WIXLIB

A Guide to Advanced TrigonometryBefore starting with Grade 12 Double and Compound Angle Identities, it is important torevise Grade 11 Trigonometry. Special attention should be given to using the generalsolution to solve trigonometric equations, as well as using trigonometric identities to simplifyexpressions. With the general solution it is important to know that in CAPS we no longer usethe ‘quadrant method’, but only the rules for general solution stated below.General Solution according to CAPS: If sin x a, -1 a 1,Then x sin 1 a k 360 or x 180 sin 1 a k 360 k Ζ If cos x a, -1 a 1,Then x cos 1 a k 360 k Ζ Ifx aa Then x tan 1 a k180 k ΖGrade 11 Identities sin 2 θ cos 2 θ 1 sin 2 θ 1 cos 2 θ cos 2 θ 1 sin 2 θsin θtan θ cos θ1cos θ tan θ sin θ Important to know and to remember If sin A sin BThen A B k 360 or A 180 B k 360 k Ζ If cos A cos BThen A B k 360 or A B k 360 k Ζ If tan A tan BThen A B k 360 k Ζ If sin A cos B Then rewrite as either sin A sin 90 B or cos 90 A cos BOnce Grade 11 has been revised we can move on to Grade 12 Trigonometry. It isrecommended that an identity or formula is taught one at a time and practised well. Startwith the Compound Angle formulas, explain how cos α β cos α cos β sin α sin β is

proved and then use it to derive the other identities, cos α β cos α cos β sin α sin β ,sin α β sin α cos β cos α sin β , sin α β sin α cos β cos α sin β . As it is stated in theCAPS document ‘Accepting cos α β cos α cos β sin α sin β derive the other compoundangle identities.’Now do examples using the Compound Angle formulas starting with basic examples andprogressing to more difficult ones.Then move on to Double Angle formulas. sin 2 A 2 sin A. cos A . Explain how to provecos 2 A cos 2 A sin 2 Aandhencethattheotherformulascanbederived,cos 2 A 2 cos A 1 , cos 2 A 1 2 sin A . Again do adequate examples only using the22Compound Angle Formula.Complete your teaching of this section by doing exercises where both Compound andDouble angle Identities are used in equations, to prove identities and to simplify expressions.It is important to encourage pupils to work through past examination papers in preparationfor their own examinations.

Video SummariesSome videos have a ‘PAUSE’ moment, at which point the teacher or learner can choose topause the video and try to answer the question posed or calculate the answer to the problemunder discussion. Once the video starts again, the answer to the question or the rightanswer to the calculation is given.Mindset suggests a number of ways to use the video lessons. These include: Watch or show a lesson as an introduction to a lesson Watch of show a lesson after a lesson, as a summary or as a way of adding in someinteresting real-life applications or practical aspects Design a worksheet or set of questions about one video lesson. Then ask learners towatch a video related to the lesson and to complete the worksheet or questions, either ingroups or individually Worksheets and questions based on video lessons can be used as short assessments orexercises Ask learners to watch a particular video lesson for homework (in the school library or onthe website, depending on how the material is available) as preparation for the next dayslesson; if desired, learners can be given specific questions to answer in preparation forthe next day’s lesson1. Revision of General Solution and IdentitiesThis video revises the general solution of trigonometric equations and trigonometricidentities.2. Identities and EquationsIn this video, the Compound Angle Identity cos α β cos α cos β sin α sin β is proved,and other identities derived from it. They are used in various examples.3. Using the Compound Angle IdentitiesExamples are done where only the Compound Angle Identities are used. Theseexamples include proving identities and simplifying expression.4. Double Angle IdentitiesThe double angle identities are introduced and proven.5. Using the Double Angle IdentitiesExamples are done where only the Double Angle Identities are used. These examplesinclude proving identities and simplifying expression.6. Revising the Sine, Cosine and Area RulesThis video revises the sine, cosine and area rules. It then applies these rules to Grade 12level problems.7. 3D Trigonometric ProblemsThis video applies all of the skills learnt in Advanced Trigonometry to three dimensionalproblems.

Resource MaterialResource materials are a list of links available to teachers and learners to enhance their experience ofthe subject matter. They are not necessarily CAPS aligned and need to be used with discretion.1. Revision of General Solutionand Identities2. Identities and Equations3. Using the Compound AngleIdentities4. Double Angle Identities5. Using ch?v w.youtube.com/watch?v px?fileticket ibmDNaU7kbA%3D&tabid 621&mid tions/video clips/trg geom/trigonometry/solving trig equations4 http://www.youtube.com/watch?v /http://www.reddit.com/r/math/comments/1cilin/i need help provingidentities involving com/watch?v /http://www.reddit.com/r/math/comments/1cilin/i need help provingidentities involving double/Notes and examples on usinggeneral solutions.Notes and examples on usinggeneral solutionsA video on using general solution.Notes on using identities to solvetrigonometric equations.A video on using identities tosolve trigonometric equations.Notes on and examples on usingidentities to solve trigonometricequations.Examples and notes on using thecompound angle formulas.Examples and notes on using thecompound angle formulas.A video on compound angleformulas.Notes and examples on using thedouble angle formulas.A video on the double angleformulas.Notes and examples on using thedouble angle formulas.Notes and examples on using thedouble angle formulas.Notes and examples on using thedouble angle formulas.A video on the double angleformulas.Notes and examples on using thedouble angle formulas.

