ME-T3 Trigonometric equations Y12



Year 12 Mathematics Extension 1ME-T3 Trigonometric equationsUnit durationThe topic Trigonometric Functions involves the study of periodic functions in geometric, algebraic, numerical and graphical representations. It extends to include the exploration of both algebraic and geometric methods to solve trigonometric problems.A knowledge of trigonometric functions enables students to manipulate trigonometric expressions to prove identities and solve equations.The study of trigonometric functions is important in developing students’ understanding of the connections between algebraic and graphical representations and how this can be applied to solve problems from theoretical or real-life scenarios, for example involving waves and signals.6-7 lessonsSubtopic focusOutcomesThe principal focus of this subtopic is to consolidate and extend students’ knowledge in relation to solving trigonometric equations and to apply this knowledge to practical situations.Students develop complex algebraic manipulative skills and fluency in applying trigonometric knowledge to a variety of situations. Trigonometric expressions and equations provide a powerful tool for modelling quantities that vary in a cyclical way such as tides, seasons, demand for resources, and alternating current.A student:applies advanced concepts and techniques in simplifying expressions involving compound angles and solving trigonometric equations ME12-3chooses and uses appropriate technology to solve problems in a range of contexts ME12-6evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7Prerequisite knowledgeAssessment strategiesStudents should have studied Year 11 Trigonometry content, MA-T2, Trigonometric functions and identities, and ME-T2, Further trigonometric identities, as well as the Year 12 subtopic, MA-T3, Trigonometric functions and graphs.Students could complete proofs on vertical whiteboards, which provides instant feedback to the teacher about their understanding. Students could also correct incorrect proofs. All outcomes referred to in this unit come from the Mathematics Extension 1 syllabus? NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2017Glossary of termsTermDescriptiondomainThe domain of a function is the set of x values of y=f(x) for which the function is defined. Also known as the ‘input’ of a function.Lesson sequenceContentSuggested teaching strategies and resources Date and initialComments, feedback, additional resources usedConverting expressions to Rcos(x±α) and Rsin(x±α) (1 lesson)convert expressions of the form a cosx+bsinx to Rcos(x±α) or Rsin(x±α) and apply these to solve equations of the form acosx+bsinx=c, sketch graphs and solve related problems (ACMSM048) Note: See the resource me-t3-unit-worked-solutions.DOCX which contains algebraic and graphical solutions for the sample questions and activities referred to throughout this unit.Converting expressions to Rcos(x±α) and Rsin(x±α) Students to review compound formulae.sin A±B=sin A cos B±cosAsin Bcos (A±B)=cos A cos B?sin A sin Btan A±B=tan A ± tan B1 ? tan A tan BTeacher to model converting expressions of the form a cosx+bsinx to Rcos(x±α) or Rsin(x±α)Note: a and b can be either positive or negative.Examples: Show 3 cosx+sinx=2cosx-π6Find 4 expressions equivalent to 3 cosx+sinxGraph a function fx=a cosx+bsinx by converting it to fx=Rcosx±α or fx=Rsinx±αTeacher can demonstrate equivalence of different forms by graphing each using graphing software such as Geogebra or Desmos (α to be inputted in terms of π). Resource: me-t3-matching-expressions.DOCX Solving equations of the form acosx+bsinx=c(1 or 2 lessons)convert expressions of the form a cosx+bsinx to Rcos(x±α) or Rsin(x±α) and apply these to solve equations of the form acosx+bsinx=c, sketch graphs and solve related problems (ACMSM048) solve trigonometric equations and interpret solutions in context using technology or otherwise Solving equations of the form acosx+bsinx=cTeacher to model:solving equations of the form a cosx+bsinx=c by converting to the form Rcos(x±α) or Rsin(x±α)adjusting the domain. Rcos(x+α)=c with a domain of 0≤x≤2π implies α≤x+α≤2π+αquestions where both a