5.4 Radian Measue So fa, ou hae measued angles in degees, with 60 being one eolution aound a cicle. Thee is anothe wa to measue angles called adian measue. With adian measue, the ac length of a cicle is compaed to the adius of the cicle in the atio ac length adius. A adian does not hae a specific unit but is a eal numbe. Radian measue has man eal-life applications fo peiodic functions. In adian measue, ac a a length dius 0 a What is the adian measue fo one complete eolution aound a cicle, that is, when 60? Once aound the cicle is the cicumfeence of the cicle. Fo a complete cicle, the ac length is equal to the cicumfeence. If the adius is, then C 2p. If a, then 2p 2p Notice that no units ae attached to this alue because it is a eal numbe. The alue 2p can be coneted to an appoimate eal numbe. Fo instance, if two decimal places ae equied, then eplace p with.14. Then, 2p 2(.14) 6.28 Obsee in these diagams how degee measue and adian measue descibe the same angle. = 2 = 60 = 2 4 = 2 = 270 48 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
0 0 = 1 2 4 = = 180 = 2 = 90 In tigonomet, it is often necessa to conet fom degee measue to adian measue, and ice esa. When coneting, the popotion ad ians d egees is often useful. When using the popotion, substitute an thee known alues to detemine the fouth alue. Thee ae two alues p can hae. If the answe is in degees, p 180. If it is in adians, p.14 adians. Eample 1 What does one adian look like and how man degees is it equialent to? One adian occus when the ac length and the adius ae the same length. Use the popotion elating adians and degees. Note that p.14 adians. ad ians d egees 1 p 18 0 0 18 p 57. Theefoe, one adian is about 57. a = = a = = 1 Eample 2 Conet 120 to adians and ound to two decimals. Use the popotion elating adians and degees. ad ians d egees p 1 20 180 1 20 p 180 2 p Notice that the numbe has no units and is now a eal numbe. Now find the appoimate alue using p.14. 5.4 RADIAN MEASURE 49
2(.14) 2.09 Check isuall. 0.09 adians 1 adian 120 1 adian Eample Change 5 p to degee measue. One method is to use the popotion elating adians and degees. ad ians d egees 5 p p 5 1 Then 5 p 00. 18 0 18 0 5 (180 ) 00 Anothe method is to simpl substitute p 180. 5 p 5(1 80 ) 5 p 00 Check isuall. 5 = 00 Eample 4 9.8 The motion of a cetain pendulum is modelled b d cos 2 t, whee d is the distance in metes of the ac length fom the elease point and t is the time in seconds since elease. (a) Make a table in 1-s incements fo 0 t 10. Round distances to the neaest mete. (b) Use the table to daw the gaph. (c) What is the length of one ccle? Eplain wh the gaph is peiodic in the contet of the question. 440 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
(d) What is the maimum displacement fom est? (e) State the amplitude of the function. 9.8 (a) Ealuate d cos 2 t, fo 0 t 10, and complete the table. t (s) 0 1 2 4 5 6 7 8 9 10 d (m) 1 0 1 0 1 0 1 0 1 0 1 (b) d(m) 1 t(s) 0 2 4 6 8 10-1 (c) One ccle is 4 s. The gaph is peiodic because it epeats itself ee 4 s. The pendulum swings 1 m awa fom est, though est to a point 1 m on the othe side of est, and then etuns, passing though est to its oiginal position. (d) The maimum displacement is 1 m. (e) The amplitude is 1. Consolidate You Undestanding 1. Show how degee measue and adian measue ae elated. 2. Eplain the diffeence between p 180 and p.14. Focus 5.4 1 0-1 0 1 Ke Ideas Angles can be measued using degees o adians. A adian has no specified unit. It is simpl a eal numbe. p adians 180 o p adians.14 adians. The popotion ad ians d egees can be used to conet between adian and degee measues. Radian measue has man pactical applications to peiodic phenomena. 5.4 RADIAN MEASURE 441
