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etryons oitacilAppnomf TrigoWhat do you remember about trig?brainstormBoard of Studies1

etrysocationApplileanomogirTfCan you remember your trig ratios?SOHCAHTOASome Old Hags Can't Always Hide Their Old Ageevbrurnda2

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soAtionpplicanof Trigometryrub and reveal4

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Angles of Depression and ElevationobserverWhen we look down at an object, the angle between theobserver's eye and the top of the object is called the angle ofdepression, shown as in the diagram.objectWhen we look up at an object, the angle between the observer'seye and the top of the object is called the angle of elevation,shown as in the diagram.objectobserver8

metryonoTrigIn senior Mathematics we also define Trigonometricns oftiocaAppliratios using the 'unit circle' (a circle with radius 1)1P(x,y)x2 y2 1yθ-1x-1From this diagram:1cos θ x x coordinate of P1sin θ y y coordinate of P1rub and revealgeogebra activity9

etrynomf Trigons ocatioilppAThere are four other trigonometric ratioswhich we define with respect to sin θ and cos θ(tangent)tan θ y sin θ cos θ 0x cos θ(cotangent) cot θ x cos θ sin θ 0y sin θ(secant)cos θ 0sec θ 1 1x cos θ(cosecant)cosec θ 1 1 sin θ 0y sin θWhy are there restrictions on each of these?question:10

etryonsitacilAppThe Fundamental Identity!mgonoirTfoPreviously we saw that in the unit circle x2 y2 1We also saw that by definition x cos θ and y sin θCombining these we get:cos2 θ sin2 θ 1Which can then be rearranged to obtain:and1 - sin2 θ 11 - cos2θ 1If you take the expression markedand divide each term by cos2θ you get:1 tan2 θ sec2 θOr if you divide each term inby sin2 θ you get:cot2 θ 1 cosec2 θManipulating expressions using these identities can be very challengingbut also very satisfying when you get them correct!11

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tionsacilpApx400metrygonoof TriConsider the diagrams below. How are the statements linked?sin 400 zcos 500 y500And in a general form:0Ɵzx00sin(Ɵ ) cos(90 - Ɵ0) AND cos(Ɵ ) sin(90 - Ɵ0)0(90 Ɵ)yThese results all apply in quadrant 1rub and reveal15

Questions:metryonoUse the new identities to complete these relationships:f Trigns ocatioilppAsin (500) cos (900 - 500) cos (400)sin (250)cos (100) sin (900 - 100) sin (800)cos (300) sin (900 - 300) sin (600) cos (900 - 250) cos (650)rub and reveal16

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catioilppAnsmgonoirTfoetrySpecial TrianglesThere are two special right-angled triangles which you must be able to remember orreproduce.The ratios must ALWAYS given as exact values rather than decimals.450300145060011Work out what the other measurements must be in these triangles. It may be usefulfor when you have to reproduce them.Use the triangles to fill in this table:ɵ 300 ɵ 450 ɵ 600sin ɵcos ɵtan ɵ20

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tryonomenogirs of TcatiAppliWe now extend these relationships toquadrants other than the first one:Drawing the graphs of the trigonometric ratios can help us to see the change insigns as angles increase.We divide the domain 00 to 3600 into 4 quadrants:geogebra activityangles of any magnitudeNOTE: angles can be expressed as either DEGREES (what you are used to) orin RADIANS. Basically π 1800, so π 900, π 450 etc.24π 1 radian23

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onscatiApplimetrygonoof TriGraphingTrigonometric functions can be graphed. The three basic trig ratios produce graphs which arecyclic (repeat in cycles). Using intervals of 100, complete the tables of values for y cos (x),y sin (x) and y tan(x), (decimals are ok for this) then use the pen tool to sketch the graphs.tangeogebra activityapplet29

ryometnogirs of TWhat about the inverse functions?cationAppliy cosec xy sec xy cot xgraph by yourself, CHECK by pulling tab30

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etryns onomf TrigocatioilppABearingsA directional compass is shown below. It is used to find a direction or bearing .The four main directions of a compass are known as cardinal points. They arenorth (N), east (E), south (S) and west (W).Sometimes, the half-cardinal points of north-east (NE), north-west (NW),south-east (SE) and south-west (SW) are shown on the compass. A compassshows degree measurements from 0 to 360 .link tobearingsactivityWhen using a directional compass, hold the compass so that the point markednorth points directly away from you. Note that the magnetic needle alwayspoints to the north.32

