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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)

Telematics Mathematics Grade 12 Resources14February to October 2019Revision: Grade 11 Geometry Theorems and ConversesThe proofs of the theorems marked with (**) must be studied because it could be examined. The part in boldin bracket is the abbreviation for the theorem, which we use as reasons when writing up geometry solutions.123Theorem**The line drawn from the centre of a circle perpendicular to a chord bisects the chord;(line from centre ٣ to chord)ConverseThe line from the centre of a circle to the midpoint of a chord is perpendicular to the chord.(line from centre to midpt of chord)The perpendicular bisector of a chord passes through the centre ofthe circle; (perp bisector of chord)Theorem** The angle subtended by an arc at the centre of a circle is double the size of the anglesubtended by the same arc at the circle (on the same side of the chord as the centre);( ס at centre 2 ס at circumference)Corollary1. Angle in a semi-circle is 900 ( ס s in semi circle)2. Angles subtended by a chord of the circle, on the same side of the chord, are equal( ס s in the same seg)3. Equal chords subtend equal angles at the circumference (equal chords; equal ס s)4. Equal chords subtend equal angles at the centre (equal chords; equal ס s)5. Equal chords in equal circles subtend equal angles at the circumference of the circles.(equal circles; equal chords; equal ס s)Corollary1. If the angle subtended by a chord at the circumference of the circleis 90q, then the chord is a diameter. (converse ס s in semi circle)Converse2. If a line segment joining two points subtends equal angles at two points on the same sideof the line segment, then the four points are concyclic.Theorem** The opposite angles of a cyclic quadrilateral are supplementary; (opp ס s of cyclic verse5Theorem6Theorem**ConverseIf the opposite angles of a quadrilateral are supplementary then the quadrilateral is a cyclicquadrilateral. (opp ס s quad sup OR converse opp ס s of cyclic quad)The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle of thequadrilateral. (ext ס of cyclic quad)If the exterior angle of a quadrilateral is equal to the interior opposite angle of thequadrilateral, then the quadrilateral is cyclic.(ext ס int opp ס OR converse ext ס of cyclic quad)The tangent to a circle is perpendicular to the radius/diameter of thecircle at the point of contact.(tan ٣ radius)If a line is drawn perpendicular to a radius/diameter at the point where the radius/diametermeets the circle, then the line is a tangent to the circle. (line ٣ radius)Two tangents drawn to a circle from the same point outside the circle are equal in length.(Tans from common pt OR Tans from same pt)The angle between the tangent to a circle and the chord drawn from the point of contact isequal to the angle in the alternate segment. (tan chord theorem)If a line is drawn through the end-point of a chord, making with the chord an angle equal toan angle in the alternate segment, then the line is a tangent to the circle.(converse tan chord theorem OR ס between line and chord)Scan the QR code for grade 11geometry revision with solutions.

Telematics Mathematics Grade 12 Resources15February to October 2019Session 3 Grade 12 GeometryThe Grade 11 geometry entails the circle geometry theorems dealing with angles in a circle, cyclicquadrilaterals and tangents. The Grade 12 geometry is based on ratio and proportion as wellas similar triangles. Grade 11 geometry is especially important in order to do the grade 12Geometry hence this work must be thoroughly understood and regularly practiced to acquire thenecessary skills. The grade 11 geometry is summarized on the previous page.Below are Grade 12 Theorems, Converse Theorems and their Corollaries which you must know. Theproofs of the theorems marked with (**) must be studied because it could be examined.1Theorem** A line drawn parallel to one side of a triangle divides the other two sides proportionally.(line one side of Δ OR prop theorem; name lines)ConverseIf a line divides two sides of a triangle in the same proportion, then the line is parallel to thethird side.(line divides two sides of in prop)Theorem** If two triangles are equiangular, then the corresponding sides are in proportion (andconsequently the triangles are similar)( s OR equiangular s)ConverseIf the corresponding sides of two triangles are proportional, then the triangles are equiangular(and consequently the triangles are similar).(Sides of in prop)Two variables are proportional if thereis a constant ratio between them.PROPORTIONALITYRatio A ratio describes the relationship between two quantities which have the same units.We can use ratios to compare lengths, age, etc. A ratio is a comparison between twoquantities of the same kind and has no units.Example 1: if the length of the base of a triangle is 200 cm and the height is 40 cm, then we can express theratio between the length of the base and the height of the triangle:A ratio written as a fraction isusually given in its simplest form.Example: IfAndIf two or more ratios areequal to each other, thenwe say that they are in thesame proportion.