Pactise, Appl, Sole 5.4 A 1. A point is otated about a cicle of adius 1. Its stat and finish ae shown. State the otation in adian measue and in degee measue. (a) (b) (c) (d) (e) (f) (g) (h) 2. Sketch each otation about a cicle of adius 1. (a) p (b) p (c) 2 p (d) 4 p (e) 5 p (f) p (g) p 2 (h) p 4. Conet to degee measue. (a) 2 p (b) 5 p (c) p 4 (d) p 4 (e) 7 p (f) p (g) 11 (h) 6 2 6p 9 2p 4. Sketch the appoimate location of each adian measue on a unit cicle. Do not conet to degee measue. (a).14 (b) 2 (c) 1.5 (d) 4.2 (e) 5. 5. Knowledge and Undestanding: Conet to adian measue. (a) 90 (b) 270 (c) 180 (d) 45 (e) 15 (f) 60 (g) 240 (h) 120 6. Sketch each angle in standad position and state its elated acute angle. (a) p 4 (b) 2 p (c) p 6 (d) p (e) 5 p (f) 5 p 2 4 B 7. (a) Gaph sin fo 2p 2p. Use a table with p 6 incements. (b) What ae the coodinates of all maimum and minimum points fo this domain? (c) State the location of all zeos of the function fo this domain. 442 CHAPTER 5 MODELLING PERIODIC FUNCTIONS
8. (a) Gaph cos fo 2p 2p. Use a table with p 4 incements. (b) What ae the coodinates of all maimum and minimum points fo this domain? (c) State the location of all zeos of the function fo this domain. 9. (a) Gaph tan fo 2p 2p. Use a table with p 6 incements. (b) State the equation of all asmptotes within this domain. (c) State the location of all zeos of the function fo this domain. 10. Sketch the gaph within the gien domain. (a) sin, 2p 2p (b) sin, 180 540 11. Sketch the gaph within the gien domain. (a) cos, 0 60 (b) cos, p p 12. Sketch the gaph within the gien domain. (a) tan, 180 180 (b) tan, p 2p 1. Detemine all alues of, to the neaest degee, fo 60 60. (a) sin 1 2 (b) cos 0.825 (c) tan 1.62 14. Detemine all alues of, to one decimal place, fo p p. (a) sin 7 8 (b) cos 0.5 (c) tan 4.25 15. A wate wheel of adius 1 m sits in a steam as shown. (a) Daw, fo one complete eolution of the wheel, a sequence of ight angle tiangles to epesent the position of a point on the wate wheel fo ee otation of p 6. (b) Make a table with inteals of p 6 to show the displacement fom the suface of the steam of the indicated point as it otates fom 0 to 2p. (c) Use the table to gaph displacement fom suface esus angle of otation. (d) Descibe the gaph and wite an equation that models the situation. 16. Application: A buo ises and falls as it ides the waes. The equation h (t ) cos p 5 t models the displacement of the buo in metes at t seconds. (a) Gaph the displacement fom 0 to 20 s, in 2.5-s inteals. (b) Detemine the peiod of the function fom the gaph. Detemine the peiod algebaicall fom the equation. (c) What is the displacement at 5 s? (d) At what time, to the neaest second, does the displacement fist each 0.8 m? wate leel diection of flow 5.4 RADIAN MEASURE 44
17. Communication: A sping bounces up and down accoding to the model d(t) 0.5 cos 2t, whee d is the displacement in centimetes fom the est position and t is the time in seconds. The model does not conside the effects of gait. (a) Make a table fo 0 t 9, using 0.5-s inteals. (b) Daw the gaph. (c) Eplain wh the function models peiodic behaiou. (d) What is the elationship between the amplitude of the function and the displacement of the sping fom its est position? 18. Check You Undestanding: Eplain how peiodic phenomena can be measued in degees o eal numbes. Gie an eample to suppot ou answe. C 19. Thinking, Inqui, Poblem Soling: A gea of adius 1 m tuns in a counteclockwise diection and dies a lage gea of adius m. Both geas hae thei cental ais along the hoizontal. (a) Which diection is the lage gea tuning? (b) If the peiod of the smalle gea is 2 s, what is the peiod of the lage gea? (c) Make a table in conenient inteals fo each gea, to show the etical displacement, d, of the point whee the two geas fist touched. Begin the table at 0 s and end it at 12 s. Gaph etical displacement esus time. (d) What is the displacement of the point on the lage wheel when the die wheel fist has a displacement of 0.5 m? (e) What is the displacement of the die when the lage wheel fist has a displacement of 2 m? (f) What is the displacement of the point on the lage wheel at 5 min? The Chapte PoblemHow Much Dalight? CP8. Reisit ou gaph of the data fo the two-ea peiod in the table. (a) Ealuate f (). (b) Show that f () f ( 12) and eplain wh this should be tue. (c) Eplain wh thee ae othe alues that ae about the same as f (). CP9. (a) Appoimate the aeage numbe of dalight hous fo f (0). (b) Etend the gaph to ealuate f (10). (c) Which months coespond to f (0) and f (10)? 444 CHAPTER 5 MODELLING PERIODIC FUNCTIONS