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Western Cape Education DepartmentTelematicsLearning Resource 2019MATHEMATICSGrade 12

Telematics Mathematics Grade 12 Resources3February to October 20192019 Mathematics Telematics ProgramDayDateTimeGradeSubjectTopicTerm 1: 9 Jan – 15 MarchTuesday12 February15:00 – 16:0012MathematicsTrigonometry RevisionWednesday13 February15:00 – 16:0012WiskundeTrigonometrieHersieningTERM 2: 2 April to 14 JuneMonday8 April15:00 – 16:0012MathematicsTrigonometryTuesday9 April15:00 – 16:0012WiskundeTrigonometrieWednesday15 May15:00 – 16:0011MathematicsGeometryThursday16 May15:00 – 16:0011WiskundeMeetkundeWednesday22 May15:00 – 16:0012MathematicsGeometryThursday23 May15:00 – 16:0012WiskundeMeetkundeTerm 3: 9 July – 20 SeptemberMonday29 July15:00 – 16:0012MathematicsDifferential CalculusTuesday30 July15:00 – 16:0012WiskundeDifferentiaalrekeningWednesday07 August15:00 – 16:0011MathematicsFunctionsMonday12 August15:00 – 16:0011WiskundeFunksiesTerm 4: 1 October – 4 DecemberTuesday15 October15:00 – 16:0011MathematicsPaper 1 RevisionWednesday16 October15:00 – 16:0011WiskundePaper 2 Revision

Telematics Mathematics Grade 12 Resources4February to October 2019Session 1: TrigonometryxDefinitions of trigonometric ratios:oIn a right-angled 'Sin eoppositSin TyrCosTxrTanTyxTadjacentoppositeadjacentTanTxo On a Cartesian Plane0 , 90 , 180 , 270 , 360 can beyx30 , 45 and 60 can beobtained from the following unit circleT.obtained from the following90qyr, the radius is1 since it is aunit circle(0 ; 1) 180q (-1 ; 0) (1 ; 0) xTTT270q3cos 45q 1T Tan is ve in the3rdquadrantSineSAllTan Cos180q TbecomesT 60q32sin 60q 2cos 60q 1 2tan 45q 13T180q- TS Sine is ve in the2nd quadrant1T 45qsin 45q 12145qT 30qsin 30q 1 2tan 30q 145q2360q1cos 30q The “CAST” rule enables you to obtain thesign of the trigonometric ratios in any of thefour quadrants.yThe trigonometric function of angles(180q T) or (360q T) or (-T)30q20q360q (0 ; -1)xrTxSpecial AnglesoTytan 60q 23A - ALL trigratios are ve inthe first quadrantxC Cos is ve inthe 4nd quadrant360q-T Trigonometric function of TThe sign is determined bythe “CAST” rule.(180q T )(180q T )(360q T )(360q T )( T )sin(180q T ) sin Tsin(180q T ) sin Tsin(360q T ) sin Tsin(360q T ) sin Tsin( T ) sin Tcos(180q T ) cos Tcos(180q T ) cos Tcos(360q T ) cos Tcos(360q T ) cos Tcos( T ) cos Ttan(180q T ) tan Ttan(180q T ) tan Ttan(360q T ) tan Ttan(360q T )tan( T ) tan Ttan T

Telematics Mathematics Grade 12 ResourcesxFebruary to October 2019TRIGONOMETRIC IDENTITIEStan Tx5sin TcosT(cosT z 0)sin 2 T cos 2 TCo-functions or Co-ratiossin(90q T )cosTcos(90q T )sin Tsin 2 T1,1 cos 2 T ,rTxDetermine theReference angleEstablish inwhich twoquadrants θ is.Calculate θ inthe interval[0q; 360q]Write down thegeneral solution2.3.4.sin(90q T ) cos Tcos(90q T ) sin TxTrigonometric Equationssin T1.90q-T ycos 2 T 1 sin 2 Tcos T0,707tan T 0,866 1 1Reference sin (0,707) 45qReference cos (0,866) 30qReference tan 1 (1) 45q? θ 45qorθ 180q - 45q? θ 180q- 30q or θ 180q 30q? θ 180q - 45q? θ 45qorθ 135q? θ 150q orθ 210q? θ r150q? θ r150 k360º where k ? θ 135q? θ 45q k360º orθ 135q k360º where k 1? θ 135q k180º & k TRIGONOMETRIC GRAPHSSine FunctionCosine FunctionTangent FunctionEquationShapea 0a 0aAmplitudePeriodaNote:“c” refers to theside of the triangleopposite to angleC that is the sideSOLUTIONS OF TRIANGLESxArea RuleArea of 'ABC x111absin C acsin B bcsin A222Sine Rulesin A sin Bsin C abcxOrabc sin A sin Bsin CACosine Rulea2b2 c 2 2bc cos Ab c a2bc2orcos A2bc2BaC

