1-3 Arc Length and DoM'S" Form Given a circle with... s θ radius of 'r' r central angle of "θ" (theta) and intercepted arc with length 's'... Then the length of the arc can be found with the formula s = rθ the angle must be in RADIANS 1 Example) Given a circle with a radius of 4 inches, find the length of the arc intercepted s by a central angle of 240o. 240o Procedure r = 4 in 1) convert angle to radians if needed!!! 2) use the formula 3) substitute for r and θ 4) simplify 5) your answer should be in the same units as the radius measure! 1) 2) s = rθ 3) 4 and 5) Example) Find the radian measure of the central angle given a radius of 27 in and an arc length of 6 in. s = 6 in Procedure θ 1) use the formula r = 27 in 2) substitute for r and s 3) simplify solve for θ 4) your answer will be an angle already in radian measure! s = rθ 6 = 27θ θ=6/27 θ = 0.222 radians 2 Note: This topic is color coded to assist your understanding. Converting DMS form to DD form Example of Degrees Minutes and Seconds (DMS) form Example of Decimal Degrees (DD) form Do M' S" Form One degree can be divided into 60 parts called minutes. Each minute can be divided into 60 parts called seconds. 135o 12' 15" is read 135 degrees 12 minutes and 15 seconds This is in Degrees Minutes and Seconds form. These are the symbols we use.... o ' " 3 Example To convert from Degrees Minute and Seconds to Decimal Degree form (DMS to DD) 1' = 1 minute = ' 1" = 1 second = " An angle of 64 degrees, 32 minutes, and 47 seconds is represented by Example Calculators are allowed but show work. Convert to Decimal Degree form. Example To convert from Decimal Degree form to Degrees Minutes and Seconds (DD to DMS ) Calculators are allowed but show work. multiply decimal Procedure portions by 60 1) Multiply the decimal portion of degrees by 60. The whole number will be the minutes. 2) Multiply the remaining decimal portion of the minutes by 60. The whole number will be the seconds. 3) Round the seconds to the nearest whole number if needed. Convert 28.54o to DMS form o 1) 28 + 0.54(60) The whole number, 28, will = 28o + 32.4' remain degrees. 2) 28o 32' + .4(60) 3) 28o 32' 24" 4 Example Change from DD to DMS. Use a Calculator but show work. o ­89.934 Negative Sign is ­89o + .934 (60) kept with the degrees. ­89o + 56.04' ­89o 56' + .04(60) ­89o 56' 2" Example Another DD to DMS. Use a Calculator. 213.875o o 213 + .875 (60) Multiply the decimal by 60 o The whole number will 213 + 52.5' be the minutes o 213 52' + .5(60) Multiply the decimal by 60 o 213 52' 30" Round the seconds to the nearest whole number 5.