Cosecant, Secant & Cotangent

Size: px

Start display at page:

Download "Cosecant, Secant & Cotangent"

Lee Webb
6 years ago
Views:

1 Cosecant, Secant & Cotangent mc-ty-cosecseccot In this unit we explain what is meant by the three trigonometric ratios cosecant, secant and cotangent. We see how they can appear in trigonometric identities and in the solution of trigonometrical equations. Finally, we obtain graphs of the functions cosec, sec and cot fromknowledgeoftherelatedfunctions sin, cosand tan. In order to master the techniques explained here it is vital that you undertake the practice exercises provided. fterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto: define the ratios cosecant, secant and cotangent plotgraphsofcosec, sec and cot Contents. Introduction 2 2. Definitions of cosecant, secant and cotangent 2. Thegraphofcosec 4 4. Thegraphofsec 5 5. Thegraphofcot 6 c mathcentre 2009

2 . Introduction This unit looks at three new trigonometric functions cosecant(cosec), secant(sec) and cotangent(cot). These are not entirely new because they are derived from the three functions sine, cosine and tangent. 2. Definitions of cosecant, secant and cotangent These functions are defined as follows: cosec = sin Key Point sec = cos cot = tan These functions are useful in the solution of trigonometrical equations, they can appear in trigonometric identities, and they can arise in calculus problems, particularly in integration. Example Consider the trigonometric identity sin 2 + cos 2 = Supposewedivideeverythingonbothsidesby cos 2.Doingthisproduces Thiscanberewrittenas thatisas sin 2 cos 2 + cos2 cos 2 = cos 2 ( ) 2 ( ) 2 sin + = cos cos tan 2 + = sec 2 This, in case you are not already aware, is a common trigonometrical identity involving sec. Example Consider again the trigonometric identity sin 2 + cos 2 = Supposethistimewedivideeverythingonbothsidesby sin 2 ;thisproduces sin 2 sin 2 + cos2 sin 2 = sin c mathcentre 2009

3 Thiscanberewrittenas thatisas + ( ) 2 ( ) 2 cos = sin sin + cot 2 = cosec 2 gain, we see one of our new trigonometric functions, cosec, appearing in an identity. Example Suppose we wish to solve the trigonometrical equation cot 2 = for 0 < 60 Webeginthesolutionbytakingthesquareroot: cot = or It then follows that Inverting we find tan = or tan = or Theanglewhosetangentis isoneofthespecialanglesdescribedintheunittrigonometrical ratios in a right-angled triangle. In fact equation tan =.Whataboutothersolutions? Werefertoagraphofthefunction tan asshowninfigure. isthetangentof 0.Sothisisonesolutionofthe tan - 0 o o 20 o o o o o o 50 0 Figure.graphof tan. Fromthegraphweseethatthenextsolutionof tan = is 20 (thatis 80 furtheralong). From the same graph we can also deduce, by consideration of symmetry, that the angles whose tangentis are 50 and 0. c mathcentre 2009

4 Insummary,theequation cot 2 = hassolutions = 0, 50, 20, 0 So, solving equations involving cosec, sec and cot can often be solved by simply turning them into equations involving the more familiar functions sin, cos and tan.. The graph of cosec Westudythegraphofcosec byfirststudyingthegraphofthecloselyrelatedfunction sin, onecycleofthegraphofwhichisshowninfigure2. sin C D 0 90o 80o o o - Figure2.graphof sin. Thegraphofcosec canbededucedfromthegraphof sin becausecosec =.Notethat sin when = 90, sin = andhencecosec = aswell.similarlywhen = 270, sin = and hencecosec = aswell. Theseobservationsenableustoplottwopointsonthegraphof cosec. Thecorrespondingpointsaremarked and inbothfigures2and. When = 0, sin = 0,butbecausewecanneverdivideby0wecannotevaluatecosec inthisway.however, notethatif isverysmallandpositive(i.e. closeto,butnotequaltozero) sin willbesmall and positive, and hence willbelargeandpositive.pointsmarked Conthegraphsrepresent sin this. Similarlywhen = 80, sin = 0andagainwecannotdividebyzerotofindcosec 80. Supposewelookatvaluesof justbelow 80.Here, sin issmallandpositive,soonceagain cosec willbelargeandpositive(pointsd). TheseobservationsenableustograduallybuildupthegraphasshowninFigure.Thevertical 4 c mathcentre 2009