etrynomf Trigons ocatioilppAThe true bearing to a point is the angle measured in degrees in a clockwisedirection from the north line. We will refer to the true bearing simply as thebearing.For example, the bearing of point P is 065º which is the number of degrees inthe angle measured in a clockwise direction from the north line to the linejoining the centre of the compass at O with the point P (i.e. OP).The bearing of point Q is 300º which is the number of degrees in the anglemeasured in a clockwise direction from the north line to the line joining thecentre of the compass at O with the point Q (i.e. OQ).33

etrynomf Trigons ocatioilppAThe conventional bearing of a point is stated as the number of degrees east orwest of the north-south line. We will refer to the conventional bearing simply asthe direction.To state the direction of a point, write: N or S which is determined by the angle being measured the angle between the north or south line and the point, measured in degrees E or W which is determined by the location of the point relative to the northsouth lineE.g. In the diagram, the direction of: A from O is N30ºE. B from O is N60ºW. C from O is S70ºE. D from O is S80ºW.rub and reveal34

etrynomf Trigons ocatioilppAState the bearing of the point P in each of the following diagrams:N 480 ES 600 WS 400 EN 700 W35

etrynomf Trigons ocatioilppAA bearing is used to represent the direction of one point relative to anotherpoint.For example, the bearing of A from B is 065º. The bearing of B from A is245º.bearings activity36

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trytionscaAppliomenogiof TrSine RuleAcsin A sin B sin CcbabhBDsine rule appletCIn triangle ABC, draw the perpendicular AD and call it hFrom triangle ABD, sin B h so h c sin BcFrom triangle ACD, sin C hbso h b sin CCombining (1) and (2), c sin B b sin Csin B sin CcbSimilarly, drawing a perpendicular from C it can be shown thatsin A sin Bba41

tryonscatiAppliomenogirof Trub and revealWorksheet42

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etrycatioilppAns onomf TrigoThe sine rule can also be rearranged to give:a b csin Csin Bsin A44

metrygonoof TritionsacilpApUse the previous example to put these steps in the correct order:sin θ44x 44sin θ sin 31065x 44 44 x sin 31065sin θ 0.3486θ 20.404.θ 20.4 to 2 sig fig45

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tionsacilpApetrymongoof TriCbApxa2 b2 c2 - 2bc cos ADraw the perpendicular CD, call it pLet AD xThen DB c - xac-xDcCosine RuleBFromFromACD, b2 p2 x2BCD, a2 p2 (c - x)2 p2 c2 - 2cx x2 p2 x2 c2 - 2cx b2 c2 - 2cxFromACD cos A xbx b cos Aso a2 b2 c2 - 2bc cos A52

etryns onomf TrigocatioilppAUse the cosine rule to find the side marked x:The cosine rule connects a side with the angle in the triangle opposite it.So in the rule, c2 a2 b2 - 2ab cos Cside you arelooking forangle opposite theside you wantLabel your diagram using a,b and c to avoid confusionSubstitute into the formula and evaluate.c2 a2 b2 - 2ab cos Cx2 92 122 - 2x9x12 cos 720cosine rule appletx2 81 144 - 216 x cos 720x2 158.25x 12.58 (to 2 decimal places)53

etrynomf Trigons ocatioilppAThe cosine rule can also be written another way, mainly for when finding themissing angle. See if you can place the steps in order!c2 a2 b2 - 2ab cos Cc2 - a2 - b2 - 2ab cos Cc2 - (a2 b2) - 2ab cos Cc2 - a2 - b2 cos C- 2ab- c2 a2 b2 cos C2aba2 b2 - c2 cos C2ab54

tryonomenogirs of TcatiAppliFind the size of θ in the diagram:Label the diagram carefully.You are looking for angle 'C'a2 b2 - c2 cos C2abSubstitute in the values:122 102 - 92 cos C2 x 12 x 10Worksheet144 100 - 81 cos C240163 cos C240Use your calculator (press 'shift' key first!)C 47.22.So θ 470 nearest degree55

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nscatioilppAcAmetryonogof TriBDbAs you already (should!) know,Area of Δ ABC ½ x base x height ½xbxhahThe area of a triangleCIn Δ BCD sin C hah a sin CSubstitute this value for hin the first formula:Area of Δ ABC ½ x b x a sin C ½ab sinC64

etryons oitacilAppnomf Trigorub and revealWorksheet65

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etrynomf Trigons ocatioilppAHow do you know which rule to use?watch a movieyoutube song68

metryonoTrigWorking in three dimensions - extensionns oftiocaApplirub and reveal69

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AttachmentsCosine Rule.ggbAngles of any magnitude.ggb

Applications of Trigonometry Special Triangles There are two special right-angled triangles which you must be able to remember or reproduce. The ratios must ALWAYS given as exact values rather than decimals. 45 0 45 0 30 0 60 0 1 1 Work out what the other measurements must be in these trian