QUESTION x = 15 answer (3) For drawing the radius vector in the correct quadrant 1/3. sin α > 0 in second quadrant

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1 Mathematics/P 7 DoE/Novemer 009() QUESTION 8 8. sin α 8 7 ( 5 ; 8) sin α > 0 in second quadrant y 8 r 7 α α 5 tanα α 8 5 ( Pythagoras) 7 α 5 For drawing the radius vector in the correct quadrant / sin(90 + α) cosα 5 7 cos α sin α Without a sketch ut correct values: 3/3 reduction () nswer only: full marks Cannot accept decimal values epansion cos α cos α cos α cos α sin α any further calculation or epansion any further calculation or epansion any further calculation or [8]

2 Mathematics/P 8 DoE/Novemer 009() QUESTION 9 NOTE: Only penalise once in the question for leaving out the Penalise once in this question for treating as an equation 9. sin(90 ).cos(80 ) + tan.cos( ).sin(80 + ) cos ( cos ) + tan (cos )( sin ) cos cos (cos sin cos sin + sin cos sin ) sin90 cos 5 tan 390 cos00 sin35 sin0 ( cos 45 ) tan 30 sin0 sin or tan 30 If using cos 80 : no penalty If the candidate stop at /7 sin + cos sin + ( sin sin sin ) + sin + 0 sin 0 (sin + )(sin ) 0 sin 90 + k.360 Or sin( 90 ) cos cos( 80 ) cos cos( ) cos sin( 80 + ) sin sin tan cos simplification sin 90 sin 0 cos 5 cos 45 tan 390 tan 30 cos 00 sin 0 sin 35 sin 45 or cos 45 o of identity standard form factorisation sin ;sin 90 + k.360 s (any two s) If k Z not included: 6/7 lso ± k. 360 ; k N0 or Z

3 Mathematics/P 9 DoE/Novemer 009() sin 0 + k.360 or k k.360 or k.360 sin + cos sin cos sin cos [ sin(90 ) ] sin 80 + (90 ) k0 k Z sin + cos sin cos sin cos cos(90 ) cos 80 (90 ) 90 k Z or or 0 + k.360 or 30 + k k.360 or 30 + k (90 ) 70 k (90 ) k0 manipulation of identity co ratios 80 + (90 ) 90 + k0 360 (90 ) 70 k360 If k Z not included: 6/7 manipulation of identity co ratios 80 (90 ) (90 ) 30 + k0 If k Z not included: 6/7 [0]

4 Mathematics/P 0 DoE/Novemer 009() QUESTION 0 0. sin( + ) sin.cos + cos.sin cos( + ) sin.sin sin.cos + cos.sin sin.sin sin.cos cos.sin + sin.sin tan + tan tan.tan epansions divisions tan and tan 0. tan + tan RHS tan. tan sin sin + cos cos sin sin cos cos sin cos + sin cos cos cos sin sin sin sin( + ) cos( + ) tan( + ) LHS tan C tan(80 ( + )) tan C tan( + ) tan + tan tan C tan.tan tan C( tan.tan ) (tan + tan ) tan C tan.tan.tan C tan tan tan + tan + tan C tan.tan.tan C sin cos multiplication epansions C tan( + ) into formula multiplication with LCD If no conclusion: 3/4

5 Mathematics/P DoE/Novemer 009() C ˆ 80 ( ˆ + ˆ) (angles in a triangle) tan C tan(80 ( + )) tan C tan((80 ) + ( )) tan(80 ) + tan( ) tan C tan(80 ).tan( ) tan C( tan(80 ).tan( )) tan(80 ) + tan( ) tan C tan C tan tan tan tan tan + tan + tan C tan.tan.tan C C rearrange angle into formula epansion QUESTION NOTE: Penalty of one for early rounding off once in this question.. Dˆ sin D sin sin D 0, Dˆ 30,58 earing of Ship from Ship 80 ( ) + 30, 58 58, 58 Dˆ sin D ˆ sin sin D ˆ 0, D ˆ 30,58 then ND ˆ (refle angles) ND ˆ 5 ut MD ˆ + ND ˆ 80 (co - interior angles/ angles around a point) MD ˆ 8 then M ˆ MD ˆ + D ˆ 30, ,58 Dˆ C 4 sine rule ˆ 30, 58 method or M ˆ D 8 (6) Dˆ C 4 sine rule N D ˆ 5 M ˆD 8 (6)

6 Mathematics/P DoE/Novemer 009().. ˆ 30, 58 E D 8 30,58 0km E sin(8 + 30,58 ) 0 E 0sin(8 + 30,58 ) E 0,4 km definition 0 P 58,58 Q R Let Q, then Q 0 PQ QR sin 58,58 sin 58,58 0 PQ.sin 58,58 QR (0 )sin 58,58 PQ + QR.sin 58,58 + (0 )sin 58,58 P R 0sin 58,58 0,4 (assume ships move at same speed) trigonometeric ratios sum trigonometeric

7 Mathematics/P 3 DoE/Novemer 009() PQ RQ Q Q 60 km PQ sin58,58 60 PQ 60sin 58,58 PR PQ 5,0 km 0,4 km (angle, angle, side) ratios 5,0 km M cos3,4 0 M 0 cos3,4 0,4 8 30,58 3,4 0 M trigonometeric ratios. C a c a + c ac cos a + a a a cos a a cos a cos a sin a a cos ( cos ) cos sin a a a / / equal sides cos rule simplification sin sin a formula [3]

8 Mathematics/P 4 DoE/Novemer 009() a + c cos ac cos rule equal sides ut a c a + a cos a. a a a a simplification QUESTION. y ( 0 ; 0) or ( 60 ; 0) ( 30 ; ) or ( 0 ; ) ()

9 Mathematics/P 5 DoE/Novemer 009() QUESTION. cos( 30 ) cos( 30 ) See points and on the graph Note: If drawn the line y and put and on the graph: 0/ manipulation () and in the correct place on the graph: full marks If and on the -ais: / If 30 and 90 : /.3 cos( 30 ) 0, g ( ) 0 is at maimum and minimum values of graph 30 ; 0.5 [ 90 ; 60 ) (0 ; 70 ] 90 < 60 or 0 < 70 If < 60 or > 0 /3 60 (ref angle) nswer only: 3/3 one for each -value () notation critical values []