part is 0.2, so multiply 0.2 by 60 to find the seconds. This gives us 12 seconds.
Importantly, rounding can be applied when necessary to ensure the seconds are a whole number. Hence, for precise angle measurements, make sure you're rounding your result to the nearest second.

Decimal Degrees

Decimal degrees are a way to express angles using just fractions of a degree rather than minutes and seconds. This system uses base-10, like standard decimal numbers, which makes calculations simpler.
Decimal degrees are used often in digital systems and can easily be converted into DMS (Degrees, Minutes, Seconds) for practical applications. For instance,

• an angle of 30.42°

is straightforward to work with mathematically, but doesn't give the same visual intuition as the DMS format.
The beauty of decimal degrees is their simplicity in computations like summation and multiplication, which makes them versatile for electronic and computational use.

DMS Conversion

The conversion of angles from decimal degrees to DMS format is a valuable skill, especially in fields such as cartography and navigation.
DMS (Degrees, Minutes, Seconds) provides a more precise description by breaking down degrees into smaller units. To convert, follow these steps:

• Identify the whole number part as the degrees.
• Convert any decimal remainder to minutes by multiplying by 60.
• If minutes aren’t whole, convert the decimal part to seconds by multiplying by another 60.

Through a DMS conversion, you rediscover the value of a number you started with differently expressed. For example, converting

• 30.42°

results in 30 degrees, 25 minutes, and 12 seconds.
This transformation not only aids in visualization but is essential for activities requiring precision in angle measurement.