Problem 56 - Basic Technical Mathematics with Calculus

Question

Change the given angles to equal angles expressed to the nearest second. $$257.019^{\circ}$$

Step-by-Step Solution

Answer

257.019° is 257° 1' 8".

Step 1: Understanding the Conversion

To convert degrees to degrees, minutes, and seconds (DMS), note that one degree = 60 minutes = 3600 seconds. Each decimal degree must be converted into minutes and seconds.

Step 2: Separate the Whole Number and Decimal Part

257.019 can be expressed as 257 degrees plus 0.019 degrees. We'll convert the decimal part to minutes and seconds.

Step 3: Convert Decimal Degrees to Minutes

Multiply the decimal point degrees by 60 to convert to minutes: 0.019 × 60 = 1.14 minutes.

Step 4: Separate Whole Minutes and Remainder

We can express 1.14 minutes as 1 minute and 0.14 of a minute. This remaining decimal, 0.14, will be converted to seconds.

Step 5: Convert Remainder to Seconds

Multiply the remainder of the minutes by 60 to convert to seconds: 0.14 × 60 ≈ 8.4 seconds. Round this to the nearest second to get 8 seconds.

Step 6: Combine all Parts

The original angle of 257.019 degrees can now be expressed as: 257 degrees, 1 minute, and 8 seconds.

Key Concepts

Decimal Degrees to Minutes

Converting decimal degrees into minutes is an essential step when transforming angle measures into the more precise degrees, minutes, seconds (DMS) form. When you have an angle in decimal degrees, the first part of the process is to handle the decimal portion. Here's how it works:

• First, look at the decimal part of the angle. For instance, in 257.019° , 257 is the whole degree, and 0.019 is the decimal portion.
• To transform this decimal into minutes, multiply it by 60 because one degree equals 60 minutes.

Following this formula:\[0.019 \times 60 = 1.14 \text{ minutes}\]By multiplying the decimal part by 60, you've effectively converted the decimal degrees into minutes. This forms the foundation for further conversion into seconds later.

Minutes to Seconds

Once you've converted the decimal portion of degrees into minutes, the next step is to deal with any remaining fraction of a minute by converting it into seconds. Here's how to perform this conversion:

• First, identify the fractional part of the minutes. In the example 1.14 minutes, 1 is the whole minute, and 0.14 is the fraction of a minute.
• To convert this fraction into seconds, multiply by 60 since each minute contains 60 seconds.

Applying this calculation gives:\[0.14 \times 60 = 8.4 \text{ seconds}\]This step will ensure that even the smallest differences in measurement are accounted for, providing an accurate and precise expression of the angle's measurement.

Angle Conversion

The process of converting a decimal degree measurement into the degrees, minutes, and seconds (DMS) format is all about breaking down and recomposing the angle. Here's a quick overview of the process:

• Start with your angle in decimal degrees, such as 257.019° .
• The whole number, here 257, represents the degrees.
• Convert the decimal part, 0.019, into minutes by multiplying by 60, giving you 1.14 minutes.
• Convert the remaining decimal of the minutes into seconds by again multiplying by 60, giving you 8.4 seconds.
• Combine these values to express the original angle; in this example, it becomes 257 degrees, 1 minute, and approximately 8 seconds.

This systematic approach ensures the angle is expressed in a widely usable and understandable format.

Rounding Seconds

After converting the remaining decimal minutes to seconds, the next step involves rounding the seconds to the nearest whole number for precision and practical use. Here's the method to follow:

• Once you've found the decimal seconds, as in our example 8.4 seconds, it becomes necessary to round to the nearest second.
• Typically, if the decimal is 0.5 or more, you round up. If it is less, you round down.

For example:\[8.4 \text{ seconds rounds to } 8 \text{ seconds}\]Rounding ensures that the final expression of the angle in degrees, minutes, and seconds remains straightforward and easier to interpret or use in further calculations. It's all about clarity without losing essential details.