Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the difference of the angles. Write your answer in \(\mathbf{D}^{\circ} \mathbf{M}^{\prime} \mathbf{S}^{\prime \prime}\) form. $$48^{\circ} 18^{\prime} \text { and } 25^{\circ} 16^{\prime} 59^{\prime \prime}$$

Step-by-Step Solution

Answer

The difference between the two given angles is \(22^{\circ} 56^{\prime} 1^{\prime\prime}\).

Step 1: Conversion of Angles into Seconds

For the first angle \(48^{\circ} 18^{\prime}\), it is converted into seconds as follows: \(48 \times 3600 + 18 \times 60 = 173280 \, seconds\). For the second angle \(25^{\circ} 16^{\prime} 59^{\prime \prime}\), it is converted into seconds as follows: \(25 \times 3600 + 16 \times 60 + 59 = 90719 \, seconds.

Step 2: Calculation of Difference

The difference between the converted values of the two angles is found by subtraction. So, \(173280 - 90719 = 82561 \, seconds\).

Step 3: Conversion of Difference Back to DMS Format

The calculated difference in seconds is then converted back into DMS format. For degrees: \(\left\lfloor \frac{82561}{3600} \right\rfloor = 22^{\circ}\). For minutes: \(\left\lfloor \frac{82561 \mod 3600}{60} \right\rfloor = 56^{\prime}\). For seconds: \(82561 \mod 60 = 1^{\prime\prime}\). Therefore, the difference is \(22^{\circ} 56^{\prime} 1^{\prime \prime}\).

Key Concepts

DMS format

The DMS format, or Degrees-Minutes-Seconds format, is a method of expressing angles using three units. It is a precise way to represent and calculate angles, especially in fields like navigation, surveying, and astronomy. - **Degrees (")**: The largest unit, symbolized by **°**, and it is the primary measure of an angle. A full circle is 360°. - **Minutes**: Each degree is divided into 60 minutes, symbolized by **'**. - **Seconds**: Each minute is further divided into 60 seconds, symbolized by **''**. This hierarchical breakdown allows for very precise angle measurement. For instance, an angle might be expressed as 48° 18' 59", meaning 48 degrees, 18 minutes, and 59 seconds. When handling calculations, converting these measurements into a single unit (like seconds) simplifies the arithmetic.

angle conversion

In angle subtractions and other operations, converting angles into a unified format is crucial. This typically means converting everything into seconds because seconds are the smallest unit and ensure precision. - **Conversion to Seconds**: Multiply the degrees by 3600 (since there are 3600 seconds in a degree), minutes by 60 (since there are 60 seconds in a minute), and add these together with any existing seconds. This gives a total in seconds. For example, converting 48° 18' into seconds would mean calculating \(48 \times 3600 + 18 \times 60\).- **Conversion from Seconds to DMS**: Once calculations are performed in seconds, the result is converted back to DMS for clarity and standard interpretation. The process involves: * Determining the degrees by dividing the total seconds by 3600 (using floor division to only capture whole degrees). * Finding the remaining seconds and then determining minutes by dividing these by 60. * The remainder are the seconds.

degree-minute-second

The system of degrees-minutes-seconds is both straightforward and immensely detailed, providing precise measurements. Here are some key insights into how each component functions: - **Degrees**: The baseline of angle measurement, degrees give the broadest sense of size. They are intuitive in dividing circles. - **Minutes and Seconds**: These provide finer subdivisions, allowing precise expression beyond whole Degrees. For example, when angles need precision for applications like astronomical calculations, minutes, and seconds provide that granularity. Understanding how to traffic between these units efficiently is crucial for executing tasks like angle addition, subtraction, and even conversions for coordinate systems. This mastery of unit conversion ensures computational accuracy and clarity, setting the foundation for more complex mathematical operations with angles.