6. Revising the Sine, Cosineand Area Rules7. 3D Trigonometric s/332 ze/standard/maths ii/trigonometry/sin cosine area Shows step by step usage of Sineand Cosine Rules.An example of how to use theArea Rule.Summary of the three rules andquestions that require the use ofall three.Questions involving all tionsoftrigonometry. It includes workedexamples of trigonometry in 3D.

TaskQuestion 1Give the general solution for:cos θ 0,766Question 2Prove that:tan x 1 cos 2 x sin xsin 2 x cos xQuestion 3Prove that:sin(30 x) sin(30 x) cos xQuestion 4Solve for x:sin(3x 50 ) cos(2 x 10 ) 0 And hence determine x if x 180 ;180 Question 5Solve for A:4 cos 2 A 2 sin A cos A 1 0Question 6 If θ 0 ;180 , solve for θ , correct to one decimal place: 3 cos 2θ 2,34Question 7Prove the identity:sin 2θ cos θcos θ sin θ cos 2θ sin θ 1Question 8If tan 40 k , express2 sin 20 . cos 20 in terms of k.2 4 cos 2 20 Question 9Find the general solution of θ , correct to one decimal place:cos 2θ 2 sin 2θ 2 0Question 10Simplify the following:sin 3x cos 3x sin xcos x

Question 11Prove the identity:cos 2 x cos 2 x 3 sin 2 x1 22 2 sin xcos 2 xQuestion 12Answer this question without using a calculator. If sin 54 0 p , express each of the followingin terms of p:12.1 tan 54 012.2 sin 306 012.3 tan 2 144 012.4 cos 1080Question 13Use the diagram to answer the questions.Determine13.1. The length of the fence needed tostretch from point A to C.13.2. The size of angle D13.3. Hence, the area or the piece of landused for the cattle. ΔADC Question 14The upper surface of the prism is an isosceles triangle withFurthermore,and.14.1. Write down an expression for AC in terms ofα and x using the cosine rule.14.2. Prove that the length of AF is given byAF x 2 1 cos cos

Question 15A surveillance camera is placedat point A. It shows two carsparked outside the building. Theangle of elevation of A from D is. Car C is equidistant from CarD and the building. Let x denotethe distance DC andProve that AB .

Task AnswersQuestion 1Give the general solution for:cos θ 0,766θ 139,99 k 360 k ΖQuestion 2Prove that:tan x 1 cos 2 x sin xsin 2 x cos xRHS1 cos 2 x sin xsin 2 x cos x1 1 2 sin 2 x sin x 2 sin x cos x cos x2 sin 2 x sin x cos x 2 sin x 1 sin x(2 sin x 1) cos x 2 sin x 1 sin x cos x tan x LHS RHS Question 3Prove that:sin(30 x) sin(30 x) cos xLHSsin(30 x) sin(30 x) sin 30 .cos x cos 30 .sin x sin 30 .cos x cos 30 .sin x 2 sin 30 .cos x1 2( ). cos x2 cos x LHS RHS

Question 4Solve for x:sin(3x 50 ) cos(2 x 10 ) 0 And hence determine x if x 180 ;180 sin(3x 50 ) cos(2 x 10 )sin(3x 50 ) sin[90 (2 x 10 )]sin(3x 50 ) sin(100 2 x)sin(3x 50 ) sin(2 x 100 ) 3x 50 2 x 100 k 360 or3x 50 180 (2 x 100 ) k 360 3x 50 180 2 x 100 k 360 x 150 k 360 5x 230 k 360 5 46 k 72 k Ζ x 46 ;118 ; 26 ; 98 ; 170 ; 150 Question 5Solve for A:4 cos 2 A 2 sin A cos A 1 04 cos 2 A 2 sin A cos A (sin 2 A cos 2 A) 03cos 2 A 2 sin A cos A sin 2 A 0 (Trinomial)3cos 2 A 2 sin A cos A sin 2 A 0(3 cos A sin A)(cos A sin A) 03cos A sin Asin A3 cos A A 71,57 k180 orcos A sin Atan A 1or A 45 k180 A 180 45 k180 A 135 k180 Question 6 If θ 0 ;180 , solve for θ , correct to one decimal place:3 cos 2θ 2,34cos 2θ 0,782θ cos 1 ( 0,78) k 360 2θ 141,26 k 360 θ 70,6 k180 k Ζk Ζk Ζ3 cos 2θ 2,34

