10Circular ONLINE PAGE PROOFS. functions

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1 Cirular funtions. Kik off with CAS. Modelling with trigonometri funtions. Reiproal trigonometri funtions. Graphs of reiproal trigonometri funtions. Trigonometri identities.6 Compound- and doule-angle formulas.7 Other identities.8 Review

2 . Kik off with CAS Modelling with trigonometri funtions Where relationships etween two variales ehiit periodi ehavior, trigonometri funtions an e used to model this ehavior. Eamples inlude tide heights, sound waves, minimum and maimum temperatures, and dail UV levels. Using CAS to assist, state the rule for eah of the funtions elow. a f() d e f.8 The Australian Radiation Protetion and Nulear Safet Agen provides dail updated UV levels for apital ities and other entres aross Australia. The graph elow is for one partiular da. It shows the predited UV level as a sinusoidal urve and the atual UV level as the da progresses. Use our knowledge of trigonometri funtions and CAS to determine a rule that models the predited UV level for Monda 6 Marh. From our rule, what is the epeted UV level at pm? Melourne UV levels Monda 6 Marh Ultraviolet radiation level f() 8 6 f() f() f().8 Foreast UV level Time of da. f(). Please refer to the Resoures ta in the Prelims setion of our ebookplus for a omprehensive step--step guide on how to use our CAS tehnolog.

3 . Units & AOS Topi Conept Trigonometr Conept summar Pratie questions WorKED EXaMPlE think a Write the funtion. Modelling with trigonometri funtions Three trigonometri funtions sin (), os () and tan () and their graphs have een studied in detail in Mathematial Methods (CAS) Units and. Trigonometri funtions an e used to model the relationships etween two variales that ehiit periodi ehaviour. Tide heights, dail UV levels, sound waves, water storage levels and ovulation les are some eamples. The independent variale () is often a measurement suh as time. When modelling with trigonometri funtions ou should work in radians unless otherwise instruted. E. oli is a tpe of aterium. Its onentration, P parts per million (ppm), at a partiular eah over a -hour period t hours after 6 am is desried the funtion P =. sin t +.. a Find i the maimum and ii the minimum E. oli levels at this eah. What is the level at pm? B skething the graph determine for how long the level is aove. ppm during the first hours after 6 pm. i The maimum value of the sine funtion is. t Sustitute sin = into the equation for P and evaluate. WRItE/dRaW a P =. sin t +. i The maimum P ours when sin Ma. P =.() +. =. t =. State the solution. The maimum E. oli level is. ppm. ii The minimum value of the sine funtion is. t Sustitute sin = into the equation for P and evaluate. ii The minimum P ours when sin Min. P =.( ) +. =. State the solution. The minimum E. oli level is. ppm. t =. 6 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

4 At pm it is 9 hours sine 6 am. Sustitute t = 9 into the equation for P and evaluate. At pm, t = 9. When t = 9, P =. sin =. sin = =. +. =. +. State the solution. The E. oli level at pm is approimatel. ppm. Skething a graph will give a etter understanding of this question. State the amplitude. Amplitude =. Calulate the period. Period = Period = Identif the asi graph. The asi graph is P =. sin t. State the translations needed. No horizontal translation is needed; the vertial translation is. units up. 6 Sketh the graph of P. 7 Draw a horizontal line through P =.. P (parts per million) t (hours) 8 8 Identif where P >. from the graph. The graph shows that P >. etween the first two points where P =.. Topi Cirular funtions 6

5 9 Solve the equation P =. to find the first two values of t. Find the differene etween the solutions t = and t =. EXERCISE. PRatise Modelling with trigonometri funtions WE The height aove the ground, h metres, of a hild on a swing at an time, t seonds, after eing released is: t h = +.6 os Find: a the maimum height of the swing the height after i seonds and ii seonds When P =., t. sin +. =.. sin sin the length of time that the swing is elow. metres, travelling from one side to the other, orret to deimal plaes. The height of a ungee jumper, h metres, aove a pool of water at an time, t seonds, after jumping is desried the funtion: h(t) = os (.8t) + a What is the initial height of the ungee jumper? When, if at all, does the ungee jumper first touh the water? Assuming the ord is perfetl elasti, how long is it until the ungee jumper returns to the lowest position? t =. t =. t = 6 or 6 = 6 or 6 t = 6 or 6 t = or P >., for = 8 hours State the solution. The E. oli level is aove. parts per million for 8 hours. 6 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

6 Consolidate The weight of a rait over a period of time is modelled the graph shown at right. a State i the amplitude and ii the period. Epress W as a funtion of t. The graph at right shows the heart rate of an athlete during a partiular hour of a workout. a Find the initial heart rate. State i the amplitude and ii the period. Epress H as a funtion of t. The temperature, T ( C), inside a uilding on a given da is given the funtion: t T = 8 sin + 8 where t is the numer of hours after 8 am. a What is the maimum temperature in the uilding and what time does it first our? Find the temperature at i 8 pm, ii 6 pm and iii am (midnight). 6 The displaement, (mm), of a harp string t seonds after it is initiall pluked is modelled the funtion: (t) = sin (t) a What is i the amplitude and ii the period of this funtion? How man virations (i.e., les) will the harp string omplete in seond? Find the displaement after.8 seonds. d At what time will its displaement first e 6 mm? 7 A list rides one lap of a irular trak at a onstant speed so that her distane, d metres, from her starting point at an time, t seonds, after starting is: t d = os Find: a the time taken to omplete one lap the radius of the trak the maimum distane from the start d the length of the trak e the distane from her starting point after i seonds and ii seonds f the times at whih she is 9. metres from her starting point, to the nearest seond. 8 The depth of water, d metres, at a port entrane is given the funtion: where t is in hours. d (t) =. +. sin W (kg) H (eats/min) t t (das) t (min) 6 Topi Cirular funtions 6

