10.1 Angles and their Measure

Transcription

1 0. Angles and thei Measue This section begins ou stud of Tigonomet and to get stated, we ecall some basic definitions fom Geomet. A a is usuall descibed as a half-line and can be thought of as a line segment in which one of the two endpoints is pushed off infinitel distant fom the othe, as pictued below. The point fom which the a oiginates is called the initial point of the a. P A a with initial point P. When two as shae a common initial point the fom an angle and the common initial point is called the vete of the angle. Two eamples of what ae commonl thought of as angles ae P An angle with vete P. Q An angle with vete Q. Howeve, the two figues below also depict angles - albeit these ae, in some sense, eteme cases. In the fist case, the two as ae diectl opposite each othe foming what is known as a staight angle; in the second, the as ae identical so the angle is indistinguishable fom the a itself. P A staight angle. The measue of an angle is a numbe which indicates the amount of otation that sepaates the as of the angle. Thee is one immediate poblem with this, as pictued below. Q

2 69 Foundations of Tigonomet Which amount of otation ae we attempting to quantif? What we have just discoveed is that we have at least two angles descibed b this diagam. Cleal these two angles have diffeent measues because one appeas to epesent a lage otation than the othe, so we must label them diffeentl. In this book, we use lowe case Geek lettes such as α (alpha), β (beta), γ (gamma) and θ (theta) to label angles. So, fo instance, we have β α One commonl used sstem to measue angles is degee measue. Quantities measued in degees ae denoted b the familia smbol. One complete evolution as shown below is 60, and pats of a evolution ae measued popotionatel. Thus half of a evolution (a staight angle) measues (60 ) = 80, a quate of a evolution (a ight angle) measues (60 ) = 90 and so on. One evolution Note that in the above figue, we have used the small squae to denote a ight angle, as is commonplace in Geomet. Recall that if an angle measues stictl between 0 and 90 it is called an acute angle and if it measues stictl between 90 and 80 it is called an obtuse angle. It is impotant to note that, theoeticall, we can know the measue of an angle as long as we The phase at least will be justified in shot ode. The choice of 60 is most often attibuted to the Bablonians.

3 0. Angles and thei Measue 695 know the popotion it epesents of entie evolution. Fo instance, the measue of an angle which epesents a otation of of a evolution would measue (60 ) = 0, the measue of an angle which constitutes onl of a evolution measues (60 ) = 0 and an angle which indicates no otation at all is measued as Using ou definition of degee measue, we have that epesents the measue of an angle which constitutes 60 of a evolution. Even though it ma be had to daw, it is nonetheless not difficult to imagine an angle with measue smalle than. Thee ae two was to subdivide degees. The fist, and most familia, is decimal degees. Fo eample, an angle with a measue of 0.5 would epesent a otation halfwa between 0 and, o equivalentl, = 6 70 of a full otation. This can be taken to the limit using Calculus so that measues like make sense. The second wa to divide degees is the Degee - Minute - Second (DMS) sstem. In this sstem, one degee is divided equall into sit minutes, and in tun, each minute is divided equall into sit seconds. 5 In smbols, we wite = 60 and = 60, fom which it follows that = 600. To convet a measue of.5 to the DMS sstem, we stat b noting( that ).5 = Conveting the patial amount of degees to minutes, we find = 7.5 = Conveting the patial amount of minutes to seconds gives 0.5 ( 60 ) = 0. Putting it all togethe ields.5 = = = = = 7 0 On the othe hand, to convet to decimal degees, we fist compute 5 ( ) 60 = and 5 ( ) 600 = 80. Then we find This is how a potacto is gaded. Awesome math pun aside, this is the same idea behind defining iational eponents in Section Does this kind of sstem seem familia?

