Question

Convert to degrees (decimal). $$118^{\circ} 33^{\prime}$$

Short answer

The decimal degree conversion for \(118^ 33'\) is \(118.55^\).

Step-by-step solution

Understand the conversion from minutes to degrees

1 degree (1^) is equal to 60 minutes (1'). Therefore, to convert minutes to degrees, you need to divide the number of minutes by 60.

Convert minutes to degrees

Divide the given number of minutes (33') by 60 to convert it into degrees. This is calculated as \(\frac{33}{60}^\) degrees.

Add the converted minutes to the whole degrees

Add the result from Step 2 to the whole number of degrees (118^) to get the decimal degrees. The converted result would be \(118 + \frac{33}{60} = 118.55^\) degrees.

Key concepts

Angle Conversion

Angle conversion is the process of changing the representation of an angle from one form to another. Commonly, angles are measured in degrees, minutes, and seconds. One degree is divided into 60 minutes, and one minute is divided into 60 seconds. Converting between these units is essential for applications ranging from navigation to mathematics.

For example, when converting an angle given in degrees and minutes to just degrees, we primarily focus on converting the minutes (and if present, the seconds) into a decimal form of degrees. It's like converting hours and minutes to just hours in a time format. This understanding is pivotal to ensure precision in fields where exact measurements are crucial, such as in engineering or astronomy.

Degrees and Minutes

Degrees and minutes are units used to measure angles. A degree, represented by the symbol \( ^\circ \), is the larger unit, and minutes, represented by the symbol \( ' \), are the smaller. There are 60 minutes in one degree, analogous to how there are 60 minutes in one hour of time. Precise angle measurements often require the use of degrees and minutes, especially on navigational or geographical charts where latitude and longitude are denoted in these units.

Understanding how degrees and minutes relate is akin to understanding how dollars and cents work. Just like 100 cents make up a dollar, 60 minutes make up a degree. When dealing with angles, converting minutes to a decimal part of a degree is a straightforward process that enhances the ease of mathematical computations and reduces complexity in solving geometrical problems.

Decimal Degrees Calculation

Decimal degrees calculation involves the conversion of angle measurements from degrees, minutes, and seconds into a decimal format. This is highly useful for digital mapping and GPS technologies since decimal degrees are easier to calculate and use in formulas. To perform this conversion, the minutes (and seconds if present) are divided by 60 (and 3600 respectively), as these are the number of minutes in a degree and the number of seconds in a minute.

In the textbook exercise, for instance, to convert \(118^\circ 33'\) into decimal degrees, we divide the 33 minutes by 60 to convert it to a fraction of a degree. The resulting decimal is then added to the whole degrees to obtain a single, easy-to-use decimal degree value. Remember that ensuring the accuracy of these conversions is critical, as even small errors can lead to significant discrepancies in practical applications such as navigation, property surveying, or any field where angular measurements are utilized.