Higher Mathematics – Vectors



TrigonometryHigher Mathematics Supplementary ResourcesSection AThis section is designed to provide examples which develop routine skills necessary for completion of this section.R1 I can convert radians to degrees and vice versa.1.Convert the following angles from degrees to radians, giving you answer as an exact value.(a)30°(b)45°(c)60°(d)90°(e)180°(f)360°(g)120°(h)135°(i)150°(j)210°(k)225°(l)240°(m)270°(n)300°(o)315°(n)330°(o)540°(p)720°2.Convert the following angles from degrees to radians, giving you answer to 3 significant figures.(a)37°(b)142°(c)226°(d)281°(e)307°(f)453°3.Convert the following angles from radians to degrees.(a)π radians(b)2π radians(c)3π radians(d)π2 radians(e)3π2 radians(f)5π2 radians(g)π3 radians(h)2π3 radians(i)4π3 radians(j)5π3 radians(k)7π3 radians(l)π4 radians(m)3π4 radians(n)5π4 radians(o)7π4 radians(p)π6 radians(q)5π6 radians(r)7π6 radians(s)11π6 radians(t)π12 radians(u)5π12 radians4.Convert the following angles from radians to degrees, giving you answer to 3 significant figures.(a)1 radian(b)2 radians(c)3 radians(d)4 radians(e)1?4 radians(f)2?7 radiansR2 I can use and apply exact values.1.Write down the exact value of(a)sin30°(b)sin60°(c)sin45°(d)sin120°(e)sin150°(f)sin135°(g)sin90°(h)sin180°(i)sin270°(j)sin210°(k)sin225°(l)sin240°(m)sin300°(n)sin330°(o)sin315°2.Write down the exact value of(a)cos30°(b)cos60°(c)cos45°(d)cos120°(e)cos150°(f)cos135°(g)cos90°(h)cos180°(i)cos270°(j)cos210°(k)cos225°(l)cos240°(m)cos300°(n)cos330°(o)cos315°3.Write down the exact value of(a)tan30°(b)tan60°(c)tan45°(d)tan120°(e)tan150°(f)tan135°(g)tan90°(h)tan180°(i)tan270°(j)tan210°(k)tan225°(l)tan240°(m)tan300°(n)tan330°(o)tan315°4.Write down the exact value of(a)sinπ6(b)sinπ4(c)sinπ3(d)sinπ(e)sin2π(f)sin3π2(g)sin5π6(h)sin3π4(i)sin2π3(j)sin7π6(k)sin5π4(l)sin4π3(m)sin11π6(n)sin7π4(o)sin5π35.Write down the exact value of(a)cosπ6(b)cosπ4(c)cosπ3(d)cosπ(e)cos2π(f)cos3π2(g)cos5π6(h)cos3π4(i)cos2π3(j)cos7π6(k)cos5π4(l)cos4π3(m)cos11π6(n)cos7π4(o)cos5π36.Write down the exact value of(a)tanπ6(b)tanπ4(c)tanπ3(d)tanπ(e)tan2π(f)tan3π2(g)tan5π6(h)tan3π4(i)tan2π3(j)tan7π6(k)tan5π4(l)tan4π3(m)tan11π6(n)tan7π4(o)tan5π3R3I can sketch or identify a basic trig graph under a single transformation.1.Write down the equation of each of the graphs3538855482608655058255(a)(b)35471100(c)897890-1270(d)34950403086102.Write down the equation of each of the graphs(a)8978902540(b)3529965194945(c)89789015875(d)3.Write down the equation of each of the graphs351853528575(a)8978903175(b)3532505306070(c)8978900(d)4.Write down the equation of each of the graphs35490158890(a)8978900(b)5.Sketch each graph showing clearly the coordinates of the maximum and minimum values and where each graph cuts the axes.