7 a Find i the maimum and ii the minimum depth at the port entrane. A ertain ship needs the depth at the port entrane to e more than metres. The ship an e loaded and unloaded, and in and out of the port, in 9 hours. Assuming that the ship enters the port just as the depth at the entrane passes metres, will the ship e ale to eit 9 hours later? How long will it have to spare, or how man minutes will it miss out? 9 The Australian dollar s value (a) in US dollars was oserved to follow the equation a =.9 +.t +. sin (.t) over a period of 8 das (t represents the numer of das). a Using a CAS alulator, sketh a graph of a for t 8. On whih da will the Australian dollar first reah US$.9? At what other times will it e worth US$.9, orret to deimal plaes? d Find all of the maimum turning points of the graph (to deimal plaes). e What is the highest value reahed? The temperature in an offie is ontrolled a thermostat. The preferred temperature, P, an e set to values etween 8 C and C. The temperature, T ( C), in the offie at time t hours after 9 am is given the rule T = P +. sin (t). If the preferred temperature on the thermostat has een set to C: a find the maimum and minimum temperatures find the temperature at: i noon ii. pm sketh the graph of the funtion etween 9 am and pm. Fredd feels thirst if the temperature is aove. C. d Find the amount of time etween 9 am and pm that Fredd feels thirst. A standing wave on a guitar string ma e approimated the funtion L =. sin, where m and m are defined on the diagram at right. a Find the period of the standing wave. If the frets oinide with the mean positions of the wave, find the value of L. If the frets were to e spaed at 6 m, what would e the equation of a similar standing wave of amplitude. m suh that a fret is at eah mean position? The numer of raits in a national park is oserved for one ear. At an time t months after oservation egins, the numer is modelled the funtion: where P is in thousands. P =.8 sin t 6 6 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

8 MASTER a Find: i the maimum numer of raits ii the minimum numer of raits iii the median numer of raits. Find i the period and ii the amplitude of the funtion. Sketh the graph of the funtion for t [, ]. d Find the population after months. e How long is the population elow 6? f How long is the population aove? The height (in m) that a lok s pendulum swings aove its ase an e 7 approimated the funtion H = +.9 os t at an time t seonds after eing released. Give all answers orret to deimal plaes. a Find i the maimum and ii the minimum heights that the pendulum reahes. Find the height after i. seonds and ii minute. Sketh the graph of the funtion for the first seonds. d On the same set of aes, sketh the median position. e Find the length of time that the pendulum is elow m travelling from one side to the other. f Find the numer of times the pendulum swings in minute. The pendulum is found to e losing time and needs its swing adjusted to 7 swings per minute. g Find the new funtion H(t) that approimates the height of the pendulum. The depth, h(t), of water in metres at a point on the oast at a time t hours after noon on a ertain da is (t + ) given h(t) =. +. os. Use a CAS alulator to answer the following. a What is the depth of the water at noon (orret to deimal plaes)? What is the period of h(t)? What is the depth of the water (and what time does eah our) at: i high tide ii low tide? d Sketh the graph of h(t) for t. e The loal people wish to uild a onfire for New Year s elerations on a rok shelf near the point. The estimate that the an pass the point safel and not get splashed waves if the depth of water is less than. m. Between what times an the work? f How long do the have? Topi Cirular funtions 6

9 . Units & AOS Topi Conept Reiproal trigonometri funtions Conept summar Pratie questions Reiproal trigonometri funtions The reiproals of the sin (), os () and tan () funtions are often used to simplif trigonometri epressions or equations. Definitions The reiproal of the sine funtion is alled the oseant funtion. It is areviated to ose and is defined as: ose () =, sin (). sin () The reiproal of the osine funtion is alled the seant funtion. It is areviated to se and is defined as: se () =, os (). os () The reiproal of the tangent funtion is alled the otangent funtion. It is areviated to ot and is defined as: or ot () =, tan () tan () os () ot () =, sin (). sin () For the right-angled triangle shown, sin () = opp hp os () = adj hp or, using the reiproal trigonometri funtions: ose () = hp opp se () = hp adj Note: In the triangle shown at right: tan () = opp adj ot () = adj opp. opp = opposite adj = adjaent hp = hpotenuse Smmetr and omplementar properties You have learned previousl aout smmetr and omplementar properties of trigonometri funtions. In summar, these were:. First quadrant: sin θ = os (θ) ose θ = se (θ) hp adj opp os θ = sin (θ) se θ = ose (θ) 66 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

10 WorKED EXaMPlE think. Seond quadrant: sin ( θ) = sin (θ) ose ( θ) = ose (θ) os ( θ) = os (θ) se ( θ) = se (θ) tan ( θ) = tan (θ) ot ( θ) = ot (θ) sin + θ = os (θ) ose + θ = se (θ) os + θ = sin (θ) se + θ = ose (θ). Third quadrant: sin ( + θ) = sin (θ) ose ( + θ) = ose (θ) os ( + θ) = os (θ) se ( + θ) = se (θ) tan ( + θ) = tan (θ) ot ( + θ) = ot (θ) sin θ = os (θ) ose θ = se (θ) os θ = sin (θ) se θ = ose (θ). Fourth quadrant: sin ( θ) = sin (θ) ose ( θ) = ose (θ) os ( θ) = os (θ) se ( θ) = se (θ) tan ( θ) = tan (θ) ot ( θ) = ot (θ) sin os + θ = os (θ) ose + θ = se (θ) + θ = sin (θ) se + θ = ose (θ) Use the triangle to find the eat value of: a sin () se () ot (). a Use Pthagoras theorem to find the magnitude of the hpotenuse of the triangle. Epress the magnitude of the hpotenuse in its simplest surd form. WRItE a hp = + Evaluate sin () rule. Simplif the ratio. = = sin () = opp hp = = or Topi CirCular functions 67

11 Evaluate se () rule. Simplif the ratio. Evaluate ot () rule. WorKED EXaMPlE think If ose () = and 9, find (to the nearest tenth of a degree). Epress the equation ose () = in terms of sin (). Write as an equation for sin (). WRItE ose () = sin () = Write the solution. = 8.9 Round off the answer to deimal plae. = 8.6 WorKED EXaMPlE think Find the eat value of se ( ). sin () = for [, 9 ] WRItE Epress se ( ) in terms of os ( ). se ( ) = os ( ) Use smmetr to simplif os ( ). = os (8 ) Sustitute the eat value for os ( ). = Simplif the ratio. se () = hp adj = = = ot () = adj opp = Simplif the ratio. = os ( ) = or 68 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