Telematics Mathematics Grade 12 Resources6February to October 2019TRIGONOMETRY SUMMARYQuestiontypeSummary ofprocedure1. Calculatethe value of atrigexpressionwithout usinga calculator.Establish whether youneed a rough sketch orspecial triangles, ASTCrules or compoundangles.Example question3, D [0q; 270q] and E [0q; 180q] .4It is given that sin(D E ) sin D. cos E sin E . cos DDetermine, without using a calculator,a) sin Db) sin(D E ) .cos( 210 ). sin 2 405 . cos14 1.2 Calculate: a)tan 120 . sin 104 1.1 If 13 cos D5 and tan E b) sin 70q cos 40q cos 70q sin 40q2. If a trigratio is givenas a variableexpressanother trigratio in termsof the samevariable.3. Simplify atrigonometrical expression.4. Prove agiven identity.5. Solve atrig equation.Draw a rough sketch2. If sin 27q q , express each of the following in terms of q.with given angle anda) sin 117qb) cos( 27q)label 2 of the sides. The3rd side can then bedetermined usingPythagoras. Express eachof the angles in questionin terms of the angle inthe rough sketch.Use the ASTC rule to3. Simplify:simplify the givencos ( 720q x) . sin ( 360q x) . tan ( x 180q)a)expression if possible.sin ( x) . cos (90q x)See if any of thesin ( 90q x) . tan ( 360q x)identities can be used tob)sin (180q x) . cos (90q x) cos(540q x). cos( x)simplify it, if not see if itcan be factorized. Checksin 2 x cos x cos 3 xagain if any identity canc)cos xbe used. This includes2using the compound andsin x cos xd)double angle identities.1 cos 2 xSimplify the one side ofProve thatthe equation usingtan x . cos 3 x1sin xa)reduction formulae and221 sin x co s x 2identities until .b)Find the reference angleSolve for x [ 180q; 360q]by ignoring the “-“signand findinga) sin x 0,435b) cos 2 x 0,435sin 1 (0,435)1Write down the twoc) tan x 1 0,435solutions in the interval2x [0q; 360q] . Thenwrite down the generalsolution of this eq. Fromthe general solution youcan determine thesolution for the specifiedinterval by using variousvalues of k.

Telematics Mathematics Grade 12 Resources7February to October 2019Question typeSummary of procedureExample question6. Sketch a trig graph.1st sketch the trig graph without thevertical or horizontal transformation.Then shift the graph in this case 1 unitup.Sketch7.Find the area of atriangle.8. Finding an unknownside or angle in atriangle.b)y2 cos 3 x 1 for x [ 90q; 120q]c)y sin( x 60)for x [ 240q; 120q]'ABC, with B 104,5q , ABIf it is a right-angled triangle then1areabase u height , otherwise use2the area rule1Area of 'ABC ab sin C2Draw a rough sketch with the giveninformation. If it is not a right-angledtriangle you will use either the sine orcosine rule.BC6cm and9cm . Calculate, correct to one decimal placearea 'ABCa)b)'ABC, with B 104,5q , AB 6cm andBC 9cm . Calculate the length of AC.'ABC, with C 43,2q , AB 4,5cm andBC 5,7cm . Calculate the size of A .SKETCHING TRIG GRAPHCalculatethe periodWrite downtheamplitude ifit is a sine orcosinegraph.Identify the shape of the graph and drawa sine, cosine or tan graph withdetermined period and amplitude. Labelthe other x-intercepts. Repeat this patternover the specified domain.SKETCHNow do the vertical or horizontaltransformation if required.2 cos 3 x 1 for x [ 90q; 120q]yyyPeriod Amplitude 2360q120q32312x-90-60-3030-1-260190x-60-3030-16090