4 696 Foundations of Tigonomet = = = = 7.65 Recall that two acute angles ae called complementa angles if thei measues add to 90. Two angles, eithe a pai of ight angles o one acute angle and one obtuse angle, ae called supplementa angles if thei measues add to 80. In the diagam below, the angles α and β ae supplementa angles while the pai γ and θ ae complementa angles. β θ α γ Supplementa Angles Complementa Angles In pactice, the distinction between the angle itself and its measue is blued so that the sentence α is an angle measuing is often abbeviated as α =. It is now fo an eample. Eample 0... Let α =.7 and β = Convet α to the DMS sstem. Round ou answe to the neaest second.. Convet β to decimal degees. Round ou answe to the neaest thousandth of a degee.. Sketch α and β.. Find a supplementa angle fo α. 5. Find a complementa angle fo β. Solution.. To convet ( ) α to the DMS sstem, we stat with.7 = ( ). Net we convet =.6. Witing.6 = + 0.6, we convet = 5.6. Hence,.7 = = +.6 = = = 5.6 Rounding to seconds, we obtain α 6.

5 0. Angles and thei Measue 697. To convet β to decimal degees, we convet 8 ( ) 60 = 7 it all togethe, we have = = = and 7 ( ) 600 = Putting To sketch α, we fist note that 90 < α < 80. If we divide this ange in half, we get 90 < α < 5, and once moe, we have 90 < α <.5. This gives us a pett good estimate fo α, as shown below. 6 Poceeding similal fo β, we find 0 < β < 90, then 0 < β < 5,.5 < β < 5, and lastl,.75 < β < 5. Angle α Angle β. To find a supplementa angle fo α, we seek an angle θ so that α + θ = 80. We get θ = 80 α = 80.7 = To find a complementa angle fo β, we seek an angle γ so that β + γ = 90. We get γ = 90 β = While we could each fo the calculato to obtain an appoimate answe, we choose instead to do a bit of seagesimal 7 aithmetic. We fist ewite 90 = = = In essence, we ae boowing = 60 fom the degee place, and then boowing = 60 fom the minutes place. 8 This ields, γ = = = 5. Up to this point, we have discussed onl angles which measue between 0 and 60, inclusive. Ultimatel, we want to use the asenal of Algeba which we have stockpiled in Chaptes though 9 to not onl solve geometic poblems involving angles, but also to etend thei applicabilit to othe eal-wold phenomena. A fist step in this diection is to etend ou notion of angle fom meel measuing an etent of otation to quantities which can be associated with eal numbes. To that end, we intoduce the concept of an oiented angle. As its name suggests, in an oiented 6 If this pocess seems hauntingl familia, it should. Compae this method to the Bisection Method intoduced in Section.. 7 Like latus ectum, this is also a eal math tem. 8 This is the eact same kind of boowing ou used to do in Elementa School when ting to find Back then, ou wee woking in a base ten sstem; hee, it is base sit.

6 698 Foundations of Tigonomet angle, the diection of the otation is impotant. We imagine the angle being swept out stating fom an initial side and ending at a teminal side, as shown below. When the otation is counte-clockwise 9 fom initial side to teminal side, we sa that the angle is positive; when the otation is clockwise, we sa that the angle is negative. Teminal Side Initial Side Teminal Side Initial Side A positive angle, 5 A negative angle, 5 At this point, we also etend ou allowable otations to include angles which encompass moe than one evolution. Fo eample, to sketch an angle with measue 50 we stat with an initial side, otate counte-clockwise one complete evolution (to take cae of the fist 60 ) then continue with an additional 90 counte-clockwise otation, as seen below. 50 To futhe connect angles with the Algeba which has come befoe, we shall often ovela an angle diagam on the coodinate plane. An angle is said to be in standad position if its vete is the oigin and its initial side coincides with the positive -ais. Angles in standad position ae classified accoding to whee thei teminal side lies. Fo instance, an angle in standad position whose teminal side lies in Quadant I is called a Quadant I angle. If the teminal side of an angle lies on one of the coodinate aes, it is called a quadantal angle. Two angles in standad position ae called coteminal if the shae the same teminal side. 0 In the figue below, α = 0 and β = 0 ae two coteminal Quadant II angles dawn in standad position. Note that α = β + 60, o equivalentl, β = α 60. We leave it as an eecise to the eade to veif that coteminal angles alwas diffe b a multiple of 60. Moe pecisel, if α and β ae coteminal angles, then β = α + 60 k whee k is an intege. 9 widdeshins 0 Note that b being in standad position the automaticall shae the same initial side which is the positive -ais. It is woth noting that all of the pathologies of Analtic Tigonomet esult fom this innocuous fact. Recall that this means k = 0, ±, ±,....