and b are positive or negative as well as where one is negative.Solving an equation using a variety of equivalent expressions.checking and solving equations graphically. Suggested software: Geogebra or Desmos.Note: This technique can be used by students to check their solutions. Questions may or may not be leading:Solve 4 cos x + 3 sin x = 3Solve 4 cos x + 3 sin x = 3 by first expressing it in the form Rcosx-αSolving equations using factorisation and/or compound angle results(1 lesson)solve trigonometric equations requiring factorising and/or the application of compound angle, double angle formulae or the t-formulaesolve trigonometric equations and interpret solutions in context using technology or otherwise Equations using factorisation and/or compound angle resultsTeacher to model/demonstrate:solving trigonometric equations involving factorising and/or compound angle formulae.Questions which include those with a specified domain and those without (general results).adjusting the domain. Rcos(x+α)=c with a domain of 0≤x≤2π implies α≤x+α≤2π+αsolving and checking equations graphically. Suggested software: Geogebra or Desmos.Note: This technique can be used by students to check their solutions. Sample equations include:tanx-π3=13sinxcosπ4+cosxsinπ4=12sinxcosπ4-cosxsinπ4=12cos2x+cosx=sin2xSample domains for equations could include:[-π, π][0, 2π]No domain specified, general results.Solving equations using double angle results(1 lesson)solve trigonometric equations requiring factorising and/or the application of compound angle, double angle formulae or the t-formulaesolve trigonometric equations and interpret solutions in context using technology or otherwise Solving equations using double angle resultsStudents to review the double angle formulae.sin 2A =2sinAcosAcos 2A=cos2A-sin2A=2 cos2A-1=1-2 sin2Atan 2A = 2 tan A1 - tan2ATeacher to model/demonstrate:solving trigonometric equations involving double angle formulae.adjusting the domain. Rcos(2x)=c with a domain of 0≤x≤2π implies 0≤2x≤4πquestions which include those with a specified domain and those without (general results).solving and checking equations graphically. Suggested software: Geogebra or Desmos.Note: This technique can be used by students to check their solutions. Sample equations might include:tan2x=3cos2x-sin2x=12cos2x=sinxsin2x=sinxSample domains for equations include:[-π, π][0, 2π]No domain specified, general results.Solving equations using the t-formulae(1 lesson)solve trigonometric equations requiring factorising and/or the application of compound angle, double angle formulae or the t-formulaesolve trigonometric equations and interpret solutions in context using technology or otherwise Solving equations using the t-formulaeStudents to review the t-formulae.sinA=2t1+t2cosA=1-t21+t2tanA=2t1-t2where t=tanx2Teacher to model/demonstrate:solving a variety of equations using the t-formulae.adjusting the domain. Solving 4cosx + 3sinx=3 with a domain of 0≤x≤2π implies 0≤x2≤πchecking solutions graphically. Suggested software: Geogebra or Desmos.Note: This technique can be used by students to check their solutions.Example:Solve 4cosx + 3sinx=3 using the result t=tanx2Proofs and applications of trigonometric identities(1 lesson)prove and apply other trigonometric identities, for example cos3x=4cos3x-3cosx (ACMSM049)Proofs and applications of trigonometric identitiesTeacher to model formal proofs of trigonometric identities.One method of proving a = b is by showing a = c and b = c.Sample trigonometric identities to prove include:cos3x=4cos3x-3cosxcos4x=8cos4x-8cos2x+1tan3x=3tanx-tan3x1-3tan2x1+cos2xsin2x-cosx=2cosx2sinx-1Sample proof and application:Prove sin3x=3sinx-4sin3x. Hence or otherwise solve 3sinx-4sin3x=32.Reflection and evaluationPlease include feedback about the engagement of the students and the difficulty of the content included in this section. You may also refer to the sequencing of the lessons and the placement of the topic within the scope and sequence. All ICT, literacy, numeracy and group activities should be recorded in the ‘Comments, feedback, additional resources used’ section. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download