Convert $5^{\circ} 52^{\prime} 6^{\prime \prime}$ to a decimal number of degrees. Do not round any intermediate computations. Round your answer to the nearest thousandth.
Answer (1)
Convert seconds to degrees: 3600 6 = 0.0016666666666666668 .
Convert minutes to degrees: 60 52 = 0.8666666666666667 .
Add the degrees, minutes (in degrees), and seconds (in degrees): 5 + 0.8666666666666667 + 0.0016666666666666668 = 5.868333333333334 .
Round to the nearest thousandth: 5.868 .
Explanation
Problem Analysis We are given an angle in degrees, minutes, and seconds, and we want to convert it to decimal degrees. The given angle is 5 ∘ 5 2 ′ 6 ′′ . To convert this to decimal degrees, we need to convert the minutes and seconds to degrees and then add them to the degrees part.
Conversion to Decimal Degrees First, convert the seconds to degrees. There are 3600 seconds in a degree, so we divide the number of seconds by 3600: 3600 6 = 0.0016666666666666668 Next, convert the minutes to degrees. There are 60 minutes in a degree, so we divide the number of minutes by 60: 60 52 = 0.8666666666666667 Now, add the degrees, minutes (in degrees), and seconds (in degrees) together: 5 + 60 52 + 3600 6 = 5 + 0.8666666666666667 + 0.0016666666666666668 = 5.868333333333334
Rounding the Result Finally, round the result to the nearest thousandth: 5.868333333333334 ≈ 5.868
Final Answer Therefore, 5 ∘ 5 2 ′ 6 ′′ is approximately equal to 5.868 degrees.
Examples
In navigation, angles are often measured in degrees, minutes, and seconds. Converting to decimal degrees makes calculations easier, such as determining distances or positions using trigonometric functions. For example, a ship's captain might need to convert a GPS coordinate from DMS to decimal degrees to input it into the ship's navigation system for plotting a course.
