Measurement!(MM5)! Applications!of!! Trigonometry! · 2!...

1! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(

!Measurement!(MM5)!

!Applications!of!!Trigonometry!

!!(((((((((((((((((((((((((((

Name(…………………………………………………………………..((((((

((

G.Georgiou(


Formula!

• Solve!Problems!using!Trigonometric!Ratios!in!One!or!More!Right@Angled!Triangles!

• Use!Appropriate!Trigonometric!Ratios!and!Formulae!in!Two@Triangle!Problems!where!One!Triangle!is!Right@Angled!and!the!Diagram!is!Given!

• Draw!Diagrams!to!Represent!Information!about!a!Given!Right@Angled!Triangle!(From(the(Preliminary(course,(you(would(remember(we(developed(trigonometric(ratios(for(rightJangled(triangles.((((((((((((((Calculate(the(value(of(θ(correct(to(the(nearest(minute.((

……………………………..................………………………………………(( ( ( (

……………………………..................………………………………………((

……………………………..................………………………………………(((((((((((((((……………………………..................………………………………………(((((Calculate(the(value(of(b(to(1(decimal(place.(

(……………………………..................………………………………………(

( (……………………………..................………………………………………(

(……………………………..................………………………………………

( (( ((( ……………………………..................………………………………………(

(((((

sin(θ(=( ( ( cos(θ(= ( ( tan(θ(= (

(This!formula!IS!given!on!the!HSC!formula!sheet!

(Example!1!

(Example!2!

(Example!3!


((( ( ( Calculate(the(value(of(x(correct(to(3(significant(figures.(

(……………………………..................………………………………………(

( ( ( (……………………………..................………………………………………(

(……………………………..................………………………………………(

(( ((( ……………………………..................………………………………………(

(((((a) Find(the(size(of ADB∠ correct(to(the(nearest(degree.((……………………………………………………..................((.…………………………………………………….................((…………………………………………………….................((.…………………………………………………….................(((b) Calculate(the(size(of CAB∠ .((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((c) Calculate(the(length(of(DC.(Answer(correct(to(the(nearest(metre.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………(!

(Example!4!

(Example!3!


!!Complete(this(statement(for(the((triangle(to(the(left.((

tan(_______(=( !!(

!!

!Find(the(value(of(x%correct(to(2(decimal(places.((( ( ( ( ( ( ( ( ( ( (( ( ( ( ( ………………………………………………............................(( ( ( ( (( ( ( ( ( ………………………………………………............................((

………………………………………………............................( ( ( ( (

( ( ( ( ( ………………………………………………............................(( ( ( ( ( (

………………………………………………............................(( ( ( ( (( ( ( ( ( ………………………………………………............................((

………………………………………………............................( ( ( ( (

( ( ( (((( (

(Example!6!

!H.S.C.!Question!(7)!

(Example!5!

x(cm(

(10(cm(

C%

45°(

10°(

D%

B% A%

Question 24 (continued)

(c) The marks in a class test are normally distributed. The mean is 100 and the standard deviation is 10.

(i) Jason’s mark is 115. What is his z-score?

(ii) Mary has a z-score of 0. What mark did she achieve in the test?

(iii) What percentage of marks lie between 80 and 110?

You may assume the following:

• 68% of marks have z-scores between −1 and 1 • 95% of marks have z-scores between −2 and 2 • 99.7% of marks have z-scores between −3 and 3.

(d) The base of a lighthouse, D, is at the top of a cliff 168 metres above sea level. The angle of depression from D to a boat at C is 28°. The boat heads towards the base of the cliff, A, and stops at B. The distance AB is 126 metres.

1

1

2

(i) What is the angle of depression from D to B, correct to the nearest minute?

(ii) How far did the boat travel from C to B, correct to the nearest metre?

End of Question 24

3

2

– 17 –

A B C

168 m168 m

126 m126 m

28°

D NOT TOSCALE


……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((! !

Activity! Ex(5.01(Q1,(2,(3,(5,(8,(11,(13,(14,(15,(16,(18!(


• Using!Compass!Bearings!(Eight!Points!Only)!and!True!Bearings!(Three@Figure!Bearings)!in!Problem@Solving!related!to!Maps!and!Charts!