(a)y=3cosx° for 0≤x≤360(b)y=sinx°+1 for 0≤x≤360(c)y=cosx-1 for 0≤x≤2π(d)y=2sinx for 0≤x≤2π(e)y=-2cosx° for 0≤x≤360(f)y=tanx-45° for 0≤x≤360(g)y=cosx-π3 for 0≤x≤2π(h)y=-3sinx for 0≤x≤2πR4I can sketch or identify a basic trig graph under combined transformations.1.Write down the equation of each of the graphs746261543053052141245745(a)(b)88569317500(c)(d)3274060163195132384249555(e)(f)31959551498601380992209803243580308610(g)(h)3132455240969(i)(j)117475593862.Sketch each graph showing clearly the coordinates of the maximum and minimum values and where each graph cuts the axes.(a)y=2cosx°-3 for 0≤x≤360(b)y=-cos2x° for 0≤x≤360(c)y=2cosx-π3 for 0≤x≤2π(d)y=sinx-π6+2 for 0≤x≤2π(e)y=4cos2x° for 0≤x≤360(f)y=-sinx-30° for 0≤x≤360(g)y=cos2x-1 for 0≤x≤2π(h)y=2-sinx for 0≤x≤2πR5I can use the addition and double angle formulae.1.Expand and use exact values to simplify(a)sinx+π6(b)sinx-60°(c)cosx-π4(d)cosx+45°(e)cosx+π3(f)sinx+60°(g)sinx-90°(h)sinx+π(i)cosx+180°2.Use an appropriate substitution (such as 45 – 30 = 15) then expand to find the exact values of(a)sin15°(b)sin75°(c)cos105°3.Given that sinx°=35 and cosx°=45 , find the exact values of:(a)sin2x°(b)cos2x°(c)sin3x° (Hint 3x=2x+x)4.Given that sinx°=513 and cosx°=1213 , find the exact values of:(a)sin2x°(b)cos2x°(c)sin4x° (Hint 4x=2(2x))5.Given that sinx°=15 and cosx°=25 , find the exact values of:(a)sin2x°(b)cos2x°(c)cos3x° 6.Given that sinx°=213 and cosx°=313 , find the exact values of:(a)sin2x°(b)cos2x°(c)cos4x° R6 I can convert acosx + bsinx to kcos(x ± α) or ksin(x ± α), where α is in any quadrant k > 0.1.A function f is defined as f(x)=5cosx°-2sinx°.Express f(x) in the form kcosx+a° where k>0 and 0≤a<360.2.Express sinx-cosx in the form ksinx-α where k>0 and 0≤α<2π.3.A function g is defined as g(x)=3cosx°+sinx°.Express g(x) in the form ksinx+α° where k>0 and 0≤α<360.4.Express sinx+2cosx in the form rcosx-a where r>0 and 0≤a<2π.5.A function Q is defined as Q(x)=2cosx°-3sinx°.Express Q(x) in the form kcosx+a° where k>0 and 0≤a<360.6.Express 3sinx-4cosx in the form asinx-b where a>0 and 0≤b<2π.7.A function f is defined as f(x)=2cosx°-sinx°.Express f(x) in the form ksinx-a° where k>0 and 0≤a<360.8.Express sinx-3cosx in the form kcosx+a where k>0 and 0≤a<2π.9.A function f is defined as f(x)=5cosx°+3sinx°.Express f(x) in the form kcosx+a° where k>0 and 0≤a<360.R7I have revised solving basic trigonometric equations in degrees and radians.1.Solve the equations:(a)5tanx°-6=2, 0≤x≤360.(b)7sinx°+1=-5, 0≤x≤360.(c)4cosx°+3=0, 0≤x≤360.(d)3tanx+3=7, 0≤x≤2π.(e)4sinx-2=-3, 0≤x≤2π.(f)9cosx-5=0, 0≤x≤2π.2.Solve the equations:(a)9tan2x°-5=3, 0≤x≤180.(b)4sin3x°+1=-2, 0≤x≤360.(c)3cos2x°+2=0, 0≤x≤360.3.Solve the equations:(a)tanx+30°=3, 0≤x≤360.(b)5sinx+10°+3=-1, 0≤x≤360.(c)4cosx+26°+3=0, 0≤x≤360.(d)3tanx+π5+1=0, 0≤x≤2π.(e)6sinx+2-2=1, 0≤x≤2π.(f)2cosx+π6+1=0, 0≤x≤2π.Section BThis section is designed to provide examples which develop Course Assessment level skillsa-bP(8, 6) Q(5, -12) R yxONR1 I can apply Trig Formulae to Mathematical Problems (excluding where trig equations have to be solved but including exact values).1.On the coordinate diagram shown, P is the point (8, 6) and Q is the point (5, -12). Angle POR = a and angle ROQ = -b.