Measurement!(MM5)! Applications!of!! Trigonometry! · 2!...
Transcript of Measurement!(MM5)! Applications!of!! Trigonometry! · 2!...
1! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
!Measurement!(MM5)!
!Applications!of!!Trigonometry!
!!(((((((((((((((((((((((((((
Name(…………………………………………………………………..((((((
((
G.Georgiou(
2! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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.((
……………………………..................………………………………………(( ( ( (
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……………………………..................………………………………………(((((((((((((((……………………………..................………………………………………(((((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!
3! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((( ( ( 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!
4! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
!!Complete(this(statement(for(the((triangle(to(the(left.((
tan(_______(=( !!(
!!
!Find(the(value(of(x%correct(to(2(decimal(places.((( ( ( ( ( ( ( ( ( ( (( ( ( ( ( ………………………………………………............................(( ( ( ( (( ( ( ( ( ………………………………………………............................((
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( ( ( ( ( ………………………………………………............................(( ( ( ( ( (
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………………………………………………............................( ( ( ( (
( ( ( (((( (
(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
5! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((! !
Activity! Ex(5.01(Q1,(2,(3,(5,(8,(11,(13,(14,(15,(16,(18!(
6! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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!
7! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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)!! ! ! ! ! ! ! ! !((((((((..........................................................( ( ..........................................................((..........................................................( ( ..........................................................(
8! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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!
9! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(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!
10! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((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!
11! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((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!
12! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((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!(
13! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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!
14! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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(
15! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(( (Calculate(the(value(of(m(correct(to(3(significant(figures.((…………………………………………………………………………….((…………………………………………………………………………….((…………………………………………………………………………….((…………………………………………………………………………….((…………………………………………………………………………….((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((!
!!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((
(Example!19!
(Example!20!
16! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((
((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((((((
(Example!21!
Activity! Ex(5.04(Q(1((a,b,c)(2,(3,(4,(5,(7,(8,(9,(10!(
17! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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!
18! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
Miscellaneous%Sine%Rule%Examples%((
((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((((((
(Example!24!
19! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(
((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((
(Example!25!
Activity! Ex(5.05(Q(1(a,b,c),(4,(5,(6,(9,(10!;;;;;(
20! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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!
((
((
This!formula!IS!given!on!the!HSC!formula!sheet!(
Angle(opposite(side(you(are(
finding(
(Example!26!
(Example!27!
21! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(((
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!(
22! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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!
(
(
!!
This!formula!IS!given!on!the!HSC!formula!sheet!(
Side(opposite(angle(you(are(trying(to(find(
(Example!29!
(Example!30!
23! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(( (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!(
24! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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(
25! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
( ((
((( (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!(
26! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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(
27! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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°
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
44
Sine and cosine rules5.4
Find the missing side (to 2 decimal places) or angle (to the nearest degree).
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
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°
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
44
Sine and cosine rules5.4
Find the missing side (to 2 decimal places) or angle (to the nearest degree).
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
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
44
Sine and cosine rules5.4
Find the missing side (to 2 decimal places) or angle (to the nearest degree).
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
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
44
Sine and cosine rules5.4
Find the missing side (to 2 decimal places) or angle (to the nearest degree).
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
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°
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
44
Sine and cosine rules5.4
Find the missing side (to 2 decimal places) or angle (to the nearest degree).
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
28! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((
((((
((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………(
(Example!36!
!H.S.C.!Question!(37)!
29! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((
((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((
!H.S.C.!Question!(38)!
30! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((
(……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((
(……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((
!H.S.C.!Question!(39)!
31! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((
((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((
!H.S.C.!Question!(40)!
32! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(
(
((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(( (……………………………………………………………………………………………........………………………………((( (((
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!(
33! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
• 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!
34! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(c)!Calculate(the(perimeter(of(ABCD.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………(!……………………………………………………………………………………………........………………………………(!(d)!Calculate(the(area(of(ABCD.((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((
35! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
( (((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(((
36! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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!
37! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
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
38! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((
((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((
!H.S.C.!Question!(46)!
39! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
((
((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………(((
!H.S.C.!Question!(47)!
40! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
(((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!
41! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((((!
Activity! Ex(5.10(Q(1J10!(
42! General(Mathematics((HSC(Course)(|(Applications(of(Trigonometry((MM5)(
!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.(((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((……………………………………………………………………………………………........………………………………((