Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Express the angle as a decimal, to the nearest ten-thousandth of a degree. $$37^{\circ} 41^{\prime}$$

Step-by-Step Solution

Answer

The angle is 37.6833°.

Step 1: Understand the Problem

We have an angle expressed in degrees and minutes: \(37^{\circ} 41^{\prime}\). To express this angle as a decimal degree, we need to convert the minutes into a fraction of a degree.

Step 2: Convert Minutes to Fraction of a Degree

Recall that 1 degree is equal to 60 minutes. Therefore, to convert 41 minutes to degrees, we divide 41 by 60. This gives us the fraction \(\frac{41}{60}\) degrees.

Step 3: Calculate Decimal Equivalent

Perform the division \(\frac{41}{60} \approx 0.6833\) (rounded to four decimal places). This is the decimal degree equivalent of 41 minutes.

Step 4: Combine with Whole Degrees

Add the decimal equivalent from the minutes to the whole degrees: \(37 + 0.6833 = 37.6833\).

Step 5: Round to the Nearest Ten-Thousandth

Since the decimal is already to the ten-thousandth place, \(37.6833\) is precisely the angle expressed as a decimal degree.

Key Concepts

Degrees and Minutes

Angles can often be expressed using degrees (\(^{\circ}\)) and minutes (\(^{\prime}\)). One degree is divided into 60 minutes. This means if an angle is given as 37 degrees and 41 minutes (\(37^{\circ} 41^{\prime}\)), it indicates 37 full degrees and a portion of a degree.
This is similar to how an hour can be broken down into minutes. Say it's 2:30 PM, the 30 minutes are a fraction of the next hour, much like the 41 minutes are a portion of a degree. Understanding this layout paves the way for converting these values into a decimal form, which is often more versatile in calculations and digital systems.

Decimal Degrees

Decimal degrees are a way to express angles using a single number, which includes fractions of a degree. This can simplify calculations and interpretations, particularly in scientific and analytical applications. When converting an angle to its decimal degree equivalent, we begin by expressing every component of the angle in terms of degrees.

• The whole number of degrees remains unchanged in the conversion.
• Minutes and seconds, if present, contribute to the decimal portion.

For example, in the angle given as \(37^{\circ} 41^{\prime}\), the 37 is the whole degree part. The 41 minutes are then converted to a fractional degree, making up the decimal component, which when added to 37, gives us the decimal degree equivalent. This form is concise and computation-friendly, especially when intricate calculations are involved.

Fraction of a Degree

Turning minutes into a fraction of a degree is a crucial step in conversion to decimal degrees. We know that:

• 1 degree equals 60 minutes.
• 41 minutes thus is a fraction of 1 degree.

To find the fractional degree, simply divide the number of minutes by 60. So, \(\frac{41}{60}\)is how you represent 41 minutes in degrees. When calculated, this yields approximately 0.6833. This value represents just how much of an additional degree the 41 minutes make up.
This fraction is what will be added to the whole number of degrees to get the decimal degree representation. Through this conversion, we see that parts of degrees are just as pivotal in precisely measuring angles.

Rounding Decimals

Rounding decimals becomes crucial when converting to decimal degrees, especially when precision to specific decimal places is needed, such as to the ten-thousandth place. The ten-thousandth place is the fourth position to the right of the decimal point.

• If the digits beyond the desired decimal place are less than 5, you leave the number as is.
• If 5 or more, you round up.

Our target angle, initially calculated as 37.6833, is already rounded to four decimal places by checking the digit right after the ten-thousandth position. Rounding ensures simpler numbers, aiding in clarity without significant loss of precision in most practical scenarios. This careful rounding helps maintain the integrity and precision of the data while simplifying presentation and communication of results.