Question

Convert each angle to a decimal in degrees. Round your answer to two decimal places. \(61^{\circ} 42^{\prime} 21^{\prime \prime}\)

Short answer

61.71°

Step-by-step solution

Understand the components

An angle given in degrees, minutes, and seconds can be converted to a decimal degree. In this problem, the angle is given as \(61^{\text{°}}\), \(42^{\prime}\) (minutes), and \(21^{\prime \prime}\) (seconds).

Convert seconds to decimal degrees

There are 3600 seconds in a degree. Therefore, to convert 21 seconds to degrees: \[ \text{degrees} = \frac{21}{3600} \]

Calculate the seconds to degrees

Solve the equation from the previous step: \[ \frac{21}{3600} = 0.0058333 \text{°} \]

Convert minutes to decimal degrees

There are 60 minutes in a degree. Therefore, to convert 42 minutes to degrees: \[ \text{degrees} = \frac{42}{60} \]

Calculate the minutes to degrees

Solve the equation from the previous step: \[ \frac{42}{60} = 0.7 \text{°} \]

Add all components together

Sum the degrees, the minutes converted to degrees, and the seconds converted to degrees: \[ 61 + 0.7 + 0.0058333 = 61.7058333 \]

Round to two decimal places

Round the final result to two decimal places: \[ 61.7058333 \rightarrow 61.71 \]

Key concepts

degrees to decimal conversion

When converting angles from degrees, minutes, and seconds (DMS) to a decimal format, we break down the components, making the calculation simpler. First, you'll need to understand the parts: degrees (°), minutes ('), and seconds (''). Each part represents a fraction of a degree, with:

• 1 degree = 60 minutes
• 1 minute = 60 seconds
• 1 degree = 3600 seconds

Using these ratios, you can convert each part to a decimal degree. For example, given 61° 42' 21'', start by converting the seconds and minutes independently to degrees before summing them up.

angle measurement

Angles are fundamental in various fields like mathematics, engineering, and astronomy. They are typically measured in degrees (°) but can also be represented in minutes (') and seconds ('') for greater precision. Each small part (minute and second) represents a fraction of a degree, allowing very precise measurements.
To understand this conceptually:

• 60 minutes make up 1 degree.
• 60 seconds make up 1 minute.

For instance, 61° 42' 21'' means 61 full degrees, 42 minutes (or 42/60 degrees), and 21 seconds (21/3600 degrees). Learning to interpret this helps in accurately converting to decimals.

rounding decimal places

After converting an angle to a decimal degree, you'll need to round it to a specific number of decimal places for practical use. Rounding helps simplify the number while maintaining a close approximation. Usually, rounding to two decimal places gives balance between precision and simplicity.
Here’s a quick method to round:

• First, look at the third decimal place.
• If it's 5 or greater, increase the second decimal place by 1.
• If it's less than 5, keep the second decimal place as is.

For example, after converting 61° 42' 21'' to a decimal (61.7058333), you would round it to 61.71 since the third decimal is 5. By following these steps and understanding the rules of rounding, you ensure your angle measurements are still accurate but more manageable.