MATHEMATICS GRADE SESSION 0 (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) (a) Rewrite cos in terms of cos. (b) () () Question tan x sin sin tan cos cos (0 minutes) cosa cosa sin A sin A cosa sin A Page of 7
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) SECTION B: SOLUTIONS AND HINTS (a) cos cos( ) cos.cos sin.sin ( cos ).cos (sin cos ).sin cos cos sin.cos cos cos ( cos ) cos cos( ) cos.cos sin.sin cos sin cos cos cos cos cos (cos cos ) (b)() (b)() cos cos cos cos cos sin x ( ) () () tan x sin sin cos cos sin sin.cos cos cos sin cos cos cos sin cos cos cos sin cos tan cos x ( ) ( ) tan x sin cos cos sin cos cos cos sin cos tan Page of 7
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) cos A sin A cos A sin A sin A cos A (cos A sin A)(cos A sin A) sin A cos A sin A cos A (cos A sin A)(cos A sin A) sin A sin A cos A cos A (cos A sin A)(cos A sin A) (sin A cos A)(sin A cos A) cos A sin A cos A sin A cos A sin A sin AcosA (cosa sina)(cosa sina) (sina cosa)(sina cosa) cosa sin A cosa sin A SECTION C: ADDITIONAL CONTENT NOTES Summary of all Trigonometric Theory sin y cos x tan y r r x sin cos tan 90 90 sin cos tan ( x; y) 80 80 60 sin cos tan sin cos tan Page of 7
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) 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 Special angles Triangle A Triangle B 45 60 45 0 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 Page 4 of 7
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) For the angles 0 ; 90 ;80 ;70 ;60 the diagram below can be used. 90 y A(0 ;) B(; ) G( ; 0) 80 r 60 45 0 C( ; ) D( ;) E( ; 0) x 0 60 F(0 ; ) 70 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 (a) (b) sin 4 4sin.cos 8sin.cos (4) () (tan x )( ) ( ) () cosx sin x () Page 5 of 7
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) Question (a) Show that (cos sin ) sin(45 ) () (b) Hence prove that sin sin 45 SECTION E: SOLUTIONS TO SESSION 9 HOMEWORK (a) sin 4 cos0 cos 4 sin0 sincos sin(4 0 ) sincos sin 4 sincos sincos sincos (b) sin( 85 ) sin 85 sin(60 75 ) ( sin 75 ) sin 75 sin(45 0 ) sin 45cos0 cos 45sin 0 6 4 sin 4 sincos sin85 sin75 sin 45 cos0 cos45 sin0 6 4 () Page 6 of 7
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) (c) cos 5 sin5 cos 75 cos 5 sin5cos5tan5 cos 5 sin5cos(90 5 ) sin5 cos 5 sin5cos5 cos5 cos 5 sin5.sin5 cos 5 sin 5 cos 5 sin 5 cos 5 sin 5 cos (5 ) cos0 cos 5sin5.sin5 sin5 cos5 cos 5 sin 5 cos0 (a) sin(45 ).sin(45 ) sin 45 cos cos 45 sin sin 45 cos cos 45 sin expansion of sin(45 ) expansion of sin(45 ) cos sin cos sin (cos sin ) (cos sin ) (cos sin )(cos sin ) 4 (cos sin ) sin 45 cos 45 (cos sin ) cos (b) cos sin 75.sin5 sin(45 0 ).sin(45 0 ) cos (0 ) cos60 4 45 0; 45 0 cos60 4 () The SSIP is supported by Page 7 of 7