Trig. Past Papers Unit 2 Outcome 3 - Mathsrevision.com

Higher MathematicsPSfrag replacementsOxy[SQA]13.( )Solve the equation 2 sin 2x − π 6= 1, 0 ≤ x < 2π. 4rag replacementsOxy[SQA] 14. (a) Solve the equation sinPSfrag 2x ◦ − cos replacements x ◦ = 0 in the interval 0 ≤ x ≤ 180. 4yy = sin 2x ◦(b) The diagram shows parts of twotrigonometric graphs, y = sin 2x ◦and y = cos x ◦ .180Use your solutions in (a) to write O 90xdown the coordinates of the point P. 1P y = cos x ◦Part Marks Level Calc. Content Answer U2 OC3(a) 4 C NC T10 30, 90, 150 2001 P1 Q5(b) 1 C NC T3 (150, −√32 )• 1 ss: use double angle formula• 2 pd: factorise• 3 pd: processpd: process• 4• 1 2 sin x ◦ cos x ◦• 2 cos x ◦ (2 sin x ◦ − 1)• 3 cos x ◦ = 0, sin x ◦ = 1 2• 4 90, 30, 150• 5 ic: interpret graphor• 3 sin x ◦ = 1 2and x = 30, 150• 4 cos x ◦ = 0 and x = 90•(150, 5 √ )− 32replacementsOxy hsn.uk.netPage 5Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

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Higher MathematicsPSfrag replacementsPSfrag replacementsOxy[SQA]18.The diagram shows the graph of acosine function from 0 to π.(a) State the equation of the graph. 1y2(b) The line with equation y = − √ 3 Oπ π xintersects this graph at point A2A Band B.y = − √ 3Find the coordinates of B.−23Part Marks Level Calc. Content Answer U2 OC3(a) 1 C NC T4 y = 2 cos 2x 2002 P1 Q8(b) 3 C NC T7 B( 7π12 , −√ 3)• 1 ic: interpret graph• 12 cos 2x• 2 ss: equate equal parts• 3 pd: solve linear trig equation inradians• 4 ic: interpret result• 1 2 cos 2x = − √ 3• 2 2x = 5π 6 , 7π 6• 3 x = 7π12[SQA]19.rag replacementsOxyrag replacementsOxyreplacementsOxy hsn.uk.netPage 8Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

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Mitch Renkow working paper for SPIA, October 2010In sum, the dearth of efforts to quantify the impacts of CGIAR research on theenvironment is striking. 6 There are several possible explanations for this. First,measuring environmental services in a consistent manner over a period of time isdifficult. It requires sampling a large number of variables that one would want to holdconstant in any meaningful statistical analysis, beginning at a very early stage in theadoption/diffusion process. And the larger the number of variables exerting influenceon a given environmental outcome, the more difficult it is to establish links betweenthat outcome and a particular agricultural practice (and thence attribution to research).Second, valuing those services poses a distinct challenge as well. With theexception of CIFOR’s work on deforestation and the work of CIP and IRRI on pesticideuse, this appears to have been an insurmountable obstacle in most CGIAR research inthis area. Particularly noticeable is the absence of non-market valuation – using eitherrevealed preference or stated preference techniques – of environmental services affectedby CGIAR technology, management or knowledge products.There are no doubt other, more prosaic reasons for the paucity of effort devoted tomeasuring the environmental impacts of the products of CGIAR research. That line ofinquiry typically requires substantial inter-disciplinary collaboration, the organizationand administration of which can be challenging. The substantial field research requiredto pursue environmental impact assessment analysis is costly as well, particularly forresearch efforts scaled at the regional, national or global levels. Such field workrequires sampling at multiple points in time, extending the duration (and cost) of theactivity. Finally, particularly in the case of assessing negative environmental impactsof CGIAR research, there is a fundamental matter of institutional appetite for pursuingthis sort of activity. Simply put, Centers have little incentive to pursue research that hassome positive probability of putting them in a bad light.Toward an Environmental Impact Assessment FrameworkFor decades there has been substantial interest among donors, policy makers, andagricultural scientists in understanding the environmental impacts of CGIAR researchoutputs (and agricultural research in general). Yet, as the review of past work in theprevious section indicated, surprisingly little headway has been made in achieving that6In order to begin bridging this gap, in 2009 SPIA recently commissioned a set of six ex-post impactassessment case studies to quantify the environmental impacts of specific Center research activities.Those case studies encompass a variety of topics, including zero tillage in the Indo-Gangetic Plains,supplemental irrigation, water control in mixed rice and shrimp areas, rubber agroforestry, potatobiodiversity preservation, and the land-saving impacts of improved bean cultivation. These studies arescheduled to be finalized by the end of 2010.12