Telematics Mathematics Grade 12 Resources16February to October 2019Triangle Proportionality Theorem.If a line parallel to one side of a triangle intersects the other two sides of the triangle, thenthe line divides these two sides proportionally.Given:Thesestatementscan bemadeStatementReasonThe theorem isthe reason,prop theorem DE BCThe proportionalitytheorem written as areason in short.SPECIAL CASE OF THE CONVERSE PROPORTIONALITY THEOREM:THE MID-POINT THEOREMA corollary of the proportion theorem is the mid-point theorem: the line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half the length ofthe third side.If AB BD and AC CE, then BC צ DE andBC ½DE.We also know thatAPPLYING THE PROPRTIONALITY THEOREM:EXAMPLE 1In the diagram below, ABC has D on AB and E on AC such that DE BC.DB 2 units, EC 3 units, AD units and AE units.Determine the value ofStatementReasonprop theorem DE BC

Telematics Mathematics Grade 12 Resources17February to October 2019CONVERSE OF THE PROPORTIONALITY THEOREM:EXAMPLE 2In the diagram : KB 7 units; AK 42 units; AM 54units and MC 9 units.Prove that KM is parallel to BC.We need to prove that KM divide the sides of theABC():Let’s investigate:This could come in handy ifyou want to prove TWOlines are parallel!EXAMPLE 3In the diagram, ABC has D and P on AB and E on AC such that DE BC and PE DCDB units, DP 3 units, AP units, AE 10 units and AE units.Determine the value ofprop theorem PE DCEXAMPLE 4In the diagramgbelow, PQR has T and S on RQ and Y on QP such that TY SP and SY PRIf; determine the ratio ofStatementReasonprop theorem TY SPprop theorem SY PR

Telematics Mathematics Grade 12 Resources18February to October 2019AREA OF TRIANGLES IN PROPORTIONALITY PROBLEMS:EXAMPLE 5with PM parallel to DF.In the diagram isPD 12 units, EP 8 units, EM 12 units and MF 18units5.1 Determine the ratio of:5.2 Determine the ratio of:x There are TWO known formulas for the area of ax We have to decide which formula works best in a given question.1) Area ofuse when two2) Area ofuse when two s have a common angle.5.15.2AlwayssimplifyScan the QR code for revision fromexamination papers on the grade 12geometry theorems with solutions.have a common height.

Telematics Mathematics Grade 12 Resources19February to October 2019VEXERCISE 142x – 10QUESTION 1In the diagram below, 'VRK has P onVR and T on VK such that PT RK.VT 4 units, PR 9 units, TK 6units and VP 2 – 10 units.Calculate the value of .TP69KRQUESTION 2In the diagram, 'ABC has P and K onAB and T and M on AC such thatPT KM BC.AP 36cm, PK 24cm, AT 48cm;MC 8cm, KBand TMCalculate the value ofandQUESTION 3In the diagram below, 'ABC has D onAB ; F on BC and E on AC such thatDE BC and EF AB .AD 12 units, EC 25 units EF 20units and FC 30 units.Calculate the value of ,andQUESTION 4O is the centre of the circlebelow. OM A AC. The radius ofthe circle is equal to 5 cm andBC 8 cm.O4.1Write down the size of B A.4.2Calculate:4.2.1The length of AM, with reasons.4.2.2Area 'AOM : Area 'ABCAMCB

Telematics Mathematics Grade 12 Resources20February to October 2019SIMILARITYTwo polygons with the same number of sides are similar if:1) All pairs of corresponding angles are equal and2) All pairs of corresponding sides are in the same ratio.The symbol forsimilarity is: ABCDE is similar to PQRST if:;1)and2)9 Both conditions must be true for two polygons to be similar.Theorem**If two triangles are equiangular, then the corresponding sides are in proportion (and consequentlythe triangles are similar)( s OR equiangular s)ThenGiven:ConverseIf the corresponding sides of two triangles are proportional, then the triangles are equiangular (and consequentlythe triangles are similar). (Sides of in prop)ThenGiven: 'ABC and 'DEF withNote:Be careful to correctly label similartriangles. The angles that are equalmust be in the same position:

Telematics Mathematics Grade 12 ResourcesEXAMPLE 1In the diagram is21February to October 2019. Prove that ܥܦܧ ܥܤܣ In ABC and DEC:1)(alt ) ܧܦצܤܣ ; ݏס 2)(alt ) ܧܦצܤܣ ; ݏס ( ס , ס , )ס Given that figures are similar; we can deduce information abouttheir corresponding parts that we didn't previously know.EXAMPLE 2In the diagram isandProve that ܻܼܺ ܴܲܳ and thatunits.Inand1)2)3) ܻܼܺ ܴܲܳ ሺ ሻ Given that figures are similar; we can deduce information abouttheir corresponding angless that we didn't previously know.EXAMPLE 3 :Given the sketch below determine the valuesof y and r without using a calculator.EXAMPLE 4:In the diagram isDetermine the value ofand.