Telematics Mathematics Grade 12 Resources8February to October 2019QUESTION .1In the figure below, the point P(–5 ; b) is plotted on the Cartesian plane.OP 13 units and RÔP D .yP(–5 ; b)x 13xROxWithout using a calculator, determine the value of the following: .2 .1.1cos D(1) .1.2tan(180q D )(3)Consider:sin(T 360q) sin(90q T ) tan( T )cos(90q T )sin(T 360q) sin(90q T ) tan( T )to a single trigonometric ratio.cos(90q T ) .2.1Simplify .2.2Hence, or otherwise, without using a calculator, solve for T if0q d T d 360q :sin(T 360q) sin(90q T ) tan( T )cos(90q T ) .3 .484 2sin A 1 cos A0,54.1 cos A .3.1Prove that .3.2For which value(s) of A in the interval 0q d A d 360q is the identity inQUESTION 5.3.1 undefined?Determine the general solution of 8 cos 2 x 2 cos x 1 0 .(5)(3)(5)(3)(6)

Telematics Mathematics Grade 12 Resources9February to October 2019QUESTION In the diagram below, the graphs of f (x) cos(x p) and g(x) q sin x are shown for theinterval 180q d x d 180q .yyg1Af0,5xx-180 -135 -90q-45q0q45q90q180q135q- 0,5B-1 .1Determine the values of p and q.(2) .2The graphs intersect at A(–22,5 ; 0,38) and B. Determine the coordinates of B.(2) .3Determine the value(s) of x in the intervalf ( x) g ( x) 0 . .4 180q d x d 180qfor which(2)The graph f is shifted 30 to the left to obtain a new graph h. .4.1Write down the equation of h in its simplest form. .4.2Write down the value of x for which h has a minimum in the interval 180q d x d 180q .(2)(1)

Telematics Mathematics Grade 12 Resources10February to October 2019QUESTION 1 2sin AaProve that in any acute-angled sin C.c(5) P̂ 132q, PQ 27,2 cm and QR 73,2 cm.P132 27,2 cmRQ .373,2 cm .2.1Calculate the size of R̂ .(3) .2.2Calculate the area of .(3)In the figure below, SP̂Q a , PQ̂S b and PQ h. PQ and SR are perpendicularto RQ.PaShbRQ .3.1Determine the distance SQ in terms of a, b and h. .3.2Hence show that RSh sin a cos b.sin( a b)(3)(3)

Telematics Mathematics Grade 12 Resources11February to October 2019Session 2: TRIGONOMETRY(rr 50/150 Marks)Compound and Double AnglesIn order to master this section it is best to learn the identities given below. These identities will also be givenon the formulae sheet in the Examination paper.xxCompound Angle Identities:(a) cos( A B)cos A cos B sin Asin Bcos( A B)cos A cos B sin Asin B(b) sin( A B)sin A cos B sin B cos Asin( A B)sin A cos B sin B cos AWhen two anglesare added orsubtracted to forma new angle, then acompound or adouble angle isformed.Double Angle Identities(c) sin 2 A2 sin A cos A(d) cos 2 Acos2 A sin 2 AReferred toas doubleangleformulae 1 2 sin 2 A 2 cos2 A 1What should you ensure you can do at the end of this section for examination purposes:A. Accepting the Compound Angle formulae cos( A B)cos A cos B sin Asin B use it to deriveThe following formulae:cos( A B)cos A cos B sin Asin Bsin( A B)sin A cos B sin B cos Asin( A B)sin A cos B sin B cos Acos 2 Acos2 A sin 2 ACo-functions or Coratiossin(90q T )cosTNegative Anglescos 2 A 1 2 sin A2sin( T )cos 2 A 2 cos2 A 1 sin Tcos( T ) cosTtan( T ) tan Tsin 2 A 2 sin A cos AYou must rememberB. Use compound angle and double angle identities to:1. Evaluate an expression without using a calculatorsin2 T cos2 T1sin2 T1 cos2 Tcos2 T1 sin2 T2. Simplifying trigonometric expressions3. Prove identities4. Solve trigonometric equations (both specific and general solutions)