(True%Bearings%Bearings(are(angles(that(are(taken(by(facing(north(and(turning(in(a(clockwise(direction(until(you(face(your(destination.(These(bearings(are(called(three@figure!bearings.!An(example(of(a(threeJfigure(bearing(has(been(included(in(the(figure(below.(((

(((((((

(In(this(diagram,(the(bearing(of(Wagga(Wagga(from(Sydney(is(249°.(I.e.(Start(at(Sydney,(face(north,(turn(249°(in(a(clockwise(direction(until(you(face(your(destination,(Wagga(Wagga.((( (

(((Calculate(the(bearing(of(each(of(the(letters(from(O.(((a)( ( ( ( ( ( (b)(((( ( ( ( ( ((((((((..........................................................( ( ..........................................................(((c)(( ( ( ( ( !!!!!!!!!!!(d)(( ( ( (((((((((..........................................................( ( ..........................................................(

(Example!8!


Recall!Corresponding,!Alternate!and!Co@Interior!Angles!(

Calculate(the(value(of(the(following.(Justify(your(answer(by(determining(whether(it(is(a(corresponding(angle,(alternate(angle,(or(coJinterior(angle.((!(a)! ! ! ! ! ! (b)!!!!!! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !!..........................................................( ( ..........................................................(!..........................................................( ( ..........................................................(!!(c)!! ! ! ! ! ! (d)!! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! !! !! ! ! ! ! ! ! ! ! ! !! ! !..........................................................( ( ..........................................................(! !..........................................................( ( ..........................................................(! !!(e)!! ! ! ! ! ! (f)!! ! ! ! ! ! ! ! !((((((((..........................................................( ( ..........................................................((..........................................................( ( ..........................................................(


Literacy!The!words!“of”!and!“from”!(

In(bearings,(an(understanding(of(the(words(“of”(and(“from”(is(essential.(((When(we(say:(“calculate(the(bearing(of!Perth(from(Sydney”,(this(actually(means:((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((When(we(say:(“calculate(the(bearing(from(Melbourne(to(Darwin”,(this(means:((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((( ((Approximate(the(true(bearing:(((a)!Of(Perth(from(Sydney(( …….………...........................…………………………………………….(

((b)!Of(Sydney(from(Perth(( …….………...........................…………………………………………….(((c)!Of(Brisbane(from(Darwin(( …….………...........................…………………………………………….(((d)!From(Darwin(to(Brisbane( (…….………...........................……………………………………………(

(((((((((((((((

(Example!9!


(Calculate(the(bearing:(((a)!Of(C(from(A( ( .………………………………………(((b)!From(C(to(B( ( .………………………………………(((c)!Of(A(from(B( ( .………………………………………(((d)!From(A(to(B( ( .………………………………………(((((By(first(finding(the(value(of(θ,(find(the(bearing(from(B(to%A.(((((((((((((..........................................( ..........................................( ..........................................((..........................................( ..........................................( ..........................................((......................... .................( ..........................................( ..........................................((((By(finding(the(value(of(α(and(β,(determine(the:(

((a)!Bearing(of(A(to(B.((………………………………………………….......................((………………………………………………….......................((………………………………………………….......................((………………………………………………….......................((

(Example!10!

(Example!11!

(Example!12!


((b)!Bearing(of(B(to(A.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((Compass%Bearings%This(links(the(concept(of(true(bearings(with(the(8(main(points(of(a(compass,(shown(below.(You(only(need(to(know(the(bearings(for(these(8(points.(((((((((((((((((((A(ship((A)(is(10(kilometres(due(east(of(a(lighthouse.(A(second(ship((B)(bears(SE(of(the(lighthouse(and(is(due(south(of(the(first(ship.(Calculate(the(distance(of(the(second(ship(from(the(lighthouse,(correct(to(1(decimal(place.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (

(Example!13!


((A(ship(sails(on(a(bearing(of(130°(for(a(distance(of(10(kilometres.(Calculate(how(far(south(of(its(starting(point(the(ship(is,(correct(to(2(decimal(places.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((( ((On(a(hike(Lisa(walked(south(for(3.5(km(and(then(turned(west(for(1.2(km.(Calculate(Lisa’s(bearing(from(her(starting(point.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((( (

(Example!14!