Telematics Mathematics Grade 12 Resources22EXAMPLES WITH CIRCLE GEOMETRY.EXERCISE 1QUESTION 1In the diagram below P, S, R and Q are pointson the circumference of the circle. QP producedmeets RS produced at T.1.1Prove that TPS TRQ1.2Show that TP. TQ1.3TS. TRHence, or otherwise, prove that PQQUESTION 2In the diagram below, CA is a tangent to thecircle.Prove, with reasons, that:2.1 ACD ABC2.2CD.AC BC.ADQUESTION 3In the diagram below SP is a tangent to thecircle. KM is the diameter of the circle.Prove, with reasons, that:3.1 KLM MLP3.2ML²KL. LPFebruary to October 2019

Telematics Mathematics Grade 12 Resources23February to October 2019QUESTION 4In the diagram below CA is a tangent to the circle with EAAD.Prove, with reasons, that:4.14.2ADC BEAAD² BE.DCQUESTION 5In the figure below, AB is a tangent to the circle with centre O. AC AO and BA CE.DC produced, cuts tangent BA at B.A1E223123144FB2O1Prove, with reasons, that:5.125.25.31ACF ADCAD4AFD3C

Telematics Mathematics Grade 12 Resources24February to October 2019QUESTION 6In the figure below, O is the centre of the circle CAKB. AK produced intersects circle AOBT at T.A B6.1Prove, with reasons, that:6.1.16.1.2 AC KB.6.1.36.2BKT CATIf AK : KT 5 : 2, determine the value of7. In the figure below, GB FC and BE CD. AC 6 cm andABBC2.AGH7.1Calculate with reasons:a)AH : EDb)BECDBF7.2 If HE 2 cm, calculate the value of AD HE.ECD

Telematics Mathematics Grade 12 Resources25February to October 2019Session 4: Grade 12 CalculusCubic Graphs. In this lesson you will work through 3 types of questions regarding graphs1. Drawing cubic graphs2. Given the graphs, answer interpretive questions3. Given the graphs of the derivative, answer interpretive questions1.1 Given:2x 3 x 2 4x 3f ( x)1.1.1 Show that ( x 1 ) is a factor of f (x).(2)1.1.2 Hence factorise f (x) completely.(2)1.1.3 Determine the co-ordinates of the turning points of f.(4)1.1.4 Draw a neat sketch graph of f indicating the co-ordinates of the turning points as wellas the x-intercepts.(4)1.1.5 For which value of x will f have a point of inflection?1.2 Given f ( x)(4)[16]x 3 x 2 5x 31.2.1 Draw a sketch graph of f (x) .(7)1.2.2 For which value(s) of x is f (x) increasing?(2)1.2.3 Describe one transformation of f (x) that, when applied, will result in f (x) havingtwo unequal positive real roots.1.2.4 Give the equation of g if g is the reflection of f in the y-axis.(2)1.2.5 Determine the average rate of change of f between the points (0 ; 3) and (1;0).(2)1.2.6 Determine the equation of the tangent to the f when x -2.1.2.7 Prove that the tangent in 1.2.6 will intersect or touch the curve of f at two places.(4)(4)(3)1.3 A cubic function f has the following properties:§1·f 2¹xf (3)f ( 1)0§ 1·f c 0 3¹ª 1 ºf decreases for x « ; 2» only 3 ¼f c(2)xDraw a possible sketch graph of f, clearly indicating the x-coordinates of the turning points(4)and ALL the x-intercepts.1.4 If f is a cubic function with:xf (3) f c (3) ,xf (0) 27 ,xfcc (x) 0 when x 3 and fcc (x) 0 when x 3draw a sketch graph of f indicating ALL relevant points.(3)

Telematics Mathematics Grade 12 Resources2.1 The graph of the function f ( x)26February to October 2019 x 3 x 2 16x 16 is sketched below.yfx02.1.1 Calculate the x-coordinates of the turning points of f.(4)2.1.2 Calculate the x-coordinate of the point at which f c(x) is a maximum.(3)2.1.3 Show that the concavity of f changes at x (3)2.2 The graphs of f ( x) ax3 bx2 cx d and g ( x) 6 x 6 are sketched below. A(– 1 ; 0)and C(3 ; 0) are the x-intercepts of f. The graph of f has turning points at A and B.D(0 ; – 6) is the y-intercept of f. E and D are points of intersection of the graphs of f and g.ygA( 1 ; 0)C(3 ; 0)xOD(0 ; 6)fBE2.2.1 Show that a 2 ; b – 2 ; c – 10 and d – 6.(5)2.2.2 Calculate the coordinates of the turning point B.(5)2.2.3 h(x) is the vertical di

This workbook however will only have the material for the grade 12 Telematics sessions. The grade 11 material you will be able to download from the Telematics website. Please make sure that you bring this workbook along to each and every Telematics session. In the grade 12 examination Trigonometry