5 dotted lines on the graph are called asymptotes. cosec C D 0 - o 90 80o o 270 o 60 Figure.graphofcosec. Notethatwhen isjustslightlygreaterthan 80 then sin issmallandnegative,sothatcosec islargeandnegativeasshowninfigure.continuinginthiswaythefullgraphofcosec can be constructed. InFigure2weshowedjustonecycleofthesinegraph. Thisgeneratedonecycleofthegraph of cosec.clearly,iffurthercyclesofthesinegrapharedrawnthesewillgeneratefurthercycles ofthecosecantgraph.weconcludethatthegraphof cosec isperiodicwithperiod 2π. 4. The graph of sec Wecandrawthegraphofsec byfirststudyingthegraphoftherelatedfunction cosonecycle ofwhichisshowninfigure4. cos C 0 90o D 80 o o o Figure4.graphof cos. Notethatwhen = 0, cos = andso sec 0 =. Thisgivesusapoint()onthegraph. Similarlywhen = 80, cos = andso sec 80 = (Point).When = 90, cos = 0 andsowecannotevaluate sec 90.Weproceedasbeforeandlookalittletotheleftandright. When cosissmallandpositive, willbelargeandpositive.thisgivespointc.when cos cos is small and negative, willbelargeandnegative.thisgivespointd.continuinginthisway cos wecanproducethegraphshowninfigure c mathcentre 2009

6 RecallthatwehaveonlyshownonecycleofthecosinegraphinFigure4.Howeverbecausethis repeatswithaperiodof 2πitfollowsthatthegraphof sec isalsoperiodicwithperiod 2π. sec C o o o o 60 D Figure5.graphof sec. 5. The graph of cot Wecandrawthegraphofcot byfirststudyingthegraphof tantwocyclesofwhichareshown infigure6. tan Figure6.graphof tan. Weproceedasbefore.When issmallandpositive(justabovezero),sotoois tan.socot willbelargeandpositive(point).when iscloseto 90 thevalueof tan isverylargeand positive,andsocot willbeverysmall(point).inthiswaywecanobtainthegraphshownin 6 c mathcentre 2009

7 Figure7.ecausethetangentgraphisperiodicwithperiod π,sotooisthegraphof cot. cot 0 90o o o o 60 Figure7.graphofcot. Insummary,wehavenowmetthethreenewtrigonometricfunctionscosec,secandcotand obtained their graphs from knowledge of the related functions sin, cos and tan. Exercises.Usethevaluesofthetrigonometricrationsofthespecialangles0 o,45 o and60 o todetermine the following without using a calculator a) cot 45 o b) cosec 0 o c) sec 60 o d) cosec 2 45 o e) cot 2 60 o f) sec 2 0 o g) cot 5 o h) cosec ( 0 o ) i) sec 240 o 2.Findallthesolutionsofeachofthefollowingequationsintherangestated(giveyour answers to decimal place) (a) cot = 0.2with 0 o < < 60 o (b) cosec = 4with 0 o < < 80 o (c) cosec = 4with 0 o < < 60 o (d) cosec = 4with 80 o < < 80 o (e) sec = 4with 0 o < < 80 o (f) sec = 4with 0 o < < 60 o (g) sec = 4with 80 o < < 80 o (h) cot = 0.5with 0 o < < 60 o (i) cosec = 0.5with 0 o < < 60 o (j) sec = 0.5with 0 o < < 60 o. Determine whether each of the following statements is true or false (a) cot isperiodicwithperiod80 o. (b) cosec isperiodicwithperiod80 o. 7 c mathcentre 2009

8 nswers (c)sincethegraphof cosiscontinuous,thegraphof sec iscontinuous. (d) cosec nevertakesavaluelessthaninmagnitude. (e) cot takesallvalues,.a) b)2 c)2 d)2 e) f) 4 g)- h)-2 i)-2 2.a)78.7 o,258.7 o b)4.5 o,65.5 o c)4.5 o,65.5 o d)4.5 o,65.5 o e)75.5 o f)75.5 o,284.5 o g)75.5 o,-75.5 o h)6.4 o,24.4 o i)nosolutions j)nosolutions.a)trueb)falsec)falsed)truee)true 8 c mathcentre 2009

Name: Period: Date: Math Lab: Explore Transformations of Trig Functions

Name: Period: Date: Math Lab: Explore Transformations of Trig Functions EXPLORE VERTICAL DISPLACEMENT 1] Graph 2] Explain what happens to the parent graph when a constant is added to the sine function.