12 WorKED EXaMPlE If os () = and a sin () < <, find the eat value of: ot (). think a Draw a right-angled triangle in the fourth quadrant, showing os () =. WRItE/dRaW a Use Pthagoras theorem to alulate the magnitude of the opposite side. EXERCISE. PRaCtISE Reiproal trigonometri funtions WE Cop and omplete the tale, using the right-angled triangles on the net page. Give eat values for: a i sin () ii os () iii tan () iv ose () v se () vi ot (). opp = = 8 Epress in simplest surd form. = State the signed value of the opposite side in the fourth quadrant. opp = in the fourth quadrant. Evaluate sin () rule. sin () = opp hp = Evaluate ot () rule. ot () = adj opp = Simplif the ratio rationalising the = denominator. = sin () os () tan () ose () se () ot () Topi CirCular functions 69

13 a Cop and omplete the tale, using the right-angled triangles elow it. Give eat values for: i sin () ii os () iii tan () iv ose () v se () vi ot (). a a sin () os () tan () ose () se () ot () WE For < a < 9, find the value of a to the nearest tenth of a degree. a sin (a) =.6 os (a) =.9 tan (a) =.8 For < a < 9, find the value of a to the nearest tenth of a degree. a ose (a) = se (a) =. ot (a) =.7 WE Find the eat value of eah of the following. a tan (6 ) ot ( ) os ( ) d se ( ) e ose ( ) f ot ( ) 6 Find the eat value of eah of the following. a ose ( ) se ( ) ot ( ) d ose ( ) e se ( ) f ot ( ) 7 WE If sin() = and < <, find the eat value of: a os () tan () ot (). 8 If tan () = and < <, find the eat value of: a sin () os () se () MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

14 CONSOLIDATE 9 Choose the orret answer for parts a to d elow given that sin () =.8 and 9 < < 8. a os () is equal to: A.6 B.67 C. D.6 E.8 ose () is equal to: A. B.6 C. D. E. ot () is equal to: A. B.7 C.7 D E. d se () is equal to: A.67 B. C.7 D. E.67 Find the eat value of the pronumeral shown in eah triangle elow. a 6 Find the eat value of the pronumeral shown in eah triangle elow. a 6 7 If os () = and < <, find the eat value of: a sin () ot () se (). If ose () =. and 8 < < 7, find the eat value of: a sin () os () ot (). If se () =. and 9 < < 8, find the eat value of: a os () sin () ot (). If ot () =.7 and 8 < < 7, find the eat value of: a tan () se () ose (). 6 If tan () =. and < <, find the eat value of: a se () os () ot (). 7 If ot () =, and is in the first quadrant, find the eat value of: a sin () se () os () + tan (). 8 If sin() = and 8 < < 7, then find: a os () ot () se (). Topi Cirular funtions 7

15 MASTER 9 Use a CAS alulator to solve the following equations over [, ]. a ose () = ot () = se () =. Use CAS tehnolog to simplif the following: a sin se + ose ot. Units & AOS Topi Conept Graphs of reiproal trigonometri funtions Conept summar Pratie questions ot + se ose + Graphs of reiproal trigonometri funtions. The graphs of the sine, osine and tangent funtions are familiar from previous work overed in Mathematial Methods. The graphs of the reiproal trigonometri funtions are otained appling the reiproal-of-ordinates method to the original funtions. The reiproal-of-ordinates method involves estimating or alulating the reiproal of the -values for an -value from the original funtion. It should e further noted that:. as,. as,. if =, is undefined that is, reiproal funtions ontain vertial asmptotes wherever -interepts our in the original funtion. the epression has the same sign as. if = ±, = ± also 6. if f(a) is a loal maimum, then is a loal minimum. f(a) Graph of = ose ( ) The graph of = sin () over [, ] is shown in the figure at right. Note that:. the graph of = sin () has turning points at = + n, n Z. sin () = at = n, n Z. period =. = sin () 7 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

16 The graph of = ose () over [, ] is shown in the figure elow. = ose () = sin () = = = = Note that:. the turning points are,,, and,. the vertial asmptotes are =, =, = and =. period =. Graph of = se ( ) The graph of = os () over [, ] is shown in the figure at right. Note that:. the graph of = os () has turning points at = n, n Z. os () = at = + n, n Z. period =. The graph of = se () over [, ] is shown in the figure elow. = se () = os () Note that: = =. the turning points are (, ), (, ), (, ) and (, ). the vertial asmptotes are =, =, and =. period =. Note: The turning points of = ose () and = se () are loated halfwa etween onseutive asmptotes. = Topi Cirular funtions 7

17 WorKED EXaMPlE 6 think Graph of = ot ( ) os () First reall that ot () = and ot () = sin () tan (). So the graph of = tan () has vertial asmptotes wherever os () =, that is, at = + n, n Z. The graph of = tan () over [, ] is shown in the figure at right. Note that:. tan () = at = n, n Z. tan () = at = + n, n Z. tan () = at = + n, n Z. period =. The graph of = ot () that is, tan () over [, ] is shown elow. = = = = = = = Note that:. the asmptotes are =, =, = and =. period =. Note: The translation of the reiproal trigonometri funtions is the same as the original trigonometri funtions; for eample, the graph of = ose ( a) + is the same as the graph of = ose () translated a units right and units up. The vertial asmptotes our at = + n, n Z. = = ot () = = tan () Sketh the graph of = tan () over the domain [, ]. WRItE/dRaW = tan () Vertial asmptotes at: = = tan () =,,, 7, 9, Divide oth sides. = 6,, 6, 7 6,, Onl values etween and are valid (the given domain). = 6,, 6 over [, ] 7 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

18 Sketh a standard tan urve, a little steeper eause of the in tan (), etween the asmptotes. 6 6 = tan () Verif the graph using a alulator. WorKED EXaMPlE 7 think Sketh the graph of = ose = 6 WRItE/dRaW The graph is the same as = ose () translated = ose units right. = = over the domain [, ]. The asmptotes are at = +, = +. Asmptotes: = + and = + = and = The -interept is = ose. -interept: = ose Sketh the graph. = si n = = = 7 (, ) 6 = ose ( ) = = Topi CirCular functions 7

