Problem 23 - Precalculus with Calculus Previews

Question

In Problems \(21-24\), express the given angle in terms of degrees, minutes, and seconds. $$ 30.81^{\circ} $$

Step-by-Step Solution

Answer

The angle is \(30^{\circ} 48' 36''\).

Step 1: Identify the Components

The given angle is \(30.81^{\circ}\). Our task is to convert this angle from a decimal form to degrees, minutes, and seconds.

Step 2: Extract the Degrees

Start by taking the whole number from the angle, which represents the degrees. Here, the degrees part is already provided as \(30^{\circ}\).

Step 3: Convert the Decimal Part to Minutes

Subtract the degrees part from the original angle to get the decimal part: \(0.81^{\circ}\). The next step is to convert \(0.81^{\circ}\) into minutes by multiplying it by 60 (since 1 degree = 60 minutes): \(0.81 \times 60 = 48.6\). Thus, we have 48 minutes and a remainder of \(0.6\) of a minute.

Step 4: Convert Remaining Decimal to Seconds

Take the remaining decimal from the minutes (\(0.6\)) and convert it into seconds by multiplying by 60 (since 1 minute = 60 seconds): \(0.6 \times 60 = 36\). That gives us 36 seconds.

Step 5: Combine All Parts

Combine the degrees, minutes, and seconds to express the original angle: \(30^{\circ} 48' 36''\).

Key Concepts

Degrees, Minutes, Seconds

Understanding angles in terms of degrees, minutes, and seconds is essential, especially in fields like astronomy, navigation, and surveying. When we express an angle, it is similar to telling the time. Instead of hours, minutes, and seconds, we use degrees, minutes, and seconds. Here is how it breaks down:

• Degrees (\(^{\circ}\)): The largest unit for measuring angles. This is similar to hours on a clock and makes up a complete rotation of 360 degrees, equivalent to a full circle.
• Minutes ('): There are 60 minutes in one degree. Just as a timepiece divides an hour into minutes, we divide degrees into smaller subunits called minutes. One arcminute is equivalent to 1/60th of a degree.
• Seconds (''): Further division of minutes into seconds occurs because precision is necessary in various applications. Specifically, 60 seconds make up one minute, similar to how seconds follow minutes on a clock. Each arcsecond is 1/60th of a minute and 1/3600th of a degree.

Generally, this system of measurement allows angles to be expressed with greater precision and is quite intuitive if we relate it to how we tell time.

Decimal to Minutes Conversion

When converting the decimal portion of an angle to minutes, you might be wondering about the relationship between degrees and minutes. To transform the decimal fraction of a degree into minutes, use the following method:Begin by examining the decimal portion. In our example, this is 0.81. The conversion process involves:

• Recognizing that each degree contains 60 minutes.
• Multiplying the decimal by 60 to find the number of minutes, giving you a result that represents a part of the 60 minutes: \(0.81 \times 60 = 48.6\)
• This result shows you have 48 full minutes and an additional fraction, 0.6 of a minute, remaining to convert into seconds.

This operation is crucial because it ensures that the minor components of the angle measurement are accurately translated into more precise units. This conversion is like determining how many minutes have passed in an hour when the clock reads "0.81."

Minutes to Seconds Conversion

Translating the remaining decimal of minutes into seconds is a straightforward task yet an essential part of converting angles. Here is a simple way to visualize this conversion:When you have a fraction of a minute, like 0.6, remember that just like in timekeeping, each minute consists of 60 seconds. Therefore, the calculation involves:

• Taking the fractional part of the minutes, which is 0.6.
• Multiplying it by 60 because each minute divides into 60 seconds: \(0.6 \times 60 = 36\)
• The product gives you the number of seconds, 36 in this scenario.

Understanding this is akin to figuring out how many seconds are in a portion of a minute on a clock. So, whether you're fine-tuning measurements in astronomy or navigating across the seas, knowing how to accurately convert decimal minutes to seconds is indispensable for precision.