Unit 2 Complete Practice.pdf

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MPM2DI& &Unit%2:%Trigonometry%I% %Lesson%2%

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Introduction*to*Trigonometry* A. Labeling*Right*Angle*Triangles& & &

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C. The&Primary&Trigonometric&Ratios! In!similar!right!triangles,!the!ratios!of!the!side!lengths!(opp,!adj,!and!hyp)!are!proportional.!These!ratios!occur!in! relation!to!an!angle!of!reference,! !(theta).!Note:!never!use!the!right!angle!as!the!reference!angle.! There+are+three+primary+trigonometric+ratios:! Sine! !=!

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MPM2DI" "Unit%2:%Trigonometry%I% %Lesson%3%

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Solving(for(a(Side(Length(in(a(Right(Angle(Triangle( We"can"solve"for"an"unknown"side"length"of"a"right"angle"triangle"by"setting"up"a"trigonometric"ratio"and" solving"using"crossXmultiplication."" " In%the%following%triangles,%solve%for%x%accurate%to%1%decimal%place.% " J" a)""""" " " " " " " b)" " " A" " x" " 9.1"cm" 7"cm" x" " " " 35°" 56°" C" " L" K" B" " " " " " c)" " " " " " " d)" " 12"m" M" N" D"" E" 53°" " " " x" " x" 29"m" " " 30°" " " " F" " " " e)""""" " " x" G"" " " " 9"mm" " " " 40°" " " " I"

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MPM2DI" "Unit%2:%Trigonometry%I% %Lesson%4%

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Solving(for(an(Unknown(Angle(in(a(Right(Angle(Triangle( A. Using(Inverse(Operations" Recall:%Inverse%operations%are%used%to%isolate%unknown%variables%in%an%equation.%Solve%each%of%the%following%using% inverse%operations:% % Equation:%

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B. Solving(for(an(Unknown(Angle(in(a(Right(Triangle" We"can"solve"for"an"unknown"angle"of"a"right"angle"triangle"by"setting"up"a"primary"trigonometric"ratio"and"using" inverse"operations!" " In%the%following%triangles,%solve%for%the$indicated$angle,$accurate%to%the%nearest%degree.% " a) """" " " " " " c)" " 7" " " " 7" 4" " 12" " " " " " " " " b)" " " " " " " d)" 8" " " " 21" " 24" 9" " " " " " " " " HW:((p.(496( (501((((#3(odds),(5(odds),(7ac,(8ef,(20ef,(21(odds),(22,(24((sketch$your$triangles$please!)

MPM2DI" "Unit%2:%Trigonometry%I% %Lesson%5%

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Solving(Right(Angle(Triangles( include"the"three"primary%trigonometric%ratios,"the"Pythagorean%Theorem,"and"Sum%of%Interior%Angles." " sin"A"=" "

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! b)!Given& GHI,&where& &H&=&90°,&side&HI&=&9.2¢imetres,&and&side&GI&=&11.3¢imetres.& & & & & & & & & & & & & & & & & & & & & & & Example&2:&Find&side&EF&to&the&nearest&tenth&of&a&metre.&& ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! HW:!!p.!500!!!!#26abdef!;!TRIGONOMETRY!WORKSHEET!II:!#1,!2,!3abc,!4abc

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MPM2DI& &Unit%2:%Trigonometry%II% %Lesson%6%

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Angles'of'Elevation'and'Depression' Two&angles&that&are&often&referenced&in&the&sighting&of&objects&(and&in&the&creation&of&trigonometry&questions!)& are&the&angle'of'elevation&and&the&angle'of'depression.& & The&angle&of&elevation&is&the&angle%between%the%horizontal% and%the%line+of+sight%up%to%the%object.&

The&angle&of&depression&is&the&angle%between%the%horizontal% and%the%line+of+sight%down%to%the%object.% % &