19 WorKED EXaMPlE 8 Sketh the graphs of eah of the following funtions over the domain [, ]. a f() = ot + + f() = se ( + ) think WRItE/dRaW a Compare the graph to = ot (). a = ot + +. This graph is the same as = ot () translated units left and units up. Loate asmptotes on domain [, ]. Asmptotes: + =, Sutrat from oth sides to loate the asmptotes eatl. = and = = and = Find the -interept. -interept: = ot + + = tan = + = Sketh the graph. 6 Verif this graph using a graphis alulator. (, ) + = ot ( + ) + (, ) = = Remove the fator from the rakets = se ( + ) so the translation is ovious. = se + Compare the graph to = se (). This graph is the same as = se () translated units left. Loate asmptotes on domain [, ]. Asmptotes: + =,,,, 76 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

20 Sutrat from oth sides. =,,,,, Divide to eatl loate the asmptotes etween and. Eerise. PRatise Graphs of reiproal trigonometri funtions WE6 Sketh the graph of eah of the following over the domain [, ]. a = tan () = tan () = tan Sketh the graph of eah of the following over the domain [, ]. a = se () = ot () = ose () d = ose () e = ot () f = se WE7 Sketh the graph of eah of the following over the domain [, ]. a f() = tan f() = ose Sketh the graph of eah of the following over the domain [, ]. a f() = ot 6 =,,,, over [, ] 6 Find the -interept. -interept: = se ( + ) = os = = 7 Sketh the graph. 8 Verif this graph using a graphis alulator. = se ( + ) (, ) (, ) = = = f() = se + 6 = f() = se + f() = ose + d f() = ot WE8 Sketh the graph of eah of the following over the domain [, ]. a = tan + + = tan = se 6 Topi Cirular funtions 77

21 6 Sketh the graph of eah of the following over the domain [, ]. a = ose + = ot + = se + Consolidate 7 Math eah of the following graphs with the orret rule elow. a = 6 = 6 = = A = ose C = se Consider the funtion f() = se + a The funtion f() has -interepts: d 7 = B = os D = ot = over the domain [, ]. A, B, C 6 onl D, The funtion f() has vertial asmptotes where is equal to: A, B, C,, D, The funtion f() has turning points given : A E 6, 6 E,,, + and, B, and, C (, E ) and, and,, D, 78 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

22 d The graph of f() is: A + B + C E = = + + = = = = = = D = = + = = 9 Sketh the graph of eah of the following over the domain [, ]. = a = ose + = ot + = se Sketh the graph of eah of the following over the domain [, ]. a = ose + = se () Sketh the graph of eah of the following over the domain [, ]. a = ose () = ot () The graph whih est represents = ose A B over [, ] is: = = = = = Topi Cirular funtions 79

23 C D = = = = = E = = For questions and onsider the funtion f : [, ] R, f() = se. f() has vertial asmptotes where is equal to: A, D onl B 8, 7 8 E, f() has -interepts where equals: A, 7 B 6 D, E, The rule for the graph elow ould e: C 8, 8 C 7, = = = A = ose + B = ose + C = se + D = se E = se + 8 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

24 6 The graph that est represents = ot + over [, ] is: A B Master. C E = = D = 6 7 Using CAS tehnolog, determine the turning points for the funtion f() = ose = = 6 = over the domain [, ]. 8 Using CAS tehnolog, sketh the graph of = ot Trigonometri identities = 6 = for < <. An identit is an epression that is true for all values of the variale in its implied domain. Reall the Pthagorean identit: sin (A) + os (A) = [] = = = Topi Cirular funtions 8

25 Units & AOS Topi Conept Trigonometri identities Conept summar Pratie questions WorKED EXaMPlE 9 Identities involving the reiproal trigonometri funtions an e derived from this identit as follows. If we divide oth sides of equation [] os (A), we otain: or sin (A) os (A) + =, os (A) os (A) tan (A) + = se (A) + tan (A) = se (A), os (A) [] If we divide oth sides of equation [] sin (A), we otain: + os (A) sin (A) =, sin (A) sin (A) + ot (A) = ose (A), sin (A) [] These identities an e used to simplif or evaluate trigonometri epressions. Simplif the epression: os () [ + tan ()]. think WRItE Write the epression. os () [ + tan ()] Simplif the epression in the rakets using identit []. = os () [se ()] Epress se () in terms of os (). = os () os () Simplif anelling down the fration. = WorKED EXaMPlE think If, and os () =, find: a sin () ot () se (). WRItE a Write down identit []. a sin () + os () = Sustitute os () = into identit []. If os () = sin () + sin () + 9 = Solve for sin (). sin () = 6 sin () = ± Retain the positive solution onl as sine is positive in the seond quadrant. = sin () =, sine is in quadrant. 8 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

26 os () Use the reiproal funtion ot () = sin(). Sustitute os () and sin () into the epression and evaluate. Note: If we did not know the value of os (), identit [] ould e used. os () ot () = sin () = Evaluate the reiproal of os (). se () = os () = WorKED EXaMPlE think a Write identit []. = = If se () = and,, use the identities to find the eat value of: a tan () sin () ose (). WRItE a + tan () = se () Sustitute se() = in the identit. If se () = + tan () = Solve for tan (). tan () = tan () = ± Sine,, is a fourth-quadrant tan () = in quadrant. angle. Retain the negative solution onl, as tan is negative in the fourth quadrant. Evaluate the reiproal of se () to os () = se () otain os (). = Write identit []. sin () + os () = = 9 Sustitute os () = into the identit. If os () = sin () + 9 = Topi CirCular functions 8