Question 7Prove the identity:sin 2θ cos θcos θ sin θ cos 2θ sin θ 1LHSsin 2θ cos θsin θ cos 2θ2 sin θ cos θ cos θ sin θ (1 2 sin 2 θ )cos θ (2 sin θ 1) 2 sin 2 θ sin θ 1cos θ (2 sin θ 1) (2 sin θ 1)(sin θ 1)cos θ sin θ 1 LHS RHSQuestion 8 If tan 40 k , express2 sin 20 . cos 20 2 4 cos 2 20 sin 2(20 ) 2(1 2 cos 2 20 )sin 40 2(2 cos 2 20 1) sin 40 2(cos 40 )tan 40 2k 2 2 sin 20 . cos 20 in terms of k.2 4 cos 2 20

Question 9Find the general solution of θ , correct to one decimal place:cos 2θ 2 sin 2θ 2 0 cos 2 θ sin 2 θ 4 sin θ cos θ 2 sin 2 θ cos 2 θ 0cos 2 θ sin 2 θ 4 sin θ cos θ 2 sin 2 θ 2 cos 2 θ 0sin 2 θ 4 sin θ cos θ 3cos 2 θ 0 sin θ 3cos θ sin θ cos θ 0sin θ 3cos θtan θ 3orθ 71,6 k180 orsin θ cos θtan θ 1θ 45 k180 Question 10Simplify the following:sin 3x cos 3x sin xcos xsin 3x cos x cos 3x sin xsin x cos xsin(3x x) sin x cos xsin 2 x sin x cos x2 sin x cos x sin x cos x 2Question 11Prove the identity:cos 2 x cos 2 x 3 sin 2 x1 22 2 sin xcos 2 xLHScos 2 x sin 2 x cos 2 x 3sin 2 x 2(1 sin 2 x) 2 cos 2 x 2 sin 2 x2(1 sin 2 x) 2(cos 2 x sin 2 x)2(1 sin 2 x)2(1)2(cos 2 x)1 cos 2 x LHS RHS k Ζ

Question 1212.1(Pythagoras)p1 p2tan 54 p 1 p212.2sin 306 0 sin 540 -p12.3tan 2 144 0sin 2 144 cos 2 144 sin 2 (90 54 )cos 2 (90 54 )sin 2 54 cos 2 54 12.41 p2p2cos 1080 cos 2(540 ) 1 2 sin 2 540 1 2 p2Question 13AC400 sin B sin 60 AC400 sin 70sin 60 AC 400.sin 70 sin 60AC 434,03md 2 a 2 c 2 2ab cos D(434,03) 2 (600) 2 (500) 2 2(600)(500) cos D188616,46 610000 600000 cos D

421383.54 600000 cos D0,7023059 cos DD 45,39 Area of ΔADC 1ac sin D21ΔADC (600)(500) sin 45,39 2ΔADC 106785,52m 2Question 14AC 2 x 2 x 2 2 x.x cos αAC 2 2 x 2 2 x 2 cos αAC 2 x 2 2 x 2 cos αAC x 2 1 cos α AFAC sin 90sin(90 θ)AF AF ACsin(90 θ)x 2(1 cos α)sin(90 θ)Question 15 CBD B B C D 180 2 BIn ΔBCDBDx sin(180 2 B) sin Bx sin 2 Bsin Bx 2 sin B cos BBD sin BBD 2 x cos BIn ΔABDAB tan θBDAB BD tan θ 2x cos B tan θBD

AcknowledgementsMindset Learn Executive HeadContent Manager Classroom ResourcesContent Coordinator Classroom ResourcesContent AdministratorContent DeveloperContent ReviewerDylan BusaJenny LamontHelen RobertsonAgness MunthaliTwanette KnoetzeHelen RobertsonProduced for Mindset Learn by TrafficFacilities CoordinatorProduction ManagerDirectorEditorPresenterStudio CrewGraphicsCezanne ScheepersBelinda RenneyAlriette GibbsNonhlanhla NxumaloSipho MdhluliJT MedupeAbram TjaleWayne SandersonThis resource is licensed under a Attribution-Share Alike 2.5 South Africa licence. When using thisresource please attribute Mindset as indicated at http://www.mindset.co.za/creativecommons

In this video, the Compound Angle Identity cos . sin is proved, and other identities derived from it. They are used in various examples. 3. Using the Compound Angle Identities Examples are done where only the Compound Angle Identities are used. These examples