7 0. Angles and thei Measue 699 α = 0 β = 0 Two coteminal angles, α = 0 and β = 0, in standad position. Eample 0... Gaph each of the (oiented) angles below in standad position and classif them accoding to whee thei teminal side lies. Find thee coteminal angles, at least one of which is positive and one of which is negative.. α = 60. β = 5. γ = 50. φ = 750 Solution.. To gaph α = 60, we daw an angle with its initial side on the positive -ais and otate counte-clockwise = 6 of a evolution. We see that α is a Quadant I angle. To find angles which ae coteminal, we look fo angles θ of the fom θ = α + 60 k, fo some intege k. When k =, we get θ = = 0. Substituting k = gives θ = = 00. Finall, if we let k =, we get θ = = Since β = 5 is negative, we stat at the positive -ais and otate clockwise 5 60 = 5 8 of a evolution. We see that β is a Quadant II angle. To find coteminal angles, we poceed as befoe and compute θ = k fo intege values of k. We find 5, 585 and 95 ae all coteminal with 5. α = 60 β = 5 α = 60 in standad position. β = 5 in standad position.

8 700 Foundations of Tigonomet. Since γ = 50 is positive, we otate counte-clockwise fom the positive -ais. One full evolution accounts fo 60, with 80, o of a evolution emaining. Since the teminal side of γ lies on the negative -ais, γ is a quadantal angle. All angles coteminal with γ ae of the fom θ = k, whee k is an intege. Woking though the aithmetic, we find thee such angles: 80, 80 and The Geek lette φ is ponounced fee o fie and since φ is negative, we begin ou otation clockwise fom the positive -ais. Two full evolutions account fo 70, with just 0 o of a evolution to go. We find that φ is a Quadant IV angle. To find coteminal angles, we compute θ = k fo a few integes k and obtain 90, 0 and 0. γ = 50 φ = 750 γ = 50 in standad position. φ = 750 in standad position. Note that since thee ae infinitel man integes, an given angle has infinitel man coteminal angles, and the eade is encouaged to plot the few sets of coteminal angles found in Eample 0.. to see this. We ae now just one step awa fom completel maing angles with the eal numbes and the est of Algeba. To that end, we ecall this definition fom Geomet. Definition 0.. The eal numbe π is defined to be the atio of a cicle s cicumfeence to its diamete. In smbols, given a cicle of cicumfeence C and diamete d, π = C d While Definition 0. is quite possibl the standad definition of π, the authos would be emiss if we didn t mention that buied in this definition is actuall a theoem. As the eade is pobabl awae, the numbe π is a mathematical constant - that is, it doesn t matte which cicle is selected, the atio of its cicumfeence to its diamete will have the same value as an othe cicle. While this is indeed tue, it is fa fom obvious and leads to a counteintuitive scenaio which is eploed in the Eecises. Since the diamete of a cicle is twice its adius, we can quickl eaange the equation in Definition 0. to get a fomula moe useful fo ou puposes, namel: π = C

9 0. Angles and thei Measue 70 This tells us that fo an cicle, the atio of its cicumfeence to its adius is also alwas constant; in this case the constant is π. Suppose now we take a potion of the cicle, so instead of compaing the entie cicumfeence C to the adius, we compae some ac measuing s units in length to the adius, as depicted below. Let θ be the cental angle subtended b this ac, that is, an angle whose vete is the cente of the cicle and whose detemining as pass though the endpoints of the ac. Using popotionalit aguments, it stands to eason that the atio s should also be a constant among all cicles, and it is this atio which defines the adian measue of an angle. s θ The adian measue of θ is s. To get a bette feel fo adian measue, we note that an angle with adian measue means the coesponding ac length s equals the adius of the cicle, hence s =. When the adian measue is, we have s = ; when the adian measue is, s =, and so foth. Thus the adian measue of an angle θ tells us how man adius lengths we need to sweep out along the cicle to subtend the angle θ. α β α has adian measue β has adian measue Since one evolution sweeps out the entie cicumfeence π, one evolution has adian measue π = π. Fom this we can find the adian measue of othe cental angles using popotions,