(Example!15!


((Texas(in(Queensland(is(160(km(north(east(of(Moree(and(Bendemeer(is(210(km(south(east(of(Moree.((((a)!How(far(is(Bendemeer(from(Texas?(

(……………………………...............……………(

(……………………………...............……………(

(…………………………...............…………….(

(……………………………...............……………(

(……………………………...............……………(

((

(b)!Find(the(size(of MTB∠ .((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((c)!What(is(the(bearing(of(Bendemeer(from(Texas?((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((!!!!

!! !

(Example!16!

Activity! Ex(5.02(Q(2J11,(13,(14!(


• Determining!the!Sign!for!the!Ratios!of!Obtuse!Angles!((((((((((((((((((((((((((((((((7.!Complete(the(following(table(relating(to(the(sign(of(obtuse(angles.((

! Acute! Obtuse!

Sin! ( (Cos! ( (Tan! ( (

(((

θ(is(an(obtuse(angle(and(sin(θ(=(0.8183.(What(is(the(size(of(θ(to(the(nearest(degree?((……………………………………………………………………………………………………………………………………((……………………………………………………………………………………………………………………………………((

(Example!17!


• Using!the!Sine!Rule!to!Find!Lengths!and!Angles((So(far(we(have(only(used(Trigonometry(in(rightJangled(triangles.(Now(we(will(look(at(ways(of(solving(problems(with(non(rightJangled(triangles.((Before(that(though,(we(need(to(make(sure(we(can(correctly(label(sides(and(angles.((((((((((((((((((%%Sine%Rule%3%Lengths%((((Calculate(the(value(of(x(to(1(decimal(place.(((……………………………………………….……………((……………………………………………….……………((……………………………………………….……………((……………………………………………….……………((……………………………………………….……………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((

Remember!Vertices/angles(are(capital(letters(and(the(sides(opposite(those(angles(

are(not(in(capitals.(

Formula!

(

(((

This!formula!IS!given!on!the!HSC!formula!sheet!(

(Example!18!

Side(length(

Angle(opposite(side(length(


(( (Calculate(the(value(of(m(correct(to(3(significant(figures.((…………………………………………………………………………….((…………………………………………………………………………….((…………………………………………………………………………….((…………………………………………………………………………….((…………………………………………………………………………….((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((!

!!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((

(Example!19!

(Example!20!


……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((

((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((((((

(Example!21!

Activity! Ex(5.04(Q(1((a,b,c)(2,(3,(4,(5,(7,(8,(9,(10!(


Sine%Rule%–%Angles%%(( ((((Calculate(the(value(of(θ(correct(to(the(nearest(degree.(((………………………………………………………….……((………………………………………………………….……((………………………………………………………….……((………………………………………………………….……((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((Calculate(the(value(of(θ(correct(to(the(nearest(minute.((………………………………………………………….……((………………………………………………………….……((………………………………………………………….……((………………………………………………………….……((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((((((

(Example!22!

(Example!23!


Miscellaneous%Sine%Rule%Examples%((

((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((((((

(Example!24!


(

((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((

(Example!25!

Activity! Ex(5.05(Q(1(a,b,c),(4,(5,(6,(9,(10!;;;;;(

Mel Boggs

Text


• Using!the!Cosine!Rule!to!Find!Lengths!and!Angles!((Cosine%Rule%–%Lengths%(((((( (((((((((Calculate(the(value(of(x(to(3(significant(figures.(((…………………………………………………………((…………………………………………………….…((…………………………………………………….…((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((( (((Calculate(the(value(of(y(to(2(significant(figures.(((……………………………………………………………….((……………………………………………………………….((……………………………………………………………….((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((

Formula!

((

((


Angle(opposite(side(you(are(

finding(

(Example!26!

(Example!27!


(((

In(Δ PQR,(PQ(=(234(mm,(QR(=(981(mm(and(∠ PQR(=(128°.(Find(the(length(of(side(PR,(correct(to(3(significant(figures.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!!!(((( (

(Example!28!

