Problem 26 - Precalculus: Functions and Graphs

Question

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$12.864^{\circ}$$

Step-by-Step Solution

Answer

The angle is \(12^\circ 51' 50''\).

Step 1: Identify the Whole Number of Degrees

The number given is \(12.864^\circ\). The whole number part is the degrees part, which is \(12\) degrees.

Step 2: Extract the Decimal Part for Conversion

Subtract the whole number of degrees from the total to find the decimal part, which is \(12.864 - 12 = 0.864\).

Step 3: Convert Decimal Degrees to Minutes

Multiply the decimal part by \(60\) to convert it to minutes: \(0.864 \times 60 = 51.84\). The whole number part is the minutes, which is \(51\) minutes.

Step 4: Convert Remaining Decimal to Seconds

Take the decimal part from the previous conversion (\(0.84\)) and multiply by \(60\) to convert it to seconds: \(0.84 \times 60 = 50.4\). Round this to the nearest second, which is \(50\) seconds.

Step 5: Combine Degrees, Minutes, and Seconds

Combine the answers from the previous steps to get the angle in degrees, minutes, and seconds: \(12^\circ 51' 50''\).

Key Concepts

Degrees to Minutes Conversion

When you need to convert degrees to a more detailed measure such as minutes, you focus on the decimal part of the degree. In the original exercise, we are given an angle of \(12.864^\circ\). The whole number, 12, represents complete degrees. But the number 0.864 represents a fraction of a degree, and that needs conversion.

To change this decimal into minutes, you multiply by 60 because there are 60 minutes in every degree. For our example, it means doing this calculation: \(0.864 \times 60 = 51.84\). The whole number, 51, becomes your minutes.

• Find the whole degree number.
• Extract the decimal part.
• Multiply by 60 to get minutes.

After converting, you gain a clearer measure of that partial degree without losing accuracy. This conversion step helps in getting angles down to finer precision.

Minutes to Seconds Conversion

Once we have the minutes part of an angle, any remaining decimal part needs converting to seconds. This follows because 1 minute can be further divided into 60 seconds. In the exercise, after determining 51 minutes, the remaining decimal was 0.84 minutes.

To convert these into seconds, multiply the decimal by 60: \(0.84 \times 60 = 50.4\). The whole number 50 makes up the seconds part. It's as simple as that—instead of leaving any part of the angle unconverted, we capture every fraction.

• Extract decimal minutes.
• Multiply by 60 to convert to seconds.
• Use the whole number as seconds.

By handling the conversion in this method, any angle can be expressed in the full degrees, minutes, and seconds format.

Mathematical Rounding

Rounding is crucial when converting decimals to whole number units like minutes and seconds. During these conversions, decimals may not fall neatly into whole numbers and need to be rounded appropriately. Let's consider the last step of the given exercise.

The seconds calculation results in 50.4, which doesn't exactly fit the 'whole number' category we seek in angles. By rounding, we simplify 50.4 to 50, providing a clean and precise measure. Following common math rules, numbers 0.5 and above are rounded up, while numbers below 0.5 are rounded down.

• Identify decimal place value.
• Compare with 0.5 to decide rounding up or down.
• Apply to obtain whole numbers for precision.

This method ensures that every component of an angle—degrees, minutes, seconds—is accurately expressed.

Angle Measurement

Angle measurement is fundamental in geometry, crucial for calculating rotations and understanding shapes. Angles are typically expressed in degrees, a standardized unit. While expressing angles, you can also refine your measure by using minutes and seconds to improve precision.

In the practical world and surveying, for example, splitting an angle into smaller units helps achieve exact measurements over distances. For the exercise, measuring an angle scientifically as \(12^\circ 51' 50''\) enhances understanding compared to just saying \(12.864^\circ\).

• Degrees are broad units for angles.
• Minutes and seconds provide finer accuracy.
• Helps in precision-demanding tasks like navigation and construction.

By breaking down angles into smaller measures, you get a more descriptive and versatile representation, aiding both everyday and scientific applications.