SECTION 6.1 Angles, Arc Length, and Circular Motion 385 2 Convert between Decimal and Degree, Minute, Second Measures for Angles Although subdivisions of a degree may be obtained by using decimals, the notion of minutes and seconds may also be used. One minute , denoted by 1 ,′ is defined as 1 60 degree. One second, denoted by 1 , ′′ is defined as 1 60 minute or, equivalently, 1 3600 degree. An angle of, say, 30 degrees, 40 minutes, 10 seconds is written compactly as 30 40 10 . ° ′ ′′ This way of writing an angle is referred to as the Degree, Minute, Second DMS ( ) ° ′ ′′ form. To summarize: It is sometimes necessary to convert from the degree, minute, second form DMS ( ) ° ′ ′′ to a decimal form, and vice versa. = ° ° = ′ ′ = ″ 1 counterclockwise revolution 360 1 60 1 60 (1) Converting between Decimal and Degree, Minute, Second Forms (a) Convert ° ′ ′′ 50 6 21 to a decimal in degrees. Round the answer to four decimal places. (b) Convert 21.256° to the degree, minute, second notation. Round the answer to the nearest second. EXAMPLE 2 Algebraic Solution (a) Because ( ) ( ) ( ) ′ = ′′ = ′ = ⋅ ° ° 1 1 60 and 1 1 60 1 60 1 60 , convert as follows: ( ) ( ) °′ ′= °+ ′+ ′′= °+ ⋅′+ ⋅′′ = ° + ⋅ + ⋅ ⋅ ≈ ° + ° + ° = ° ° ° 50621 50 6 21 50 61 211 50 6 1 60 21 1 60 1 60 50 0.1 0.0058 50.1058 Graphing Solution (a) Figure 10 shows the solution using a TI-84 Plus CE graphing calculator. So ° ′ ′′ = ° 50621 51.1058 rounded to four decimal places. Convert minutes and seconds to degrees. Figure 10 Converting ° ′ ′′ D M S to a decimal Figure 11 Converting a decimal to ° ′ ′′ DMS (b) Figure 11 shows the solution using a T1-84 Plus CE graphing calculator. (b) Because ° = ′ 1 60 and ′ = ′′ 1 60 , proceed as follows: ( ) ( ) ( ) ( ) ( ) ( ) ° = ° + ° = ° + ⋅ ° = ° + ′ = ° + ′ = ° + ′ + ′ = ° + ′ + ⋅ ′ = ° + ′ + ′′ = ° + ′ + ′′ ≈ ° ′ ′′ 21.256 21 0.256 21 0.256 1 21 0.256 60 21 15.36 21 15 0.36 21 15 0.36 1 21 15 0.36 60 21 15 21.6 21 15 22 So ° = ° ′ ′′ 21.256 211522 rounded to the nearest second. Convert fraction of degree to minutes; ° = ′ 1 60 . Convert fraction of minute to seconds; ′ = ′′ 1 60 . Round to the nearest second. Now Work PROBLEMS 59 AND 65
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