Activity! Ex(5.06(Q(1(a,b,c),(2,(3,(4,(7,(8,(9!(


Cosine%Rule%–%Angles%%((((!((((((((((Calculate(the(size(of(angle(B(to(the(nearest(minute.(((………………………………………….…………………((………………………………………….…………………((………………………………………….…………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((Calculate(the(size(of(angle(T%correct(to(1(decimal(place.((((…………………………………………………………………………((…………………………………………………………………………((…………………………………………………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((

Formula!

(

(

!!


Side(opposite(angle(you(are(trying(to(find(

(Example!29!

(Example!30!


(( (A(soccer(goal(is(8(m(wide.(((a)(A(player(is(directly(in(front(of(the(goal(such(that(he(is(12(m(from(each(post.(Within(what(angle(must(he(kick(the(ball(to(score(a(goal?((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((b)(A(second(player(takes(an(angled(shot.(This(player(is(12(m(from(the(nearest(post(and(17(m(from(the(far(post.(Within(what(angle(must(this(player(kick(to(score(a(goal?((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((!!!! !

(Example!31!

Activity! Ex(5.07(Q(1(a,b),(2,(4,(5,(7,(8,(10!(


• Calculating!the!Area!of!a!Triangle!(

Up(to(this(point,(we(have(used( !"!" (to(find(the(area(of(a(triangle.(But(what(if(we(

don’t(have(the(base(or(perpendicular(height?(((The(formula(below(allows(us(to(find(the(area(of(a(triangle(given(any(two(sides(and(the(angle(in(between(them.((((((!!!! !!!!!Calculate(the(area(of(the(triangle(below.(((………………………………………………………………………..((………………………………………………………………………..((………………………………………………………………………..((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((( ((Calculate(the(area(of(the(triangle(below.((……………………………………………………..……………..((……………………………………………………..……………..((……………………………………………………..……………..((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( ((

Formulae!

(

!This!formula!IS!given!on!the!HSC!formula!sheet!

(

(Example!32!

(Example!33!

Angle(between(the(

2(sides(


( ((

((( (The(plan(for(a(new(subdivision(showed(two(new(corner(blocks.(!(a)(Calculate(the(area(of(triangle(A.((………………………………….……………...((………………………………….……………...((………………………………….……………...((………………………………….……………...((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( ((b)(What(is(the(area(of(block(B?((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (!!! !

(Example!34!

(Example!35!

Activity! Ex(5.08(Q(2J10!(


• Solving!Problems!involving!Non!Right@Angled!Triangles!• Selecting!and!Using!Appropriate!Trigonometric!Ratios!and!Formulae!to!Solve!

Problems!(The(main(problem(with(the(questions(in(any(exam(is(that(you(need(to(be(able(to(select(and(apply(the(correct(formula(to(use.(In(many(instances,(the(question(will(not(indicate(which(formula(to(use.(((Here(is(an(overview:((Trigonometric!Ratios!These(can(be(used(to(find(a(missing(side(or(angle(in(a(rightJangled(triangle.((

sin(θ(=( !""!#$%&'("!%&)*#&

( ( cos(θ(= !"#!$%&'()*+'%&,-%

( ( tan(θ(= !""!#$%&'()*(+&,%

(

(!The!Sine!Rule!The(Sine(rule(can(be(used(in(any(triangle.(There(are(two(types(of(applications(of(the(sine(rule.(

1.(To!find!a!side:!you(need(the(angle(opposite(the(unknown(side(and(another(side(and(angle(opposite(each(other.(2.!To!find!an!angle:!you(need(the(side(opposite(the(unknown(angle(and(another(side(and(angle(opposite(each(other.(

!!

!"#"=

#!"#$

=%

!"#&(

((The!Cosine!Rule!The(Cosine(rule(can(be(used(in(any(triangle.(There(are(two(versions(of(this(rule.((

1.(To!find!a!side:!you(need(the(other(two(sides(and(the(angle(between(those(two(sides.(

!! ="! +#! !!"#"$%&$ (((

2.(To!find!an!angle:!you(need(all(three(sides.((

!"#! = "$ +#$ !$$

$"#(

((You(need(to(be(able(to(apply(bearings(to(all(of(the(above(three(instances,(and(your(familiarisation(with(bearings(is(essential(if(you(are(to(succeed(in(your(HSC(course.

