GAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 19 (LEARNER NOTES)

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1 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) TRIGONOMETRY () Learner Note: Trigonometry is an extremely important and large part of Paper. 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 Trigonometry which involves compound and double angles. These Grade concepts will be integrated with the Trigonometry you studied in Grade. Before attempting the typical exam questions, familiarise yourself with the basics in Section C. SECTION A: TYPICAL EXAM QUESTIONS Question (0 minutes) Simplify the following without using a calculator: (a) tan( 60 )cos( 56 )cos 94 sin 49 (b) cos 75 cos ( 75 ) sin( 50 )sin 0 sin 40 cos0 Question (0 minutes) (a) Show that cos60 cos 60 sin (5) (b) Hence, evaluate cos05 cos5 without using a calculator. (5) SECTION B: SOLUTIONS AND HINTS Page of 9

2 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) (a) tan( 60 )cos( 56 )cos 94 sin 49 ( tan 60 )(cos56 )( cos 66 ) (sin ) ( )( cos 4 )( sin 4 ) (sin 48 ) ( )( cos 4 )( sin 4 ) sin 4cos 4 (b) cos 75 cos ( 75 ) sin( 50 )sin 0 sin 40cos0 cos 5 cos 75 ( sin 50 )( sin 50 ) (sin 40 )(cos50 ) cos 5 sin 5 sin 50 (cos50 )(cos50 ) cos 5 sin 5 sin 50 cos 50 cos (5 ) cos0 cos 60 cos 60 (a) cos60.cos sin 60.sin cos60.cos sin 60.sin.cos.sin.cos.sin sin ( tan 60 )(cos56 ) cos66 sin 48 sin 4 sin4cos4 cos 5 cos 75 sin 50 cos 50 cos0 cos60.cos sin 60.sin cos60.cos sin 60.sin sin (5) Page of 9

3 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) (b) cos05 cos5 cos cos sin 45 6 cos60 45 cos sin 45 6 (5) Page of 9

4 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) SECTION C: ADDITIONAL CONTENT NOTES Summary of all Trigonometric Theory sin y cos x tan y r r x sin cos tan sin cos tan ( x; y) sin cos tan sin cos tan Reduction rules sin(80 ) sin sin(80 ) sin sin(60 ) sin cos(80 ) cos cos(80 ) cos cos(60 ) cos tan(80 ) tan tan(80 ) tan tan(60 ) tan sin(90 ) cos sin(90 ) cos cos(90 ) sin cos(90 ) sin sin( ) sin cos( ) cos tan( ) tan Whenever the angle is greater than 60, keep subtracting 60 from the angle until you get an angle in the interval 0 ;60. Identities sin cos sin tan cos Page 4 of 9

5 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) Special angles Triangle A Triangle B From Triangle A we have: From Triangle B we have: sin 45 sin0 and sin60 cos45 tan 45 cos0 and tan0 and cos60 tan 60 For the angles 0 ; 90 ;80 ;70 ;60 the diagram below can be used. 90 y A(0 ;) B(; ) G( ; 0) 80 r C( ; ) D( ;) E( ; 0) x 0 60 F(0 ; ) 70 Page 5 of 9

6 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) The following identities are important for tackling Grade Trigonometry: Compound angle identities sin(a B) sin Acos B cos Asin B sin(a B) sin Acos B cos Asin B cos(a B) cos Acos B sin Asin B cos(a B) cos Acos B sin Asin B Double angle identities cos sin cos sin sin cos cos sin cos cos sin SECTION D: HOMEWORK Question Determine the value of the following without using a calculator. (a) sin 4 cos0 cos4 sin0 sincos () (b) sin( 85 ) (5) (c) cos 5 sin5cos75 cos 5 sin5cos5tan5 (6) Question (a) Prove that sin(45 ).sin(45 ) cos (5) (b) Hence determine the value of sin 75.sin5 () Page 6 of 9

7 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) SECTION E: SOLUTIONS TO SESSION 8 HOMEWORK (a) 5 A ( 0, ) A R65 56 (b) 5 A ( 0,6) A R40 068, correct formula answer () correct formula answer () (c) Sinking fund , Sinking fund 54 5, answer () (d) Draw a time line T0 T T T 0,8 0,05 T60 (d) 6 x (,05) 54 5, 0, , 0,05 x 6 (,05) x R59,55 correct formula F 54 5, n 6 0,8 0,05 x R59,55 (4) (e)() Draw a time line T0 T T T 4 6 T48 60 T Page 7 of 9

8 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) (e)() (e)() Future value of the withdrawals: ,8 0, ,8 0, R, The reduced value of the sinking fund will be: R54 5, R, R 99, If we add R, to the original sinking fund amount of R54 5,, then it will be possible to not only receive the sinking fund amount of R54 5, at the end of the five year period, but also be able to make the service withdrawals at the end of each year for the five year period. 6 x (, 05) 54 5,, 0,05 6 x (, 05) ,55 0, ,550, 05 x 6 (, 05) x R87,90 services R, reduced value () correct formula F ,55 n 6 0,8 0,05 x R87,90 (5) (a) 000 n (, 08) , ,08 n (, 08) ,08 (,08) n (, 08),08 n log 5 n n 9,9788 n 0 half-years n 0 years (since there are 0 half years in a ten-year period) correct formula 0,6 0,08 time period n 5 (,08) n log 5 n,08 n 9,9788 n 0 years Page 8 of 9

9 MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) (b) 7000 n (, 04) , ,04 (,04) n 7000 n ,04 (, 04) 7000 (, 04) 0, , (, 04) log 0, n,04 n n 7, years months n correct formula 0,6 0,04 4 0, (,04) n log,05 n 0,065 n 7, answer (6) The SSIP is supported by Page 9 of 9