(a)Find the exact value of sin(a-b).(b)Find the exact value of cos2a.pq221ACB2.In triangle ABC, show that:(a)The exact value of sin2p=223(b)The exact value of cos(p+q)=22-115A3.For the shape shown, find the exact value of cos(ABC)°x°x°ABC2523PQRa4.The diagram shows the right angled triangle PQR, with dimensions given.(a)Find the exact value of sin2a° .(b)By expressing sin3a° as sin2a+a° , find the exact value of sin3a°.5.If cos2x=-3149 and 0<x<π2, find the exact values of cosx and sinx.6.Using the fact that π4+π6=5π12 , find the exact value of cos5π12.7.It is given that cosa=35 and sinb=23.(a)Find the exact value of sina+b and cosa+b.(b)AHence find the exact value of tana+b.PQRS8.Triangles PSQ and RSQ are right angled with dimensions as shown in the diagram.(a)Show that cosa+b is -1385 .(b)ACalculate the value of sina+b.(c)Hence calculate the value of tana+b.NR2 I have experience of using wave functions to find the maximum and minimum values.xy0360y=f(x)A1.A function f is defined as f(x)=5cosx°-2sinx°.(a)Express f(x) in the form kcosx+a° where k>0 and 0≤a<360.(b)Part of the graph of y=f(x) is shown in the diagram. Find the coordinates of the minimum turning point A.2.(a)Express sinx-cosx in the form ksinx-a where k>0 and 0≤a<2π.(b)Hence state the maximum and minimum values of sinx-cosx and determine the values of x, in the interval 0≤x<2π , at which these maximum and minimum values occur.3.(a)Express 12sinx+5cosx in the form ksinx+a where k>0 and 0≤a<2π.(b)Hence state the maximum value of 4+12sinx+5cosx and determine the value of x, in the interval 0≤x<2π , at which the maximum occurs.4.A function f is defined as f(x)=4cosx°+3sinx°.Find the maximum and minimum values of f(x) and the values of, in the range 0≤x<360 , at which the maximum and minimum values occur.5.A function g is defined as g(x)=3cosx-2sinx.Find the maximum and minimum values of g(x) and the values of, in the range 0≤x<2π , at which the maximum and minimum values occur.NR3 I can solve trigonometric equations in the context of a problem.1.Solve the equation sin2x°-cosx°=0, in the interval 0≤x<180.2.Solve the equation sinx°-sin2x°=0, in the interval 0≤x<360.3.Solve the equation 3cos2x+10cosx-1=0, in the interval 0≤x<2π.4.Solve the equation cos2x°+2sinx°=sin2x°, in the interval 0≤x<360.5.Solve the equation 2cos2x-5cosx-4=0, in the interval 0≤x<2π.6.Solve the equation tan2x=3, in the interval 0≤x<π.7.Solve the equation sinθ=4cosθ, in the interval 0≤x<2π.8.(a)Express 3sinx+4cosx in the form ksinx+a where k>0 and 0≤a<2π.(b)Hence solve the equation 3sinx+4cosx-3=0 in the interval 0≤x<2π .9.(a)Express 5sinx°+3cosx° in the form kcosx-a° where k>0 and 0≤a<360.