10 70 Foundations of Tigonomet just like we did with degees. Fo instance, half of a evolution has adian measue (π) = π, a quate evolution has adian measue (π) = π, and so foth. Note that, b definition, the adian measue of an angle is a length divided b anothe length so that these measuements ae actuall dimensionless and ae consideed pue numbes. Fo this eason, we do not use an smbols to denote adian measue, but we use the wod adians to denote these dimensionless units as needed. Fo instance, we sa one evolution measues π adians, half of a evolution measues π adians, and so foth. As with degee measue, the distinction between the angle itself and its measue is often blued in pactice, so when we wite θ = π, we mean θ is an angle which measues π adians. We etend adian measue to oiented angles, just as we did with degees befoehand, so that a positive measue indicates counte-clockwise otation and a negative measue indicates clockwise otation. Much like befoe, two positive angles α and β ae supplementa if α + β = π and complementa if α + β = π. Finall, we leave it to the eade to show that when using adian measue, two angles α and β ae coteminal if and onl if β = α + πk fo some intege k. Eample 0... Gaph each of the (oiented) angles below in standad position and classif them accoding to whee thei teminal side lies. Find thee coteminal angles, at least one of which is positive and one of which is negative.. α = π 6. β = π. γ = 9π. φ = 5π Solution.. The angle α = π 6 is positive, so we daw an angle with its initial side on the positive -ais and otate counte-clockwise (π/6) π = of a evolution. Thus α is a Quadant I angle. Coteminal angles θ ae of the fom θ = α + π k, fo some intege k. To make the aithmetic a bit easie, we note that π = π 6, thus when k =, we get θ = π 6 + π 6 = π 6. Substituting k = gives θ = π 6 π 6 = π 6 and when we let k =, we get θ = π 6 + π 6 = 5π 6.. Since β = π (π/) is negative, we stat at the positive -ais and otate clockwise π = of a evolution. We find β to be a Quadant II angle. To find coteminal angles, we poceed as befoe using π = 6π, and compute θ = π + 6π k fo intege values of k. We obtain π, 0π and 8π as coteminal angles. The authos ae well awae that we ae now identifing adians with eal numbes. We will justif this shotl. This, in tun, endows the subtended acs with an oientation as well. We addess this in shot ode.

11 0. Angles and thei Measue 70 α = π 6 β = π α = π 6 in standad position. β = π in standad position.. Since γ = 9π is positive, we otate counte-clockwise fom the positive -ais. One full evolution accounts fo π = 8π of the adian measue with π o 8 of a evolution emaining. We have γ as a Quadant I angle. All angles coteminal with γ ae of the fom θ = 9π + 8π k, whee k is an intege. Woking though the aithmetic, we find: π, 7π 7π and.. To gaph φ = 5π, we begin ou otation clockwise fom the positive -ais. As π = π, afte one full evolution clockwise, we have π o of a evolution emaining. Since the teminal side of φ lies on the negative -ais, φ is a quadantal angle. To find coteminal angles, we compute θ = 5π + π k fo a few integes k and obtain π, π and 7π. φ = 5π γ = 9π γ = 9π in standad position. φ = 5π in standad position. It is woth mentioning that we could have plotted the angles in Eample 0.. b fist conveting them to degee measue and following the pocedue set foth in Eample 0... While conveting back and foth fom degees and adians is cetainl a good skill to have, it is best that ou lean to think in adians as well as ou can think in degees. The authos would, howeve, be