' ' Example+1:%%An%observer%on%the%top%of%a%radar%tower%measures%the%angle%of%depression%to%a%buoy%on%the%water%to% be%28°.%%If%the%observer%is%40%metres%above%the%water,%how%far,%to%the%nearest%metre,%is%the%buoy%from%the%base%of% the%tower?& & & & & & & & & & Example+2:%% head%is%70°.%%Find%the%height%of%the%dinosaur%to%the%nearest%metre.% & & & % % % % % % Example+3:%The%angle%of%depression%to%a%disabled%ship%from%an%approaching%helicopter%is%55°.%%If%the%helicopter%is% 500%metres%above%the%level%of%the%water,%what%is%the%horizontal%distance%from%the%helicopter%to%the%ship?% & & & & & & & & & ' HW:''TRIGONOMETRY'WORKSHEET'III:'#3,'5,'6,'9,'12'(record'your'eyeball'height'here:'_____________'cm)

MPM2DI& &Unit%2:%Trigonometry%II% %Lesson%7%

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The$Cosine$Law$ A. What$is$the$Cosine$Law?& The&Pythagorean&Theorem&only&applies&to&right&angle&triangles.&&& Extending&this&theorem&to&apply&to&any&type&of&triangle&(including&right& angle&triangles)&results&in&a&new&trigonometric&law:&the$Cosine$Law.&&& The&cosine&law&can&be&expressed&in&two&different&forms:& & % % % % % % % % % % % % % To&use&the&cosine&law,&we&will&have&three&pieces&of&information&about&the&triangle,&but&will&be&unable&to&set&up&the& Sine&Law&ratios.&Basically,%when&given%three(sides((SSS)%and&no&angles%or%two(sides(and(a(contained(angle((SAS),%we& will&use&the&cosine&law.& & i) Set&up&the&Cosine&Law&to&solve&for&side&f:& & & & & % ii) Set&up&the&Cosine&Law&to&solve&for&angle&D:& & & % % %

B. Using$the$Cosine$Law$Given$Two$Sides$and$the$Contained$Angle$(SAS)& Example%1:%%Given% & & & & & &

where% D%=%50°,%e%=%10,%f%=%8%m,%find%side%d,%accurate%to%one%decimal%place.&

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C. Using(the(Cosine(Law(Given(Three(Sides((SSS)! When(given(a(choice,(find(the(largest(angle(first((hint:(the(largest(angle(corresponds(to(the(largest(side)!!( Example(2:((Given( where((j(=(5(cm,(e(=(6(cm,(r(=(8(cm,(find( R,(accurate(to(the(nearest(degree.( ( ( ( ( ( ( ( ( ( ( (

( D. Solving(Triangles(Using(the(Cosine(Law( Example(3:( nearest(degree.( (

,(given( Q(=(130°,(r(=(8,(p(=(10;(solve(sides(accurate(to(1(decimal(place(and(angles(to(the(

( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( HW:((Complete(the(following.(Sketch(the(triangles(for(#6F8.(When(given(a(choice,(find(the(largest(angle(first!( 1)!Find!b,!in! ABC,!given! B!=!36°,!a!=!8,!c!=!10! ! 2)!Find!e,!in! DEF,!!given! E!=!75°,!d!=!15,!f!=!9! 3)!Find!h,!in! GHI,!!given! H!=!120°,!g!=!6.3,!i!=!4.7! ! 4)!Find! U,!in! STU,!given!s!=!4,!t!=!7,!u!=!8! 5)!Find! X,!in! XYZ!,!given!x!=!1.78,!y!=!11.3,!z!=!12.3! 6)!Solve! JKL,!given! K!=!136°,!j!=!10,!l!=!12! 7)!Solve! PQR!,!given!p!=!9,!q!=!8,!r!=!5! ! ! 8)!Solve! ABC,!given!a!=!8.8,!b!=!6.3,!c!=!8.4! % Answers:! 1)!b!=!5.9! 2)!e!=!15.4! 3)!h!=!9.6! 4)! U!=!89°! 5)! X!=!7°! 7)! P!=!84°,!! Q!=!62°,!! R!=!34°! 8)! A!=!72°,!!! B!=!43°,!!! C!=!65°! ! !

6)!k!=!20,!!! J!=!20°,!!! L!=!24°!