(b)Hence solve the equation 5sinx+3cosx=4 in the interval 0≤x<360 .10.Two curves have equations y=6cosx° and y=sin2x°. Find the coordinates of the points of intersection in the range 0≤x<360.11.Two curves have equations y=-3cos2x° and y=cosx°+1. Find the coordinates of the points of intersection in the range 0≤x<180.12.A curves has the equation =cos2x°-3cosx°+2 . Find the coordinates of the points where the curve cuts the x-axis in the range 0≤x<360.13.A curves has the equation y=sin2x°+cosx° . Find the coordinates of the points where the curve cuts the x-axis in the range 0≤x<360.14.The graph shows two curves which have equations y=2cos2x° and y=sin2x° in the range 0≤x<180.AByxO18OFind the coordinates of A and B, the points of intersection between the two curves.NR4 I have experience of cross topic exam standard questions.Trigonometry and integration1.(a)The expression 2sinx-5cosx can be written in the form ksinx+α where k>0 and 0≤α<2π.Calculate the values of k and α.(b)Hence find the value of p for whichp0(2sinx-5cosx)dx=3 .2.A curve for which dydx=5cos2x passes through the point π12, 1.Find y in terms of x.y=3cos2xy=-4cos2x+3xy3.Find the area enclosed by y=3cos2x and y=-4cos2x+3.Trigonometry and differentiation1.Two functions are defined as f(x)=7cosx and g(x)=2sinx.(a)Write f(x)+g(x) in the form kcosx-α where k>0 and 0≤α<π2.(b)Hence find f'(x)+g'(x) as a single trigonometric expression.2.A curve has equation y=7cosx-24sinx. (a)Write 7cosx-24sinx in the form ksinx-α where k>0 and 0≤α<π.(b)Hence find, in the interval 0≤x≤π, the x-coordinate of the point on the curve where the gradient is 1.3.Find the equation of the tangent to the curve y=3cosx-π6 at the point where x=π3Trigonometry, functions and graphs1.A function f is defined as f(x)=3cosx°+sinx°.(a)Express f(x) in the form kcosx-a° where k>0 and 0≤a<360.(b)Sketch the graph of y=f(x) between 0≤x<360, showing clearly the coordinates of the maximum and minimum turning points.2.(a)Express 3sinx°+4cosx° in the form ksinx+a° where k>0 and 0≤a<360.(b)Sketch the graph of y=3sinx°+4cosx°+1 between 0≤x<360, showing clearly the coordinates of the maximum and minimum turning points and where the curve cuts the axes.3.Functions a(x)=sinx, b(x)=cosx and c(x)=x-π4 are defined on a suitable set of real numbers.(a)Find expressions for;(i)a(c(x));(ii)b(c(x)).(b)(i)Show that a(c(x))=12sinx-12cosx.(ii)Find a similar expression for b(c(x)) and hence solve the equation a(c(x))+b(c(x))=1 for 0≤x≤2π.4.Functions f and g are defined on suitable domains by f(x)=sinx° and g(x)=2x.(a)Find expressions for;(i)f(g(x));(ii)g(f(x)).(b)Solve 3f(g(x))=g(f(x)) for 0≤x≤360.Trigonometry and straight linea°P(6, 5)Oyx1.P is the point (6, 5). The line OP is inclined at an angle of a° to the x-axis.(a)Find the exact values of sin2a° and cos2a°.O2a°Qyx(b)The line OQ is inclined at an angle of 2a° to the x-axis. Write down the exact value of the gradient of OQ.AnswersR11.(a)π6(b)π4(c)π3(d)π2(e)π(f)2π(g)2π3(h)3π4(i)5π6(j)7π6(k)5π4(l)4π3(m)3π2(n)5π3(o)7π4(p)11π6(q)3π(r)4π2.(a)0?646(b)2?48(c)3?94(d)4?90(e)5?36(f)7?913.