Make(sure(the(side(length(that(goes(here(is(opposite(the(angle(you(are(trying(to(find(


Miscellaneous!Trigonometry!Questions!!!a)(

(!!!!!!!!!!!!!!

!b)((

!

!c)!

( !

!d)!

( !

!e)!

( (((((((((((((((

((

!f)!

( (

!g)!

( (

!h)((

(

!

NEW CENTURY MATHS 12 GENERAL – HSC COURSE: TEACHER RESOURCE PACK© Nelson Thomson Learning 2002. This page may be photocopied for educational use within the purchasing institution

42

Trigonometry review5.2

Find the missing side (to 2 decimal places) or angle (to the nearest degree).

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 22. 23. 24.

x

16°

24 cm

h

57°

7 cm

θ

5 cm

16 cm

12 cm θ

θ

16 cm14 cm

8 cm

11 cm

7 cm

5 cm

8 cm

θ

d

65°

33°

p

r20°

3 cm

3 cm

12 cm

7 cmθ

13 cm

θ 5 cm

y

55°18 cm

a

60°

10 cmθ

4 cm

θ 9 cmf

21°

4 cm

w71°

5 cm

θ

11 c

m

θ

5 cm

c29° 15 cm

θ

13 cm

θ

16 cm

u

45°

k40°

2 cm

15 cm

1 cm

3 cm

8 cm

16 cm

9 cm

Mixed answers:14°16°1610.7848°23°72°13.16939°30°9.0241°

12.29 5.2037°6.1335°9.973.2354°6.626°7.716.1079°


44

Sine and cosine rules5.4


Note:

θ

is obtuse in some diagrams.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

x

50°

u

92°

9 cm

12 cm

9 cm

θ

5 cm

10 cm4 cm

7 cm

5 cmθ

θ 15° θ

h52°

8 cm

5 cm

θ7 cm

b

126°

4 cm

θ

10 cm

θ

12 cm

7 cm4 cm

20°

9.4 cm

7 cm 11 c

m

47°

10 cm25°

h

100°

5 cm

6 cm

3 cm

14 cm108°

5 cm

54°d

15 cm

9 cm 110°

42°e θ

5 cm

5 cm

13 c

m

θ

13 cm

20°

50°

44°x

9 cm

4 cm

4 cm

6 cm

8 cm

33°

28° m

19. 20.5 cm

70° f

60°

θ

5 cm

3 cm

3 cm

Mixed answers:10.3758°35.1563°33°6.598.43115°19.27117°4.5490°54°124°60°8.2966°14.286.0570°9.83


42



1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 22. 23. 24.

x

16°

24 cm

h

57°

7 cm

θ

5 cm

16 cm

12 cm θ

θ

16 cm14 cm

8 cm

11 cm

7 cm

5 cm

8 cm

θ

d

65°

33°

p

r20°

3 cm

3 cm

12 cm

7 cmθ

13 cm

θ 5 cm

y

55°18 cm

a

60°

10 cmθ

4 cm

θ 9 cmf

21°

4 cm

w71°

5 cm

θ

11 c

m

θ

5 cm

c29° 15 cm

θ

13 cm

θ

16 cm

u

45°

k40°

2 cm

15 cm

1 cm

3 cm

8 cm

16 cm

9 cm

Mixed answers:14°16°1610.7848°23°72°13.16939°30°9.0241°12.29 5.2037°6.1335°9.973.2354°6.626°7.716.1079°


44



Note:

θ


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

x

50°

u

92°

9 cm

12 cm

9 cm

θ

5 cm

10 cm4 cm

7 cm

5 cmθ

θ 15° θ

h52°

8 cm

5 cm

θ7 cm

b

126°

4 cm

θ

10 cm

θ

12 cm

7 cm4 cm

20°

9.4 cm

7 cm 11 c

m

47°

10 cm25°

h

100°

5 cm

6 cm

3 cm

14 cm108°

5 cm

54°d

15 cm

9 cm 110°

42°e θ

5 cm

5 cm

13 c

m

θ

13 cm

20°

50°

44°x

9 cm

4 cm

4 cm

6 cm

8 cm

33°

28° m

19. 20.5 cm

70° f

60°

θ

5 cm

3 cm

3 cm

Mixed answers:10.3758°35.1563°33°6.598.43115°19.27117°4.5490°54°124°60°8.2966°14.286.0570°9.83


