Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Convert each angle measure to decimal degrees. Use a calculator, and round to the nearest thousandth of a degree if necessary. $$20^{\circ} 54^{\prime}$$

Step-by-Step Solution

Answer

20°54' is 20.9 degrees in decimal format.

Step 1: Understanding minutes to degrees conversion

First, recognize that the problem involves converting a measurement that is given in degrees and minutes into a decimal degree. We know that 1 degree is equivalent to 60 minutes.

Step 2: Conversion from minutes to decimal degrees

Since \(54^{\prime}\) is in minutes, we convert it to degrees by dividing by 60, the number of minutes per degree. Thus, \( 54^{\prime} = \frac{54}{60} \) degrees.

Step 3: Calculating the decimal equivalent

Calculate \( \frac{54}{60} = 0.9 \). Therefore, \(54^{\prime}\) is equivalent to 0.9 degrees.

Step 4: Adding the converted minutes to the degrees

Add the converted decimal degrees to the whole degrees part. Thus, \( 20^{\circ} + 0.9^{\circ} \).

Step 5: Final calculation

Perform the addition: \( 20 + 0.9 = 20.9 \). Thus, the angle in decimal degrees is 20.9 degrees.

Key Concepts

Degrees and Minutes

Every geographical location or angular measure is often expressed in degrees and minutes. Think of degrees just like hours on a clock, and minutes as the subdivisions within those hours, akin to how there are 60 minutes in an hour of time. Unlike time, in angle measure, we use this system to define positions, directions, and sizes of angles. A full circle is 360 degrees, where each degree is split into smaller divisions called minutes.

• 1 degree = 60 minutes

So, when you see an angle written as \(20^{\circ} 54^{\prime}\), it means 20 whole degrees plus 54 minutes. To handle these properly in some calculations, especially in trigonometry, it becomes useful to convert them to a simpler decimal form.

Angle Measure Conversion

Converting angle measures is crucial for many fields such as navigation, astronomy, and physics. Our task is to transform degrees and minutes into a format that's easy to use in calculations -- the decimal degrees. This method allows us to work with angles in the same way we handle regular numbers. The conversion from minutes to decimal degrees generally involves a simple division, as there are 60 minutes in one degree. So, for any number of minutes you have, you divide them by 60 to get how many degrees that will be. Here’s how you can remember:

• Divide minutes by 60 to convert to degrees

This small step simplifies many angular calculations, allowing for more precise results, especially when rounding is needed.

Decimal Degree Calculation

Once you’ve converted minutes into degrees, you add this decimal figure to the number of whole degrees you have. In our example, we converted \(54^{\prime}\) to degrees, resulting in 0.9 degrees. After this conversion, add the results to the original degree value. In simple terms, you add your decimal part to the integer part:

• Add the decimal degrees to the whole degrees

This process fortifies the angle in a decimal form, which is easier to read and calculate, particularly for various mathematical or scientific applications. Using arithmetic addition, your angle, now expressed only in degrees, balances the detail (minutes) and simplicity (decimal) for practical use. Thus, \(20^{\circ} 54^{\prime}\) becomes \(20.9^{\circ}\) in decimal degree form.