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GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 4 (LEARNER NOTES) TOPIC 1: TRIGONOMETRY (A) Learner Note: Trigonometry is an extremely important and large part of Paper 2. You must ensure that you master all the basic rules and definitions and be able to apply these rules in many different types of questions. In this session, you will be concentrating on Grade 12 Trigonometry which involves compound and double angles. These Grade 12 concepts will be integrated with the trigonometry you studied in Grade 11. Before attempting the typical exam questions, familiarise yourself with the basics in Section B. SECTION A: TYPICAL EXAM QUESTIONS QUESTION 1 Simplify the following without using a calculator: (a) tan(60)cos(156)cos 294 sin 492 (7) (b) cos2 375 cos2 (75) sin(50)sin 230 sin 40 cos310 (7) [14] QUESTION 2 (a) Show that cos 60 cos 60 3 sin (5) (b) Hence, evaluate cos105 cos15 without using a calculator. (5) [10] QUESTION 3 Rewrite cos3 in terms of cos . [6] Page 1 of 6 GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 4 (LEARNER NOTES) SECTION B: ADDITIONAL CONTENT NOTES Summary of all Trigonometric Theory sin sin y r cos x r tan y x sin 90 90 cos cos tan tan ( x ; y) 180 360 180 sin sin cos cos tan tan Reduction rules sin(180 ) sin sin(180 ) sin sin(360 ) sin cos(180 ) cos tan(180 ) tan sin(90 ) cos cos(180 ) cos tan(180 ) tan sin(90 ) cos cos(360 ) cos tan(360 ) tan cos(90 ) sin sin() sin cos(90 ) sin cos() cos tan() tan Whenever the angle is greater than 360 , keep subtracting 360 from the angle until you get an angle in the interval 0 ;360 . Identities cos 2 sin 2 1 tan sin cos Page 2 of 6 GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 4 (LEARNER NOTES) Special angles Triangle A Triangle B 45 2 60 2 1 1 30 45 1 3 From Triangle A we have: sin 45 1 cos 45 1 2 2 From Triangle B we have: 2 2 sin 30 2 2 cos30 3 2 tan 30 1 1 tan 45 1 1 1 2 and sin 60 3 2 and cos 60 1 2 and tan 60 3 3 3 1 For the angles 0; 90;180;270;360 the diagram below can be used. 90 y A(0 ;2) B(1 ; 3) C( 2 ; 2) 2 60 2 r2 G( 2 ; 0) 45 D( 3 ; 1) 30 180 E(2 ; 0) x 0 360 F(0 ; 2) 270 Page 3 of 6 GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 4 (LEARNER NOTES) The following identities are important for tackling Grade 12 Trigonometry: Compound angle identities sin(A B) sin A cos B cos A sin B sin(A B) sin A cos B cos A sin B cos(A B) cos A cos B sin A sin B cos(A B) cos A cos B sin A sin B Double angle identities cos 2 sin 2 cos 2 sin 2 2sin cos cos 2 sin 2 cos 2 2 cos 2 1 2 1 2sin SECTION C: HOMEWORK QUESTION 1 Determine the value of the following without using a calculator: (a) sin 34 cos10 cos34 sin10 sin12 cos12 (3) (b) sin(285) (5) (c) cos 2 15 sin15 cos 75 cos 2 15 sin15 cos15 tan15 (6) [14] QUESTION 2 1 cos 2 2 (a) Prove that sin(45 ).sin(45 ) (b) Hence determine the value of sin 75.sin15 (5) (3) [8] QUESTION 3 Prove that: sin 4 4sin .cos 8sin 3 .cos [4] Page 4 of 6 GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 4 (LEARNER NOTES) SECTION D: SOLUTIONS AND HINTS TO SECTION A 1(a) tan( 60) cos( 156) cos 294 sin 492 ( tan 60)(cos156)( cos 66) (sin132) ( 3)( cos 24)( sin 24) (sin 48) ( 3)( cos 24)( sin 24) 2sin 24 cos 24 3 2 cos 2 375 cos 2 (75) sin(50)sin 230 sin 40 cos 310 1(b) cos 2 15 cos 2 75 ( sin 50)( sin 50) (sin 40)(cos 50) cos 2 15 sin 2 15 2 sin 50 (cos 50)(cos 50) cos 2 15 sin 2 15 sin 2 50 cos 2 50 cos 2(15) 1 cos 30 ( tan 60)(cos156) cos 66 sin 48 3 sin 24 2sin 24 cos 24 3 2 (7) cos2 15 cos 2 75 sin 2 50 cos 2 50 cos30 1 3 2 (7) 3 2 [14] Page 5 of 6 GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS 2(a) GRADE 12 SESSION 4 cos 60 cos 60 (LEARNER NOTES) cos60.cos sin 60.sin cos60.cos sin 60.sin cos 60.cos sin 60.sin cos 60.cos sin 60.sin 3 3 1 1 .cos .sin .cos .sin 2 2 2 2 3 sin 2(b) cos105 cos15 cos 60 45 cos 60 45 (5) cos 60 45 cos 60 45 3 sin 45 3 sin 45 2 3 2 3 3 2 1 2 3 sin 2 2 6 2 6 2 (5) [10] cos 3 cos(2 ) cos 2.cos sin 2.sin cos(2 ) cos 2.cos sin 2.sin (2 cos 2 1).cos (2sin cos ).sin 2 cos3 cos 2sin 2 .cos 2 cos3 cos 2(1 cos 2 ) cos 2cos 2 1 2sin cos 1 cos2 4cos3 3cos (6) 2 cos3 cos 2(cos cos3 ) 2 cos3 cos 2 cos 2 cos3 4 cos3 3cos [6] The SSIP is supported by Page 6 of 6