27 Solve for sin (). sin () = 9 sin () = ± Retain the negative solution onl, as sine is negative in the fourth quadrant. sin() = Evaluate the reiproal of sin (). ose () = sin () = Rationalise the denominator. = WorKED EXaMPlE think If ot () = Use the reiproal of ot () to epress the equation in terms of tan ()., solve for over the interval [, ]. WRItE tan () = ot () = = in quadrant. State the solution for in the first quadrant. = in the first quadrant Use smmetr to identif the solution in the third quadrant where tan is also positive. or = + = State the two solutions in the domain [, ]. = or WorKED EXaMPlE think in the third quadrant over [, ] Solve ose () =.8 over the interval. Give our answer(s) orret to deimal plaes. WRItE Use the reiproal of ose () to find sin (). sin () = ose () =.8 =.6 Write the solution. Solving ose () =.8 over the interval gives =.89,. 6, 6.87, Round the answers to deimal plaes. =.9,., 6.87, MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

28 Eerise. PRatise Consolidate Trigonometri identities WE9 Simplif eah of the following epressions. a tan () se () sin () ot () Simplif eah of the following epressions. tan () + tan () a sin () os () ose () ot () os () sin () + sin () WE If, and sin () =.8, find: a os () tan () se () d ose () e ot (). If, and os() =, find: a sin () ot () se (). WE If, and se () =, use the identities to find eat values for: a os () sin () tan () d ose () e ot (). 6 If, and ot () =., use the identities to find eat values for: a tan () sin () os () d ose () e se (). 7 WE Solve for in sin () = over the interval [, ]. 8 Solve for in tan () = over the interval [, ]. 9 WE Solve ose () =. over the interval [, ]. Give our answer orret to deimal plaes. Solve se () = over the interval [, ]. Give our answer orret to deimal plaes. If tan () = and sin () >, then the value of ose () is: A B 6 C 9 If ose () = and,, find eat values for: D 6 a os () tan () se (). If tan () =. and,, find eat values for: a se () sin () os (). If os () =.9 and,, find eat values for: a sin () ot () ose (). Solve for in eah of the following equations over the interval [, ]. a os () = E 9 se () = ose () = d ot () = Topi Cirular funtions 8

29 Master.6 Units & AOS Topi Conept Compoundand douleangle formulas Conept summar Pratie questions 6 Solve eah of the following equations over the interval [, ]. Give our answers orret to deimal plaes. a ot () =.7 se () =.9 ose () =. d ot () =. tan () 7 When simplified, + os () is equal to: se () A se () B se () C ose () D sin () + os () E ose () 8 If sin () = 7, then ot () is equal to: A B C 6 D E If se () = 8, then is nearest to: A 7 B 8 C 97 D 7 E 66 Given that se () =, ot () is equal to: A onl B or D or E onl C or Prove the following. se () ose () a = tan () tan () os () = ose () se () sin () Prove the following. os () a tan () + se () = sin () tan (θ) sin (θ) = tan (θ) sin (θ) Compound- and doule-angle formulas Compound-angle formulas Consider the right-angled triangles shown in the figure elow. Let AD =, BAC = and DAC =. Then AC = os () CD = sin () DCE = (sine BCA = 9 and ECB = 8 ) DE = sin () sin () CE = os () sin () BC = sin () os () AB = os () os () Now BE = sin ( + ) (as BE = FD) = BC + CE = sin () os () + sin () os () sin ( + ) = sin () os () + os () sin () A D F E C B 86 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

30 and AF = os ( + ) = AB BF = AB DE (sine DE = BF) = os () os () sin () sin () os ( + ) = os () os () sin () sin (). Using a similar approah, or replaing with, the following identities an also e derived:. sin ( ) = sin () os ( ) + os () sin ( ) = sin () os () os () sin () sine os ( ) = os () and sin ( ) = sin ().. os ( ) = os () os ( ) sin () sin ( ) = os () os () + sin () sin (). Furthermore: sin ( + ) tan ( + ) = A os ( + ) sin () os () + os () sin () = os () os () sin () sin () Dividing the numerator and denominator os () os (), this simplifies to: tan () + tan () tan ( + ) = tan () tan () Similarl: tan () tan () tan ( ) = + tan () tan () Note: These identities an also e derived using a unit irle approah. In summar, the ompound-angle formulas are: sin (A + B) = sin (A) os (B) + os (A) sin (B) os (A + B) = os (A) os (B) sin (A) sin (B) tan (A) + tan (B) tan (A + B) = tan (A) tan (B) sin (A B) = sin (A) os (B) os (A) sin (B) os (A B) = os (A) os (B) + sin (A) sin (B) tan (A) tan (B) tan (A B) = + tan (A) tan (B). os ( + ) D F E C sin ( + ) B Topi Cirular funtions 87

31 WorKED EXaMPlE a Epand, and simplif where possile, eah of the following. i sin ( ) ii os ( + ) Simplif the epression sin () os () + os () sin (). think a i Write the appropriate ompound-angle formula. Doule-angle formulas If A replaes B in the ompound-angle formula: then sin (A + B) = sin (A) os (B) + os (A) sin (B) sin (A + A) = sin (A) os (A) + os (A) sin (A) or sin (A) = sin (A) os (A) [] Similarl, os (A + A) = os (A) os (A) sin (A) sin (A) or os (A) = os (A) sin (A) [] Two other forms of os (A) are otained using the Pthagorean identit. that is, and Finall, sin (A) + os (A) = os (A) = [ sin (A)] sin (A) = sin (A) os (A) = os (A) [ os (A)] = os (A) tan (A) + tan (A) tan (A + A) = tan (A) tan (A) tan (A) or tan (A) = tan (A) WRItE a i sin (A B) = sin (A) os (B) os (A) sin (B) Sustitute A = and B =. sin ( ) = sin () os () os () sin () ii Write the appropriate ompound-angle formula. ii os (A + B) = os (A) os (B) sin (A) sin (B) Sustitute A = and B =. os ( + ) = os ( ) os ( ) sin ( ) sin ( ) Replae sin ( ) and os ( ) with their eat values. Write the appropriate ompoundangle formula. Sustitute A = and B = to reveal the answer. = os( ) sin( ) sin (A) os (B) + os (A) sin (B) = sin (A + B) sin () os () + os () sin () = sin ( + ) [] [] [] 88 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