12 70 Foundations of Tigonomet deelict in ou duties if we ignoed the basic convesion between these sstems altogethe. Since one evolution counte-clockwise measues 60 and the same angle measues π adians, we can π adians use the popotion 60, o its educed equivalent, π adians 80, as the convesion facto between the two sstems. Fo eample, to convet 60 to adians we find 60 ( ) π adians 80 = π adians, o simpl π 80. To convet fom adian measue back to degees, we multipl b the atio π adian. Fo eample, 5π 6 adians is equal to ( 5π 6 adians) ( 80 π adians) = Of paticula inteest is the fact that an angle which measues in adian measue is equal to 80 π We summaize these convesions below. Equation 0.. Degee - Radian Convesion: To convet degee measue to adian measue, multipl b π adians 80 To convet adian measue to degee measue, multipl b 80 π adians In light of Eample 0.. and Equation 0., the eade ma well wonde what the allue of adian measue is. The numbes involved ae, admittedl, much moe complicated than degee measue. The answe lies in how easil angles in adian measue can be identified with eal numbes. Conside the Unit Cicle, + =, as dawn below, the angle θ in standad position and the coesponding ac measuing s units in length. B definition, and the fact that the Unit Cicle has adius, the adian measue of θ is s = s = s so that, once again bluing the distinction between an angle and its measue, we have θ = s. In ode to identif eal numbes with oiented angles, we make good use of this fact b essentiall wapping the eal numbe line aound the Unit Cicle and associating to each eal numbe t an oiented ac on the Unit Cicle with initial point (, 0). Viewing the vetical line = as anothe eal numbe line demacated like the -ais, given a eal numbe t > 0, we wap the (vetical) inteval [0, t] aound the Unit Cicle in a counte-clockwise fashion. The esulting ac has a length of t units and theefoe the coesponding angle has adian measue equal to t. If t < 0, we wap the inteval [t, 0] clockwise aound the Unit Cicle. Since we have defined clockwise otation as having negative adian measue, the angle detemined b this ac has adian measue equal to t. If t = 0, we ae at the point (, 0) on the -ais which coesponds to an angle with adian measue 0. In this wa, we identif each eal numbe t with the coesponding angle with adian measue t. 5 Note that the negative sign indicates clockwise otation in both sstems, and so it is caied along accodingl.

13 0. Angles and thei Measue 705 θ s t t t t On the Unit Cicle, θ = s. Identifing t > 0 with an angle. Identifing t < 0 with an angle. Eample 0... Sketch the oiented ac on the Unit Cicle coesponding to each of the following eal numbes.. t = π. t = π. t =. t = 7 Solution.. The ac associated with t = π is the ac on the Unit Cicle which subtends the angle π in adian measue. Since π is 8 of a evolution, we have an ac which begins at the point (, 0) poceeds counte-clockwise up to midwa though Quadant II.. Since one evolution is π adians, and t = π is negative, we gaph the ac which begins at (, 0) and poceeds clockwise fo one full evolution. t = π t = π. Like t = π, t = is negative, so we begin ou ac at (, 0) and poceed clockwise aound the unit cicle. Since π. and π.57, we find that otating adians clockwise fom the point (, 0) lands us in Quadant III. To moe accuatel place the endpoint, we poceed as we did in Eample 0.., successivel halving the angle measue until we find 5π 8.96 which tells us ou ac etends just a bit beond the quate mak into Quadant III.

14 706 Foundations of Tigonomet. Since 7 is positive, the ac coesponding to t = 7 begins at (, 0) and poceeds counteclockwise. As 7 is much geate than π, we wap aound the Unit Cicle seveal s befoe finall eaching ou endpoint. We appoimate 7 π as 8.6 which tells us we complete 8 evolutions counte-clockwise with 0.6, o just sh of 5 8 of a evolution to spae. In othe wods, the teminal side of the angle which measues 7 adians in standad position is just shot of being midwa though Quadant III. t = t = Applications of Radian Measue: Cicula Motion Now that we have paied angles with eal numbes via adian measue, a whole wold of applications awaits us. Ou fist ecusion into this ealm comes b wa of cicula motion. Suppose an object is moving as pictued below along a cicula path of adius fom the point P to the point Q in an amount of t. Q θ s P Hee s epesents a displacement so that s > 0 means the object is taveling in a counte-clockwise diection and s < 0 indicates movement in a clockwise diection. Note that with this convention the fomula we used to define adian measue, namel θ = s, still holds since a negative value of s incued fom a clockwise displacement matches the negative we assign to θ fo a clockwise otation. In Phsics, the aveage velocit of the object, denoted v and ead as v-ba, is defined as the aveage ate of change of the position of the object with espect to. 6 As a esult, we 6 See Definition. in Section. fo a eview of this concept.

