Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the angle-conversion capabilities of a graphing utility to convert the angle measure to decimal degree form. Round your answer to three decimal places, if necessary. $$-408^{\circ} 16^{\prime} 25^{\prime \prime}$$

Step-by-Step Solution

Answer

-408.274 degrees

Step 1: Identify the Degree Part

Note down the degree, \( -408^{\circ} \), as it is.

Step 2: Convert the Minute Part to Degrees

Convert the 'minutes' part into a degree by dividing it by 60 (since 1 degree = 60 minutes). It gives \( \frac{16}{60}=0.267 \) degrees.

Step 3: Convert the Seconds Part to Degrees

Then, convert 'seconds' into a degree by dividing by 3600 (since 1 degree = 3600 seconds). This results in \( \frac{25}{3600}=0.007 \) degrees.

Step 4: Sum the Parts

Now, let's add the values obtained in the previous steps. Remember that we're working with a negative angle, so you will subtract the minutes and seconds from the degree part: \( -408 - 0.267 - 0.007 = -408.274 \) degrees.

Step 5: Round to Three Decimal Places

The final step is to round the answer to three decimal places to give \( -408.274 \) degrees.

Key Concepts

Decimal Degrees

Understanding degrees in decimal form is essential for precise calculations. Decimal degrees (DD) simplify the traditional degree-minutes-seconds (DMS) format, making computations more consistent and easier to use in mathematical equations or graphing tools.

The main advantage of decimal degrees is that it allows you to express angles without using the parts of degrees, which is common in GPS and other digital mapping systems. It eliminates the need for additional conversions while performing calculations.

With decimal degrees, each degree is equal to 60 minutes, and each minute is equal to 60 seconds. Therefore, when converting from DMS to DD, you focus on converting those minutes and seconds into a fraction or decimal of the degree. Once in decimal form, it becomes straightforward to use these values in calculations without further complication.

Minute to Degree Conversion

Every angle degree is composed of 60 minutes. Therefore, to convert minutes into degrees, you divide by 60. This conversion is vital when dealing with DMS formats where both minutes and seconds need to be expressed as part of a whole degree.

In practical terms, if you have 16 minutes, you'd convert it to degrees by calculating
\[ \frac{16}{60} = 0.267 \].

Thus, 16 minutes equate to approximately 0.267 degrees. This conversion is necessary for various applications in both mathematics and real-world navigation, enabling a clearer understanding and computation of angles.

Second to Degree Conversion

Seconds are the smallest unit in the DMS system. Each minute contains 60 seconds, and each degree contains 3600 seconds, given that there are 60 minutes to a degree. As a result, converting seconds to degrees is an essential step in moving to decimal degrees.

For the conversion, you simply divide the number of seconds by 3600. Using the exercise, 25 seconds would convert to
\[ \frac{25}{3600} = 0.007 \] degrees.

This step is crucial whenever you need precise angles in navigation, astronomy, or any field requiring accurate angle measurements. By accounting for every second in terms of its degree value, calculations become more straightforward and results more accurate.