Higher MathematicsPSfrag replacementsOxy[SQA]25.rag replacementsOxyrag replacementsOxy[SQA]26.If cos θ = 4 5 , 0 ≤ θ < π 2, find the exact value of(a) sin 2θ 2(b) sin 4θ. 3rag replacementsOxy[SQA] 27. Given that tan α =√113 , 0 < α < π 2, find the exact value of sin 2α. 3rag replacementsOreplacements xyOxy hsn.uk.netPage 13Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

Higher MathematicsPSfrag replacementsOxy[SQA] 28. Given that cos D = √ 2 and 0 < D < π 5 2, find the exact values of sin D and cos 2D. 3rag replacementsOxy[SQA] 29. Given that sin A = 3 4 , where 0 < A < π 2, find the exact value of sin 2A. 3rag replacementsOxy[SQA] 30. For acute angles P and Q, sin P = 1213 and sin Q = 3 5 .Show that the exact value of sin(P + Q) is 6365 . 3rag replacementsOxyreplacementsOxy hsn.uk.netPage 14Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

Higher MathematicsPSfrag replacementsOxy[SQA] 31. Find the exact value of sin θ ◦ + sin(θ ◦ + 120 ◦ ) + cos(θ ◦ + 150 ◦ ). 3rag replacementsOxy[SQA] 32. If x ◦ is an acute angle such that tan x ◦ =3 4 , show that the exact value ofsin(x ◦ + 30 ◦ ) is 4√ 3 + 3. 310rag replacementsOxyreplacementsOxy hsn.uk.netPage 15Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

Higher MathematicsPSfrag replacementsOxy[SQA]33.rag replacementsOxyrag replacementsOxy[SQA] 34. A and B are acute angles such that tan A =4 3 and tan B = 12 5 .Find the exact value of(a) sin 2A 2(b) cos 2A 1(c) sin(2A + B). 2rag replacementsOxyreplacementsOxy hsn.uk.netPage 16Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

Higher MathematicsPSfrag replacementsOxy[SQA]35.Functions f (x) = sin x, g(x) = cos x and h(x) = x + π 4set of real numbers.(a) Find expressions for:(i) f (h(x));are defined on a suitable(ii) g(h(x)). 2(b) (i) Show that f (h(x)) = 1 √2sin x + 1 √2cos x.(ii) Find a similar expression for g(h(x)) and hence solve the equationf (h(x)) − g(h(x)) = 1 for 0 ≤ x ≤ 2π. 5Part Marks Level Calc. Content Answer U2 OC3(a) 2 C NC A4 (i) sin(x + π 4), (ii) 2001 P1 Q7cos(x + π 4 )(b) 5 C NC T8, T7 (i) proof, (ii) x = π 4 , 3π 4• 1 ic: interpret composite functions• 2 ic: interpret composite functions• 3 ss: expand sin(x + π 4 )• 4 ic: interpret• 5 ic: substitute• 6 pd: start solving processpd: process• 7• 1 sin(x + π 4 )• 2 cos(x + π 4 )• 3 sin x cos π 4+ cos x sin π 4andcomplete• 4 g(h(x)) = √ 1 2cos x − √ 12sin x• 5 ( 1 √2sin x + 1 √2cos x) − ( 1 √2cos x − 1 √2sin x)• 6 2 √2sin x• 7 x = π 4 , 3π 4accept only radiansreplacementsOxy hsn.uk.netPage 17Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

Higher MathematicsPSfrag replacementsOxy[SQA]36.On the coordinate diagram shown, A is theypoint (6, 8) and B is the point (12, −5). PSfrag Angle replacementsA(6, 8)AOC = p and angle COB = q.Find the exact value of sin(p + q). 4pCO qxB(12, −5)Part Marks Level Calc. Content Answer U2 OC34 C NC T96365 2000 P1 Q1• 1 ss: know to use trig expansionsin p cos q + cos p sin q• 2 pd: process missing sides PSfrag replacements • 2 10 and 13• 3 ic: interpret data• 3 8• 410 · 1213 + 10· 6 513pd: processO• 1• 4 126130xyA[SQA]37.In triangle ABC, show that the exactC2value of sin(a + b) is √ . 45 1abB1 3Part Marks Level Calc. Content Answer U2 OC34 C NC T9 proof 2002 P1 Q5pd: process the missing sides• 2 ss: expand• 3 pd: substitutepc: process and complete proof• 1• 4• 1 AC = √ 2 and BC = √ 10stated or implied by • 3• 2 sin(a + b) = sin a cos b + cos a sin b• 3 √ 1 √32· 10+ √ 1 12 · √10• 4 √ 420= . . . = √ 2 5replacementsOxy hsn.uk.netPage 18Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes

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Higher MathematicsPSfrag replacementsOxy[SQA]39.rag replacementsOxyrag replacementsOxy[END OF WRITTEN QUESTIONS]replacementsOxy hsn.uk.netPage 20Questions marked ‘[SQA]’ c○ SQAAll others c○ Higher Still Notes