15 0. Angles and thei Measue 707 have v = displacement = s length. The quantit v has units of t and conves two ideas: the diection in which the object is moving and how fast the position of the object is changing. The contibution of diection in the quantit v is eithe to make it positive (in the case of counte-clockwise motion) o negative (in the case of clockwise motion), so that the quantit v quantifies how fast the object is moving - it is the speed of the object. Measuing θ in adians we have θ = s v = s t = θ t = θ t thus s = θ and The quantit θ is called the aveage angula velocit of the object. It is denoted b ω and is t ead omega-ba. The quantit ω is the aveage ate of change of the angle θ with espect to and thus has units adians. If ω is constant thoughout the duation of the motion, then it can be shown 7 that the aveage velocities involved, namel v and ω, ae the same as thei instantaneous countepats, v and ω, espectivel. In this case, v is simpl called the velocit of the object and is the instantaneous ate of change of the position of the object with espect to. 8 Similal, ω is called the angula velocit and is the instantaneous ate of change of the angle with espect to. If the path of the object wee unculed fom a cicle to fom a line segment, then the velocit of the object on that line segment would be the same as the velocit on the cicle. Fo this eason, the quantit v is often called the linea velocit of the object in ode to distinguish it fom the angula velocit, ω. Putting togethe the ideas of the pevious paagaph, we get the following. Equation 0.. Velocit fo Cicula Motion: Fo an object moving on a cicula path of adius with constant angula velocit ω, the (linea) velocit of the object is given b v = ω. We need to talk about units hee. The units of v ae length, the units of ae length onl, and the units of ω ae adians length. Thus the left hand side of the equation v = ω has units, wheeas the ight hand side has units length adians = length adians. The supposed contadiction in units is esolved b emembeing that adians ae a dimensionless quantit and angles in adian measue ae identified with eal numbes so that the units length adians educe to the units length. We ae long ovedue fo an eample. Eample Assuming that the suface of the Eath is a sphee, an point on the Eath can be thought of as an object taveling on a cicle which completes one evolution in (appoimatel) hous. The path taced out b the point duing this hou peiod is the Latitude of that point. Lakeland Communit College is at.68 noth latitude, and it can be shown 9 that the adius of the eath at this Latitude is appoimatel 960 miles. Find the linea velocit, in miles pe hou, of Lakeland Communit College as the wold tuns. Solution. To use the fomula v = ω, we fist need to compute the angula velocit ω. The eath π adians π makes one evolution in hous, and one evolution is π adians, so ω = hous = hous, 7 You guessed it, using Calculus... 8 See the discussion on Page 6 fo moe details on the idea of an instantaneous ate of change. 9 We will discuss how we aived at this appoimation in Eample 0..6.

16 708 Foundations of Tigonomet whee, once again, we ae using the fact that adians ae eal numbes and ae dimensionless. (Fo simplicit s sake, we ae also assuming that we ae viewing the otation of the eath as counteclockwise so ω > 0.) Hence, the linea velocit is v = 960 miles π miles 775 hous hou It is woth noting that the quantit evolution hous in Eample 0..5 is called the odina fequenc of the motion and is usuall denoted b the vaiable f. The odina fequenc is a measue of how often an object makes a complete ccle of the motion. The fact that ω = πf suggests that ω is also a fequenc. Indeed, it is called the angula fequenc of the motion. On a elated note, the quantit T = is called the peiod of the motion and is the amount of it takes fo the f object to complete one ccle of the motion. In the scenaio of Eample 0..5, the peiod of the motion is hous, o one da. The concepts of fequenc and peiod help fame the equation v = ω in a new light. That is, if ω is fied, points which ae fathe fom the cente of otation need to tavel faste to maintain the same angula fequenc since the have fathe to tavel to make one evolution in one peiod s. The distance of the object to the cente of otation is the adius of the cicle,, and is the magnification facto which elates ω and v. We will have moe to sa about fequencies and peiods in Section.. While we have ehaustivel discussed velocities associated with cicula motion, we have et to discuss a moe natual question: if an object is moving on a cicula path of adius with a fied angula velocit (fequenc) ω, what is the position of the object at t? The answe to this question is the ve heat of Tigonomet and is answeed in the net section.