Problem 25 - Precalculus: Functions and Graphs

Question

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$63.169^{\circ}$$

Step-by-Step Solution

Answer

63.169° is 63° 10' 8" in degrees, minutes, and seconds.

Step 1: Convert Decimal Degrees to Minutes and Seconds

Start with the angle given in decimal degrees, which is 63.169°. The whole number, 63°, represents the degrees. To convert the decimal part (0.169°) into minutes and seconds, first separate the whole number from the decimal by focusing only on 0.169°.

Step 2: Convert Decimal Degrees to Minutes

Take the decimal part of the degrees, 0.169°, and multiply it by 60 (since there are 60 minutes in one degree): \[ 0.169 imes 60 = 10.14 \]This calculation gives you 10.14 minutes. The whole number, 10, is the number of minutes.

Step 3: Convert Remaining Decimal to Seconds

Now take the decimal part of the minutes, 0.14, and convert these into seconds by multiplying by 60 (since there are 60 seconds in a minute):\[ 0.14 imes 60 = 8.4 \]This calculation gives approximately 8.4 seconds. Rounding 8.4 to the nearest whole number, you get 8 seconds.

Step 4: Combine to Form the Final Answer

Combine the values determined through your calculations: 63 degrees, 10 minutes, and 8 seconds. Hence, 63.169° converts to 63° 10' 8", to the nearest second.

Key Concepts

Degrees

Degrees are a unit of measurement used to express angles. In a full circle, there are 360 degrees. This means that one degree represents 1/360th of a complete circle.

• Degrees are usually denoted by the symbol °.
• They are commonly used in various fields such as navigation, engineering, and astronomy.
• To understand how degrees work, imagine a pie. A full pie is 360°, and slicing it into smaller pieces divides the full 360° into various parts.

When dealing with angles, especially in real-life applications, it might not always be convenient to work with whole degrees. That’s where minutes and seconds come in handy, offering more precision by subdividing degrees into smaller units.

Minutes

Minutes in the context of angle measurement are not related to the time unit but are a subdivision of a degree.

• Each degree can be divided into 60 minutes.
• Minutes help to provide a more precise measurement when an angle is not a whole number.
• The symbol used for minutes is a prime (') — for example, 10 minutes is written as 10'.

For instance, if you have an angle measurement like 63.169°, you use decimal minutes and seconds to get an exact value: - First, multiply the decimal part of the degree (0.169°) by 60 to get the minutes. In this case, you get approximately 10.14 minutes - The integer part, 10, becomes the minutes. This step bridges the gap between whole degrees and the need for precision by converting decimal values into minutes.

Seconds

Seconds are the smallest unit in angle measurement related to degrees and minutes. They are further divisions of a minute.

• Each minute is divided into 60 seconds.
• This allows angles to be measured very precisely, especially useful in fields requiring high accuracy, such as astronomy.
• Seconds are denoted by a double prime ("). An example is 8 seconds, written as 8".

After you have found the minutes from the previous calculation, convert the remaining decimal into seconds. - From the 10.14 minutes calculated from 0.169°, take the decimal 0.14 and multiply by 60 to get about 8.4 seconds. - After rounding, you have 8 seconds. Combining these results gives a complete angle measurement with a high level of precision, expressed as degrees, minutes, and seconds. This completes the conversion from a decimal angle measurement to a more precise format.