Problem 28 - Precalculus: Functions and Graphs

Question

Exer. \(25-28:\) Express the angle in terms of degrees, minutes, and seconds, to the nearest second. $$81.7238^{\circ}$$

Step-by-Step Solution

Answer

The angle is 81° 43' 26''.

Step 1: Separate the Whole Degrees

The whole number part of the angle is already in degrees and needs no further conversion. In this case, the integer part is 81. So, we have the whole degrees: \[ 81^{\circ} \]

Step 2: Calculate the Minutes

To find the minutes, take the fractional part of the angle (i.e., 0.7238) and multiply it by 60, since there are 60 minutes in a degree.\[ 0.7238 \times 60 = 43.428 \] This yields 43 whole minutes and a remainder of 0.428.

Step 3: Calculate the Seconds

Convert the fractional part of the minutes (0.428) into seconds by multiplying by 60.\[ 0.428 \times 60 = 25.68 \] Since we are instructed to round to the nearest second, we round 25.68 to 26 seconds.

Step 4: Compile the Angle in DMS Form

Now, compile the angle as degrees, minutes, and seconds (DMS): \[ 81^{\circ} \ 43' \ 26'' \] This is the final representation of the angle in DMS format.

Key Concepts

Understanding Degrees

Degrees are used to measure angles. Imagine a circle, which is 360 degrees all the way around. So, when we talk about 1 degree, it corresponds to 1/360th of a full circle. Think of it like a slice of pie – if you cut the whole pie into 360 pieces, each slice is one degree.

Degrees are often used in everything from navigation to mathematics. They create a clear way to talk about how far around something is. It’s important to remember that degrees are the main part in angle measurement, as they establish the primary unit.

When dealing with conversions or more granular measurements, degrees are broken into smaller parts, which leads us to the next smaller units - minutes and seconds.

What are Minutes in Angles?

In angle measurements, minutes are much smaller than degrees. There are 60 minutes in each degree. This is similar to how hours are divided into minutes, making it easier to handle precise measurements.

To convert the fractional part of degrees into minutes, you multiply the fraction by 60. For example, if you have 0.7238 of a degree, you’d calculate by multiplying 0.7238 by 60, which results in 43.428. This calculation means that your angle has 43 minutes in addition to its whole degree part.

• 1 degree = 60 minutes
• To determine minutes from a fraction of a degree, multiply by 60

Minutes help make measurements more precise, going deeper into accuracy when degrees alone don't provide enough detail.

Explaining Seconds in Angles

Seconds are a smaller segment of angles, breaking down minutes further. Just like how a minute of time can be divided into 60 seconds, each minute of an angle contains 60 seconds. This helps in reaching the highest level of precision in angular measurements.

To find seconds, you take any leftover fraction of a minute and multiply it by 60. For instance, if the fraction left over from minutes is 0.428, you multiply 0.428 by 60, resulting in 25.68 seconds. In practical terms, you'd round this to the nearest whole number, giving you 26 seconds.

• 1 minute = 60 seconds
• To find seconds from a fractional minute, multiply by 60 and round if needed

Using seconds allows for a high level of precision necessary for fields like navigation and astronomy, and it's crucial for expressing angles exactly.