32 WorKED EXaMPlE think In summar, the doule-angle formulas are: sin (A) = sin (A) os (A) os (A) = os (A) sin (A) = sin (A) = os (A) tan (A) tan (A) = tan (A) Epand the following using the doule-angle formulas. a tan () a Epress in doule-angle notation as tan [()]. Epand using the appropriate formula. Epress in doule-angle notation as sin θ sin θ WRItE. sin a tan () = tan [()] tan () = tan () θ = sin θ Epand using the appropriate formula. = sin WorKED EXaMPlE 6 think Simplif: a sin (7 C) a Write the appropriate ompound-angle formula. WRItE se θ. θ os a sin (A B) = sin (A) os (B) os (A) sin (B) Sustitute A = 7 and B = C. sin (7 C) = sin (7 ) os (C ) os (7 ) sin (C ) Simplif. = ( ) os (C ) () sin (C ) = os (C ) Epress in terms of os using the reiproal identit se () = os (). Epand the denominator using the appropriate angle formula. se = os θ = os θ os (θ) + sin θ sin (θ) Topi CirCular functions 89

33 Simplif the denominator. = () os (θ) + () sin (θ) = sin (θ) Epress as a reiproal funtion. = ose (θ) WorKED EXaMPlE 7 think Find the eat value of ot. WRItE Epress as the sum of and 6. ot = ot + 6 Epress ot in terms of its reiproal, tan. = Use the appropriate ompound-angle formula to epand the denominator. = tan + 6 tan tan + tan 6 tan 6 tan tan 6 Epress in simplest fration form. = tan + tan 6 Simplif. = () + = + = + = + ( ) ( ) 6 Rationalise the denominator. = ( + ) ( ) 9 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

34 7 Simplif. = + = = WorKED EXaMPlE 8 think If sin (θ) = and θ,, find the eat values of: a sin (θ) os (θ) sin a Write the identit sin (θ) + os (θ) = to find os (θ). WRItE a sin (θ) + os (θ) = Sustitute sin (θ) = into the identit. If sin (θ) = + os (θ) = Solve for os (θ). Retain the positive solution onl, as θ is in the first quadrant. 9 + os ( θ) = os (θ) = 9 = 9 os (θ) = ± θ. Sine θ is in the first quadrant, os (θ) =. Epand sin (θ) using the appropriate douleangle formula. sin (θ) = sin (θ) os (θ) 6 Sustitute sin (θ) = and os (θ) =. = 7 Simplif. = 9 Epand os (θ) using the appropriate douleangle os (θ) = os (θ) sin (θ) formula. Sustitute for sin (θ) and os (θ). = 9 9 Simplif. = 9 Use the alternative doule-angle formula os (A) = sin (A). os (A) = sin (A) Topi CirCular functions 9

35 Replae A with θ. os (θ) = sin θ Sustitute os (θ) =. os (θ) = so = θ sin Solve for sin Eerise.6 PRatise θ. θ sin Compound- and doule-angle formulas WE Epand eah of the following, simplifing if possile. a sin ( + ) os ( ) tan ( + ) Epand eah of the following, simplifing if possile. a sin ( ) os ( + 6) tan ( ) WE Epand eah of the following using the doule-angle formula. a sin (6) os () tan (8) Epand eah of the following using the doule-angle formula. B a sin (A) os tan (A) WE6 Simplif eah of the following epressions. a os (9 A) sin (7 + A) tan (8 + B) 6 Simplif eah of the following epressions. a sin (9 + B) os (7 A) tan (6 + A) 7 WE7 Find the eat value of eah of the following. a tan os 8 Find the eat value of eah of the following. a ose ot 7 sin = θ sin = ± 6 Retain the positive solution onl, as θ θ, implies θ, sin =. 6 or 6 6 sine θ is in the first quadrant. = sin se 9 WE8 If os () =.6 and,, find the values of: a os () sin () tan (). θ = MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

36 Consolidate If tan () = and,, find the eat values of: a tan () sin () os (). Use ompound- and doule-angle formulas to epress eah of the following in a different form. a os () os () + sin () sin () sin (A + B) os (A B) os (A + B) sin (A B) os (z) os ( z) sin (z) sin ( z) d tan () tan ( + ) e tan (A + B) + tan (A B) Consider the reiproal funtion ose ( + A). a When epanded, ose ( + A) is equal to: A B sin () os (A) os () os (A) sin () sin (A) C os () sin(a) sin () os (A) D sin () sin (A) + os () os (A) E sin () os (A) + os () sin (A) The epression ose ( + A) simplifies to: A sin (A) B os (A) C ose (A) D se (A) E se (A) Simplif eah of the following. a sin se + tan A d ot + A e ose B f se + B g ot + h ose ( A) a The epression is equal to: A 6 B C The eat value of os is: 6 + A B D E + and tan () =,, D C 6 + E + 7, find the eat value of If sin () =,, eah of the following. a tan () os () se () d ose () e sin ( + ) f os ( ) g tan ( ) h tan ( + ) Topi Cirular funtions 9

37 Master.7 6 If os () =.,, and ose () =.,,, find the value of eah of the following, orret to deimal plaes. a sin () tan ( + ) os () d os ( ) 7 Simplif eah of the following epressions using doule-angle formulas. tan a sin () os () sin () os () tan d sin () sin () os () e f (sin os ) sin () sin () os () 8 If sin (A) = and,, find the eat values of: a sin (A) os (A) sin (A) d sin 9 If os (B) =.7 and, deimal plaes., then find eah of the following, orret to B B B a os sin tan Use the doule-angle formulas to find the eat values of: a sin 8 Prove the following identities. os a ot () se () = ose () [ + ot ()] [ os ()] = [ + sin ()] [ sin ()] = se () d ose () + se () = ose () se () 8 tan d ot 8. a Write os (t) as os (t + t) and use ompound-angle and doule-angle formulas to show that os (t) = os (t) os (t). Hene solve os (t) = os (t), t [, ] using CAS tehnolog. Solve for in sin () = os (), [, ]. Give answer to deimal plaes. Other identities Fatorisation identities From our work with the addition identities, we have: os ( + ) = os () os () sin () sin () sin ( + ) = sin () os () + os () sin () os ( ) = os () os () + sin () sin () sin ( ) = sin () os () os () sin () A. B 9 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