(a)180(b)360(c)540(d)90(e)270(f)450(g)60(h)120(i)240(j)300(k)420(l)45(m)135(n)225(o)315(p)30(q)150(r)210(s)330(t)15(u)754.(a)57?3(b)115(c)172(d)229(e)80?2(f)155R21.(a)12(b)32(c)12(d)32(e)12(f)12(g)1(h)0(i)-1(j)-12(k)-12(l)-32(m)-32(n)-12(o)-122.(a)32(b)12(c)12(d)-12(e)-32(f)-12(g)0(h)-1(i)0(j)-32(k)-12(l)-12(m)12(n)32(o)123.(a)13(b)3(c)1(d)-3(e)-13(f)-1(g) Undefined(h)0(i) Undefined(j)13(k)1(l)3(m)-3(n)-13(o)-14.(a)12(b)12(c)32(d)0(e)0(f)-1(g)12(h)12(i)32(j)-12(k)-12(l)-32(m)-12(n)-12(o)-325.(a)32(b)12(c)12(d)-1(e)1(f)0(g)-32(h)-12(i)-12(j)-32(k)-12(l)-12(m)32(n)12(o)126.(a)13(b)1(c)3(d)0(e)0(f)Undefined(g) -13(h)-1(i)-3(j)13(k)1(l)3(m)-13(n)-1(o)-3R31.(a)y=sinx°(b)y=cosx°(c)y=tanx°(c)y=2sinx°2.(a)y=3cosx°(b)y=4sinx°(c)y=-sinx°(c)y=-cosx°3.(a)y=-2sinx°(b)y=sinx°+1(c)y=cosx°-3(c)y=cosx°+44.(b)y=sinx+30°(c)y=tanx-90°5.Sketches can be check by using a graph drawing packageR41.(a)y=-4cosx°(b)y=-sinx°+3(c)y=3cosx°-1(d)y=2sinx+45° or y=2cosx-45°(e)y=-tanx+30°(f)y=sinx-30°+2(g)y=12cos2x°+3?5(h)y=2sin3x°(i)y=-sin3x°(j)y=3cosx°-22.Sketches can be check by using a graph drawing packageR51.(a)32sinx+12cosx(b)12sinx°-32cosx°(c)12cosx+12sinx(d)12cosx°-12sinx°(e)12cosx-32sinx(f)12sinx°+32cosx°(g)-cosx°(h)-sinx(i)-cosx°2.(a)3-122(b)3+122(c)1-3223.(a)2425(b)725(c)1171254.(a)120169(b)119169(c)28560285615.(a)45(b)35(c)25256.(a)1213(b)513(c)-119169R61.29cosx+21?8°2.2sinx-π43.10sinx+71.6°4.5sinx-0?465.13cosx+56?3°6.5sinx-0?937.5sinx-243?4°8.2cosx+7π69.14cosx+307°R71.(a)x=58, 238(b)x=239, 301(c)x=139, 221(d)x=0?93, 4?1(e)x=3?4, 6?0(f)x=0?98, 5?32.(a)x=20?8, 110.8(b)x= 76?2, 103.8, 196.2, 223.8, 316.2, 343.8(c)x=65?9, 114.1, 245.9, 294.13.(a)x=41?6, 221.6(b)x=223, 297(c)x=112.6, 195.4(d)x=3?0, 6?2(e)x=0?62, 4?8(f)x=7π12, 13π12NR11.(a)-3365(b)7252.(a)Proof(b)Proof3.-20294.(a)1213(b)4613135.cosx=±37, sinx=±4076.(a)3-1227.(a)sina+b=45+615, cosa+b=35-8157.(b) 45+635-88.(a)Proof(b)8485(c)-8413NR21.(a)29cosx+21.8°(b)158?2, -292.(a)2sinx-π4(b)min -2 at x=7π4, max 2 at x=3π43.(a)13sinx+0.395(b) max 17 at x=1.184.min -5 at x=216?9, max 5 at x=36?95.min -13 at x=2?55, max 13 at x=5?69NR31.x=30, 90, 1502.x=0, 60, 180, 3003.x=1?23, 5?054.x=90, 199?5, 340?55.x=2?42, 3?866.x=π3, 2π37.θ=1?33, 4?478(a)5sinx+0?93(b)x=1?57, 69(a)34cosx-59°(b)x=12?3, 105?710.x=90, 27011.x=60, 131?912.x=0, 60, 30013.x=90, 210, 270, 33014.A45, 1 and B90, 0NR4Trigonometry and integration1(a)29sinx+5?09(b)p=-4?72.y=52sin2x-143.Area = 12.4 square unitsTrigonometry and differentiation1(a)11cosx-0?65(b)-11sinx-0?652(a).25sinx-3.43(b)x=1?893.y-332=-32x-π3Trigonometry and functions and graphs1(a)2cosx-30°(b)Sketches can be check by using a graph drawing package2(a).5sinx+53?1°(b)Sketches can be check by using a graph drawing package3(a).(i)sinx-π4(ii)cosx-π4(b)(i)Proof(ii)x= π4, 3π44(a).(i)sin2x(ii)2sinx(b)x=0, 70?5, 180, 289?5, 360Trigonometry and the straight line1(a)sin2a=6061, cos2a=1161(b)tan2a=6011 ................
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