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MPM2DI& &Unit%2:%Trigonometry%II% %Lesson%8%

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The$Sine$Law$$ A. Proving$the$Sine$Law& i) For&ANY&triangle,&the&following&equivalent&ratios&can&be&observed:&& & % % % % % % % % & This&relationship&is&called&the$Sine$Law.&To&use&the&sine&law,&we&must&have&one&complete&ratio&(both&values&known)& and&one&ratio&with&one&value&missing.&Basically,%we&need%one$corresponding$side,angle$pair%(e.g.% A%and%side%a)%+$ one$other$piece$of$information%to&use&the&sine&law.% & ii) Prove&the&ratios&in&the&Sine&Law&are&equivalent&using&the&following&triangles:& & & & & & & & & & & & & & & & & & & &

B. Using$the$Sine$Law$Given$Two$Angles$and$One$Opposite$Side& Example%1:%% &

L%=%58°,% F%=%22°,%e%=%63.8%m,%find%side%f,%accurate%to%one%decimal%place.&

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C. Using(the(Sine(Law(Given(Two(Sides(and(One(Opposite(Angle! Example(2:((Given( ( ( ( ( ( ( ( ( ( ( ( ( ( (

where( J(=(42°,(j(=(10.2(cm,(e(=(8.5(cm,(find( E,(accurate(to(the(nearest(degree.(

( D. Solving(Triangles(Using(the(Sine(Law( ( Example(3:( G(=(27°,(p(=(17,(g(=(13,(for(all(unknown(values;(solve(sides(accurate(to(1(decimal( place(and(angles(to(the(nearest(degree.( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( HW:((p.(549( (550((((#1,(2,(3,(4,(5,(6,(8,(10,(11(

MPM2DI& &Unit%2:%Trigonometry%II% %Lesson%9%

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Sine%&%Cosine%Law%Applications% Recall:%For&non0right&triangles,%to&use&the&Sine&Law,&we&need%one(corresponding(side/angle(pair%(e.g.% A%and%side%a)%% +(one(other(piece(of(information.%Otherwise,&we&will&use&the&Cosine&Law!& % Example%1:%%A%greenhouse%is%10%metres%wide%and%the%rafters%make%angles%of%25°%and%60°%with%the%joists.%%Find%the% length%of%each%type%of%rafter,%accurate%to%one%decimal%place.& & & & & & & & & & & Example%2:%%From%a%point%A,%the%angle%of%elevation%to%the%top%of%a%transmission%tower%is%61°.%%At%a%point%B,%52% metres%closer%to%the%base%of%the%tower,%the%angle%of%elevation%to%the%top%is%71°.%%How%high%is%the%tower,%accurate%to% one%decimal%place?% % % % % % % % % % % % % % % Example%3:%%A%triangular%piece%of%land%is%bounded%by%32%metres%of%brick%wall,%50%metres%of%fencing,%and%28%metres%of% road%frontage.%%What%angle%does%the%fence%make%with%the%road?%

% % % % % % % % HW:%%p.%555% %556%%%%#1:5;%%%%%%p.%568%%%%#15:18%

MPM2DI& &Unit%2:%Trigonometry%II% %Lesson%10% &

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Solving(Problems(Using(Trigonometry( Based&on&the&type&of&triangle&involved&in&a&given&problem,&we&have&different&tools&that&are&available&for&us&to&use:& %

Right& Triangles&

NonFRight& Triangles&

% % Example(1:%%Fiona%is%installing%a%guy%wire%to%stabilize%a%tower%that%is%61.0%metres%tall.%After%attaching%the%wire%to%the% top%of%the%tower,%Fiona%is%lowered%in%the%basket%of%the%arm%on%the%repair%truck.%Part%way%down,%the%arm%malfunctions% and%she%is%stuck.%Looking%up%at%the%top%of%the%tower,%the%angle%of%elevation%is%42°;%looking%down%at%the%bottom%of%the% tower,%the%angle%of%depression%is%32°.%Her%horizontal%sightline%%is%40.0%metres%away%from%the%tower.% & a) How&much&wire&has&been&let&out?& & & b)&How&much&more&wire&is&needed&to&reach&the&ground?& % %

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! Example!2:!!The!Queen!Charlotte!Islands!in!BC!form!a!fairly!triangular!region!in!the!Pacific!Ocean.!Two!sides!are! about!256!km!and!288!km!and!form!an!angle!of!19°.!! ! a) What!is!the!length!of!the!third!side?!! ! ! ! ! ! ! ! ! ! ! ! ! ! b) What!is!the!approximate!area!bounded!by!the!Queen!Charlotte!Islands?!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! HW:!!p.!574! !576!!#1,!3(assume!a!right!triangle),!4=6,!9,!15,!18;!!p.!588!#6!