44



Note:

θ


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

x

50°

u

92°

9 cm

12 cm

9 cm

θ

5 cm

10 cm4 cm

7 cm

5 cmθ

θ 15° θ

h52°

8 cm

5 cm

θ7 cm

b

126°

4 cm

θ

10 cm

θ

12 cm

7 cm4 cm

20°

9.4 cm

7 cm 11 c

m

47°

10 cm25°

h

100°

5 cm

6 cm

3 cm

14 cm108°

5 cm

54°d

15 cm

9 cm 110°

42°e θ

5 cm

5 cm

13 c

m

θ

13 cm

20°

50°

44°x

9 cm

4 cm

4 cm

6 cm

8 cm

33°

28° m

19. 20.5 cm

70° f

60°

θ

5 cm

3 cm

3 cm

Mixed answers:10.3758°35.1563°33°6.598.43115°19.27117°4.5490°54°124°60°8.2966°14.286.0570°9.83


44



Note:

θ


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

x

50°

u

92°

9 cm

12 cm

9 cm

θ

5 cm

10 cm4 cm

7 cm

5 cmθ

θ 15° θ

h52°

8 cm

5 cm

θ7 cm

b

126°

4 cm

θ

10 cm

θ

12 cm

7 cm4 cm

20°

9.4 cm

7 cm 11 c

m

47°

10 cm25°

h

100°

5 cm

6 cm

3 cm

14 cm108°

5 cm

54°d

15 cm

9 cm 110°

42°e θ

5 cm

5 cm

13 c

m

θ

13 cm

20°

50°

44°x

9 cm

4 cm

4 cm

6 cm

8 cm

33°

28° m

19. 20.5 cm

70° f

60°

θ

5 cm

3 cm

3 cm

Mixed answers:10.3758°35.1563°33°6.598.43115°19.27117°4.5490°54°124°60°8.2966°14.286.0570°9.83


42



1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 22. 23. 24.

x

16°

24 cm

h

57°

7 cm

θ

5 cm

16 cm

12 cm θ

θ

16 cm14 cm

8 cm

11 cm

7 cm

5 cm

8 cm

θ

d

65°

33°

p

r20°

3 cm

3 cm

12 cm

7 cmθ

13 cm

θ 5 cm

y

55°18 cm

a

60°

10 cmθ

4 cm

θ 9 cmf

21°

4 cm

w71°

5 cm

θ

11 c

m

θ

5 cm

c29° 15 cm

θ

13 cm

θ

16 cm

u

45°

k40°

2 cm

15 cm

1 cm

3 cm

8 cm

16 cm

9 cm

Mixed answers:14°16°1610.7848°23°72°13.16939°30°9.0241°12.29 5.2037°6.1335°9.973.2354°6.626°7.716.1079°


44



Note:

θ


1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

x

50°

u

92°

9 cm

12 cm

9 cm

θ

5 cm

10 cm4 cm

7 cm

5 cmθ

θ 15° θ

h52°

8 cm

5 cm

θ7 cm

b

126°

4 cm

θ

10 cm

θ

12 cm

7 cm4 cm

20°

9.4 cm

7 cm 11 c

m

47°

10 cm25°

h

100°

5 cm

6 cm

3 cm

14 cm108°

5 cm

54°d

15 cm

9 cm 110°

42°e θ

5 cm

5 cm

13 c

m

θ

13 cm

20°

50°

44°x

9 cm

4 cm

4 cm

6 cm

8 cm

33°

28° m

19. 20.5 cm

70° f

60°

θ

5 cm

3 cm

3 cm

Mixed answers:10.3758°35.1563°33°6.598.43115°19.27117°4.5490°54°124°60°8.2966°14.286.0570°9.83


((

((((

((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………(

(Example!36!