38 Units & AOS Topi Conept 6 Other trigonometri identities Conept summar Pratie questions WorKED EXaMPlE 9 If we selet the two sine identities and add them, we otain: sin ( + ) = sin () os () + os () sin () + sin ( ) = sin () os () os () sin () sin ( + ) + sin ( ) = sin () os () Now, if we sutrat the same two sine identities, we have: sin ( + ) = sin () os () + os () sin () sin ( ) = sin () os () os() sin () sin ( + ) sin ( ) = os () sin () Repeating this proedure for the two osine identities we have, in the ase of addition: os ( + ) = os () os () sin () sin () + os ( ) = os () os () + sin () sin () os ( + ) os ( ) = os () os () And, in the ase of sutration: os ( + ) = os () os () sin () sin () os ( ) = os () os () + sin () sin () os ( + ) os ( ) = sin () sin () In summar, we have the following four results: sin ( + ) + sin ( ) = sin () os () sin ( ) sin ( ) = os () sin () os ( + ) + os ( ) = os () os () os ( + ) os ( ) = sin () sin () These results enale us to epress the sum or differene of two sines and the sum or differene of two osines, as a produt. B writing these results in refleive form, we have the following fatorisation identities. sin () os () = sin ( + ) + sin ( ) os () sin () = sin ( + ) sin ( ) os () os () = os ( + ) + os ( ) sin () sin () = os ( + ) os ( ) Epress sin () os () as a sum or differene. think Note the produt sin os. This indiates a sum of two sines. Write the appropriate identit. Replae with and with to write the equation for sin () os (). WRItE sin () os () = sin ( + ) + sin ( ) sin () os () = sin ( + ) + sin ( ) Simplif. = sin (7) + sin () Topi CirCular functions 9

39 WorKED EXaMPlE Epress os () os () as a sum or differene. think Note the produt os os. This indiates a sum of two osines. Write the appropriate identit. Replae with and with to write an equation for os () os (). WRItE os () os () = os ( + ) + os ( ) os () os () = os ( + ) + os ( ) Simplif. = os () + os () Multipl oth sides of the equation to otain an epression for os () os (). WorKED EXaMPlE Epress os () sin () as a sum or differene. os () os () = (os () + os ()) think WRItE Note the produt os sin. This indiates os () sin () = sin ( + ) sin ( ) a differene of two sines. Write the appropriate identit. Replae with and with to write an os () sin () = sin ( + ) sin ( ) equation for os () sin (). Simplif. = sin (6) sin () Multipl oth sides of the equation to otain an epression for os () sin (). WorKED EXaMPlE Epress sin () sin () as a sum or differene. os () sin () = (sin (6) sin ()) think WRItE Note the produt sin sin. This indiates sin () sin () = os ( + ) os ( ) a differene of two osines. Write the appropriate identit. Replae with and with to write an sin () sin () = os ( + ) os ( ) equation for sin () sin (). Simplif. = os () os () Multipl oth sides of the equation to otain an epression for sin () sin (). sin () sin () = (os () os ()) 96 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

40 WorKED EXaMPlE think r sin ( ± α) or r os ( ± α) The funtion = a sin () + os () ma e onvenientl epressed in one of two forms: = A sin ( + α) or = A os ( + α) given appropriate restritions on α and taking A >. To do this we onsider the addition identities as follows: A sin ( + α) = A sin () os (α) + A os () sin (α) = A os (α) sin () + A sin (α) os () Note: In the formulas elow, α and a represent different angles. Comparing = a sin () + os () with = A os (α) sin () + A sin (α) os () gives: A os (α) = a and A sin (α) = So A os (α) + A sin (α) = a + B appling the Pthagorean identit: Also: A (os (α) + sin (α)) = a + A = a + A = a +, A > A sin (α) A os (α) = a tan (α) = a Therefore: α = tan a Epress os () sin () in the form A os ( α) where A > and < α < 6. WRItE Write the appropriate addition identit. A os ( α) = A os () os (α) + A sin () sin (α) Compare the given epression with the A os (α) = and A sin (α) = identit and assign appropriate values for the A os (α) + A sin (α) = ( ) + variales. Appl the Pthagorean identit to find A. A = A =, A > Find α otaining a tan relationship. Deide in whih quadrant α is loated. Rememer that < α < 6. A sin (α) A os (α) = tan (α) = α = tan Topi CirCular functions 97

41 As os (α) = and sin (α) = WorKED EXaMPlE think a Epress < α <. os () + sin () in the form A sin ( + α) where A > and Determine the etreme values of the given funtion. WRItE a Write the appropriate addition identit. a A sin ( + α) = A sin () os (α) + A os () sin (α) Compare the given epression with the identit and assign appropriate values for the variales. A sin (α) = and A os (α) = Determine the unknown quantities A and α. A os (α) + A sin (α) = ( ) + A sin (α) A os (α) = tan (α) = α = tan ( ) A = A =, A > Sine α lies in the first quadrant, α =. Write the answer. os () + sin () = sin +. The etreme values are the greatest and least values of the funtion. then α must lie in the fourth quadrant. Hene, α = tan α α. Write the answer. So os () sin () = os ( ). The etreme values follow from the fat that sin + osillates etween ±, sine sin +. The greatest value is and the least value is. 98 MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

42 Eerise.7 PRatise Consolidate Other identities Epress the following as a sum or differene for questions to 8. WE9 sin () os () sin () os () WE os () os () os () os () WE os () sin () 6 os () sin () 7 WE sin () sin () 8 sin () sin () 9 WE Epress os () sin () in the form A os ( α) where A > and < α < 6. Epress sin () + os () in the form A sin ( + α) where A > and < α < 9. WE a Epress os () sin () in the form A sin ( α) where A > ; and < α <. Find the maimum and minimum values of os () sin (). a Epress sin () os () in the form A sin ( α) where A > and < α <. Find the maimum and minimum values of sin () os (). Epress the following as a sum or differene for questions to. os () sin () sin () sin () os ( + ) os ( ) 6 sin (a + ) os (a ) 7 a os () os () 8 os ( ) sin ( + ) 9 sin ( + ) os ( ) sin ( ) sin ( + ) a Epress os () sin () in the form A os ( + α) where A > and < α <. Epress os () 8 < α < 7. Epress sin () + < α < 6. sin () in the form A sin ( α) where A > and Master os () in the form A os ( α) where A > and Epress eah of the following produts as a sum or differene. a sin () os () a sin () os () os () os () Solve sin () + sin () =. If 8 os () sin () = r os ( + α): a find the values of r and α where r > and < α < 6 use CAS tehnolog to determine the least value of for whih 8 os () sin () = 9 over < < 6. Topi Cirular funtions 99

