Problem 93 Perform each computation without... [FREE SOLUTION] (2024)

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Understanding the degrees-minutes-seconds (DMS) format is essential in trigonometry. This format is used to express angles in more precise terms compared to decimal degrees.
Here is a quick breakdown:

• Degrees (°) are whole numbers representing the large angle portion
• Minutes (') are finer measurements, where 1 degree equals 60 minutes
• Seconds (''), even finer, where 1 minute equals 60 seconds

As a result, 1 degree equals 3600 seconds since 60 minutes per degree multiplied by 60 seconds per minute equals 3600 seconds.

Converting DMS to decimal degrees and vice versa is critical. Here's a step-by-step method:

• Seconds to Degrees: Divide the number of seconds by 3600, because there are 3600 seconds in one degree. For example, 15'' means \(\frac{15}{3600} = 0.0042\) degrees.
• Minutes to Degrees: Divide the number of minutes by 60, as 60 minutes make one degree. For instance, 40' equates to \(\frac{40}{60} = 0.6667\) degrees.
• Adding Converted Values: Sum up the degree values. So, 34° 40' 15'' is 34° + 0.6667° + 0.0042° = 34.6709°.
• Degrees Back to DMS: For reverse conversion, separate the whole number part as degrees. Multiply the decimal part by 60 to get minutes. Any leftover decimals converted to seconds by multiplying again by 60.

These conversions allow you to shift seamlessly between DMS and decimal degrees.

Performing manual trigonometry computations requires careful attention to each step.
For calculations without a calculator:

• Step 1: Convert seconds to degrees. For example, \(\frac{15}{3600} = 0.0042° \).
• Step 2: Convert minutes to degrees. Here, \(\frac{40}{60} = 0.6667° \).
• Step 3: Add all degree values together: 34° + 0.6667° + 0.0042° = 34.6709°.
• Step 4: Divide total degrees by the desired value, e.g., \(\frac{34.6709}{3} = 11.55697° \).
• Step 5: Convert decimal degrees back to DMS: 11.55697° means 11° and 0.55697° for decimals. Multiply 0.55697 by 60 to get minutes: 33.4182'. Convert the remaining decimal to seconds: 25.092''.
• Step 6: Present the final result: \(11° 33' 25'' \).

These manual steps ensure a deep understanding of the trigonometric operations and resulting angle formats.