Telematics Mathematics Grade 12 Resources12February to October 2019The sketches below gives a visual of compound and double angles.Sketch 1Sketch 3Sketch 2Sketch 1: The compound angle AB̂C is equal to the sum of D and β. eg. 75q45q 30qSketch 2: The compound angle EĜH is equal to the difference between D and β. eg. 15q 60q 45q or15q 45q 30qSketch 3: The double angle PT̂R is equal to the sum of D and D. eg. 45q 22.5q 22.5qGiven any special angles D and β, we can find the values of the sine and cosine ratios of the anglesα β , α β and 2α .Are you clear on the difference betweena compound and double angle?Please note:0q ; 30q ; 45q ; 60q and 90q are special angles, you are able to evaluate any trigonometric function ofthese angles without using a calculator.Exercises: Do not use a calculator.A. Derive each of the compound and double angle formulae in the box on the previous page.B.1.1.1 Evaluate each of the following without using a calculator.b) cos15qc) cos105qd) sin 165qa) sin 75qe) sin 36q.cos 54q cos 36q sin 54qf) cos 42q. cos18q sin 42q sin18qg) sin 85q. sin 25q cos 85q cos 25qh) sin 70q. cos 40q cos 70q sin 40q2 sin 40q. cos 40qi) 2 sin 30q. cos 30qj)cos10q21.2 If sin α, tan β2 and D and β are acute angles determine the value of sin(α β ) .31.3 Iftan A23and90q A 360q , determine without using a calculator2.Simplify the following expression to a single trigonometric function:3.Prove that4 cos( x). cos(90q x)sin(30q x). cos x cos(30q x). sin xa) cos 75q2 ( 3 1)4b) cos(90q 2 x). tan(180q x) sin 2 (360q x)c) (tan x 1)(sin 2 x 2 cos 2 x)4.cos 2 A .3 sin 2 x2(1 2 sin x cos x)Determine the general solution for x in the following:a) sin 2x. cos10q cos 2x. sin10q cos 3xb)cos2 x3 sin 2 xc)2 sin xsin( x 30q)Scan the QR code for revision from examinationpapers on this section with solutions.

Telematics Mathematics Grade 12 Resources13February to October 2019ADDITIONAL QUESTIONS1.8Given sin α; where 900 d α d 270q17With the aid of a sketch and without the use of a calculator, calculate:a) tan αb) sin(90q α)c) cos 2α2.a) Using the expansions for sin( A B) and cos( A B) , prove the identity of:sin( A B)cos( A B)tan A tan B1 tan A. tan B(3)sin( A B), prove in any ΔABC thatcos( A B)tan A. tan B. tan C tan A tan B tan Cb) If tan( A B)3.4.If sin 36q cos12qa) sin 48qb) sin 24qc) cos 24qp and6.cos 36q sin12q q ,determine in terms of p and q :32and 80q 60q 20qShow that sin 2 20q sin 2 40q sin 2 80q(7)Given: f ( x) 1 sin x and g ( x) cos 2 xCalculate the points of intersection of the graphs f and g for x [180q ; 360q]Given that sin T1, calculate the numerical value of sin 3T , WITHOUT using a3calculator.7.(4)(3)(3)(3)(HINT: 40q 60q 20q5.(3 2 3)(7)(5)Prove that, for any angle A:4 sin A cos A cos 2 A sin 15qsin 2 A(tan 225q 2 sin 2 A)8.Solve for x if 2 cos x9.If cos β6 22(6)tan 2 x and x [ 90q ; 90q] . Show ALL working details.(8)p; where p 0 and β [00 ; 900 ] , determine, using a diagram, an5expression in terms of p for:a) tan βb)cos 2 β10.1 If sin 28 a and cos 32 b, determine the following in terms of a and/or b :a) cos 28qb) cos 64qc) sin 4 10.2 Prove without the use of a calculator, that if sin 28 a and cos 32 b, thenb 1 a2 a 1 b21.2(4)(3)(2 3 4)(4)