43 ONLINE ONLY.8 Review the Maths Quest Review is availale in a ustomisale format for ou to demonstrate our knowledge of this topi. the Review ontains: Multiple-hoie questions providing ou with the opportunit to pratise answering questions using CAS tehnolog Short-answer questions providing ou with the opportunit to demonstrate the skills ou have developed to effiientl answer questions using the most appropriate methods studon is an interative and highl visual online tool that helps ou to learl identif strengths and weaknesses prior to our eams. You an then onfidentl target areas of greatest need, enaling ou to ahieve our est results. Etended-response questions providing ou with the opportunit to pratise eam-stle questions. a summar of the ke points overed in this topi is also availale as a digital doument. REVIEW QUESTIONS Download the Review questions doument from the links found in the Resoures setion of our ebookplus. Units & Cirular funtions Sit topi test MaTHs QuEsT specialist MaTHEMaTiCs VCE units and

44 Answers Eerise. a.6 m i m ii.7 m. s a m s s a i kg ii 6 das t W = os + a eats/min i eats/min ii 6 min t H = sin + a 6 C at pm i 8 C ii C iii (8 ) C 6 a i mm ii s ( + ) mm; if the displaement is positive to the right, then the string is ( + ) mm to the left (or vie versa). d s 7 a 6 s m m d.6 m e i m ii 7 m f s and s 8 a i 6 m ii m 9 a Yes, appro. minutes a ($A) (8,.98) t (das) t =.8 ( d.p.), on the th da t = 6.8; t = 7 d (.6,.9) and (.66,.97) e $.98 when t = 8 a. C,.6 C i C ii. C T ( C) d hours, minutes a m 8 m. sin t (hours) 6 a i 8 ii iii i months ii 8 raits P (thousands) d 6.8. e months f Appro. months a i 9.9 m ii 8. m i.7 m ii 8. m H d See t (months) t t Topi Cirular funtions

45 e f s 7. times g H = +.9 os a.7 m hours i m at 9 pm 8t ii m at : pm d h(t) (m). (,.7) (9, )... (., ) (,.) 6 9 t (hours) e : pm and : pm f h min Eerise. a a a a. 7.. a d e f 6 a d e f sin() os() tan() ose() se() ot() sin() os() tan() ose() se() ot() 7 a 8 a 9 a D E C d A a a a a a a 6 6 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and 6 a 7 a 6 7 6

46 8 a 9 a, a os, 7 sin (), d = ose () Eerise. a a = tan () 7 8 = tan ) ) = se () = ot () = ose () = tan () e f a a = ot () ( ) = se 7 = tan ( ) = ose ( ) = se ( + ) = ot ( 6 ) Topi Cirular funtions

47 = se ( + 6 ) 6 a = ose ( + ) 6 6 d a = ose ( + ) = ot ( ) 7 6 ( + ) = tan + ( ) = tan 6 = se ( 6 ) = ot + ( ) = se ) ) 7 a C A D d B 8 a B C E d D 9 a = ose + ( ) 6 ( ) = ot MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

48 = se ( ) 8 ( ) = ot + 7 (, ) 6 a a B B C E 6 C 7 ( + ) = ose 6 = se () = ose () = ot (),, Eerise. a os () sin () a sin () os () ose () a.6 a d. e.7 a d e 6 a. d e 9 9 Topi Cirular funtions

49 7 =, 8 =, 9 =.9,.86 =.8,.8 D a a a a =, 7 =, = 6, 6 d = 6, 6, 7 6, 6 6 a =.9,. =.,. =.,. d =.96,.8 7 D 8 A 9 B D Answers will var. Answers will var. Eerise.6 a sin () os () + os () sin () os () os () + sin () sin () tan () + tan () tan () tan () a sin () os () os () sin () os () sin () tan () + tan () a sin () os () os () sin () tan () tan () a sin A os A os B B sin tan (A) tan (A) a sin (A) os (A) tan (B) 6 a os (B) sin (A) tan (A) 7 a a a.8.96 a a os () sin (B) d tan () (tan + ) tan () tan () e a E C tan (A) ( + tan B) tan (A) tan (B) 7 a os () ose () ot (A) d tan (A) e se (B) f ose (B) g tan () a D C a d 6 e 6 f 6 6 g 6 h 86 6 a.9..6 d.9 h ose (A) 6 MATHS QUEST SPECIALIST MATHEMATICS VCE Units and

50 7 a os () sin () tan () d sin () e f sin () 8 a d 9 a.9.9 a. d.8 + a ot () se () = ose () ot () se () = tan () os () os () = sin () os () = sin () = ose () [ + ot () os ()] = [ + ot ()] sin () = sin () + sin () ot () = sin () + sin () os () sin () = sin () + os () os () = sin () = [ + sin ()][ sin ()] = se () = ( sin ()) = os () os () = se () se () = os () So, [ + sin ()][ sin ()] = se () d ose () + se () = ose () se ) ose () + se () = a t =,,,, =.79,. Eerise.7 sin () + sin () sin () sin () os () + os () os () + os () [sin (7) sin ()] 6 sin () sin () 7 [os (8) os ()] 8 os () os (6) 9 os ( ) sin ( + ) a 7 sin (.8) sin () + os () = os () + sin () sin () os () = sin () os () = sin () os () = ose () se () The maimum is 7 and the minimum 7. a sin (.6) Maimum, minimum sin () sin () os () os () [os ( + 6 ) + os ()] 6 sin (a) + sin () 7 a [os (6) + ] 8 [sin ( + ) sin ( )] 9 (sin + sin ) [os () os ()] a os ( +.7) sin ( ) os ( 6 ) a sin ( + ) + sin ( ) a (os os ) = ± n, n Z a r = 7 and α = 6 6 = Topi Cirular funtions 7