((

((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((



((

(……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((

(……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((



((

((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((



(

(

((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((( (((

1

2

1

!H.S.C.!Question!(41)!!

Activity! Ex(5.09(Q(1((a,b,c),(2((a,b,c)(,(4,(6,(7,(8,(11,(12!(


• Conducting!Radial!(both!Plane!Table!and!Compass)!Surveys!and!Solving!Problems!involving!Non!Right@Angled!Trigonometry,!Pythagoras’!Theorem!and!Area!in!Offset!and!Radial!Surveys!

((Offset%Surveys%This(was(covered(in(the(Preliminary(course.(Here(is(a(recap.((((((((((((((((((((( (((a)!Write(a(field(book(entry(for(the(above(shape(of(land.(((((((((((((b)!Calculate(the(length(of(AB.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(

The(line(that(runs(through(the(middle(of(the(diagram(is(

called(the(((

………………………………((

The(lines(that(go(from(the(corner(of(the(land(to(the(traverse(line(are(

called((

………………………………((

(Example!42!


(c)!Calculate(the(perimeter(of(ABCD.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………(!(d)!Calculate(the(area(of(ABCD.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((


( (((a)!Draw(a(field(diagram(that(corresponds(to(the(notebook(entry(below.(

((((((((

(((((

((((

((

(b)!Calculate(the(area(of(this(land.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((! !

(Example!43!

A

C 0

7 12

17 8

30 12

40 B(((

E(((

D(((


Plane%Table%Radial%Surveys%A(plane(table(radial(survey(involves(measuring(the(distance(from(the(centre(of(the(land(to(each(corner.(The(angles(between(these(measurements(are(also(taken.((( ((((((a)!Calculate(the(size(of(angle(CTA.((………………………………………………………………………….(( (………………………………………………………………………….((((b)!Calculate(the(area(of(triangle(CTB.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((c)(Calculate(the(length(of(CA.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(

(Example!44!


Compass%Radial%Surveys%A(compass(radial(survey(involves(measuring(the(distance(from(the(centre(of(the(land(to(each(corner.(The(bearing(from(north(to(each(corner(is(also(taken.((((((((((((((((((((a)(Calculate(the(area(AOD.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((( ( ( ( ( ( ((b)(Calculate(the(length(of(CB.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((( (((((

(Example!45!

300o19km

020o

25km

150o

33km

230o

29km

A

BC

D

O


((

((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((



((

((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((



(((The(diagram(shows(a(compass(radial(survey(of(Dylan’s(property.((((((((((((((((((Line(AED(is(the(boundary(of(his(property.(BE(and(EC(are(proposed(fence(lines(on(his(property(and(∠ BEC(=(81o.((((a)!What(is(the(bearing(of(C(from(E?(!(b)!What(is(the(area(of(the(triangular(section(BEC?(Answer(to(the(nearest(square(metre.(!(c)(Calculate(the(perimeter(of(BEC(correct(to(the(nearest(metre.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((

(Example!48!


……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((!

Activity! Ex(5.10(Q(1J10!(


!Literacy!(

(1.(Which(of(the(words(“sometimes”,(“always”(or(“never”(is(missing?(((a)!The(sine,(cosine,(and(tangent(ratios(of(angles(between(0(and(90(degrees(are(……………………………….(positive(!(b)!The(cosine(ratio(of(an(angle(between(90(and(180(degrees(is(……………………(negative.(((c)!The(largest(angle(in(a(triangle(is(……………………………(opposite(the(largest(side.(!(d)!The(tangent(ratio(of(an(angle(between(90(and(180(degrees(is(………………………..(negative.((((e)!The(sine(ratio(of(an(angle(between(90(and(180(degrees(is(………………….(negative.((!!2.(Explain(the(difference(between(a(Plane(Table(Survey(and(a(Compass(Radial(Survey.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((3.(If(a(question(asks(you(to(take(a(bearing(from(A(to(B,(how(you(would(go(about(this?(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((4.(When(calculating(the(area(of(a(triangle,(by(using(either(formula,(explain(what(information(is(required(from(the(triangle.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((

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