15 LO3 Rules for Triangles (3)

8/4/2019 15 LO3 Rules for Triangles (3)

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Grade 12 Mathematical Literacy

LO3 Rules for Triangles: Area Rule, Sine Rule andCosine Rule

Maths & Science Marketing

Sine, Cos, Tan,Pythagoras

A = bxh

Sine Rule,Cosine Rule,Area Rule

Solving Triangles:

Right Angled Non-right Angled

Use the area rule: A = bc sin A to calculate the area of anytriangle of which two sides and the angle between them areknown. (SAS)

Use the sine rule to calculate an unknown side or angle in anytriangle. The sine rule compares the ratio between the sine ofan angle and the side opposite that angle in any triangle. Thesine rule is written in one of the following two forms:

Use the cosine rule to calculate the third side of a triangle if theother two sides and the angle opposite to the unknown side isknown. (SAS or SSS) The cosine rule is written as follows:

c=a+b - 2ab cos C

Sin, Cos, Tan,Pythagoras

hbA =2

1

Sine Rule,Cosine Rule,Area Rule


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1. Apply the area rule to determine the area of each of the followingtriangles:

1.1

1.2



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2. Use the sine rule to calculate the value of the unknown angles andsides in each of the drawings.

2.1

2.2



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3. Use the cosine rule to calculate the value of x rounded off to onedecimal place in each of the drawings below.

3.1

3.2

3.3



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4. Mrs Nortje lives on the slopesof Table Mountain. She is nothappy with her television receptionso she has decided to install a 45metre high pole on top of whichshe will have a satellite dish installed.An anchor wire will be attachedto the top of the pole and anchoredto a point 22 metres downhillfrom the base of the pole. Thedrawing on the left illustratesthe situation. Study the drawingand answer the questions that follow.

4.1 Calculate the size of BCA if ACD isa straight line.

4.2 Use the cosine rule to calculate the lengthof the anchor wire, AB.


Im finally getting thehang of trigonometry!Just one morequestion to go!


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5. A soccer player aims towards a goal post which is 15 metres fromthe back line CH on a soccer field. The angle from the left goalpost, FG, to the soccer player, S, is 116. The goal posts are7,32 m wide.

The diagram below represents the above situation.

5.1 Calculate how far the soccer player is from the left goal postFG.

5.2 Calculate how far the soccer player is from the right goal postEH.

5.3 Calculate the approximate size of HSG , the angle within whichthe soccer player could possibly score a goal.


116


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ANSWERS

1.1 A = bc sin A

A = x 9 x 7 x sin 96A = 31,33 m

1.2 A = bc sin AA = x 22 x 10 x sin 27A = 49,94 cm

2.1

r

R

q

Q

p

P sinsinsin==

cmp

p

R

Prp

R

r

P

p

,sin

sin

sin

sin

sinsin

195834

2480

=

=

=

=

.ofsum... == 1223424180Q

cm,

sin

sin

sinsin

sinsin

32121

34

12280

=

=

=

=

q

q

RQrq

R

r

Q

q



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2.2ofsum... == 1512936180B

cmb

b

CBcb

C

c

B

b

9382

129

15249

,

sin

sin

sinsin

sinsin

=

=

=

=

3.1

cm,

cos))((

cos))((x

Ccos

2

913

27302423024

27302423024

2

22

22

222

=

+=

+=

+=

x

x

accab

3.2



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cm

x

abb

abbac

924

123101821018

2

2

22

2

222

,x

cos

Ccosax

Ccos

2

=

+=

+=

+=

3.3

=

=

=

+=

123,9

360

201-xcos

201-xcos360

xcos))((

x

18102181025222



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4.1 ACD is a straight line and thus 180== 12258180BCA

4.2

cm

c

abb

abbac

6559

122224522245

2

2

22

2

222

,c

cos

Ccosac

Ccos

2

=

+=

+=

+=



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5.1

=

=

=

64

116180

180

SGC

SGC

SGHSGC

In CGS

mGS

SinGS

SinGS

GSSin

GS

CSSGCSin

Hypotenuse

OppositeSGCSin

6916

64

15

1564

1564

,

)(

)(

)(

)(

)(

=

=

=

=

=

=

The soccer player is 16,69 metres from the goalpost FG, sinceGS is the distance from FG to the player.

5.2 Use the Cos Rule: a = b + c - 2bc Cos

Substitute: a = SH; b = GS; c = GH and = SH

mSH

SH

CosSH

HGSCosGHGSGHGSSH

9620

25439

116327691623276916

2

222

222

,

,

)(),)(,(),(),(

)())((

=

=

+=

+=

The distance from the soccer player to the goal post is 20,96 m.



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5.3 In SGH, by the sine rule, we haveb

BSin

a

ASin =

Substitute: = SGH; B = HSG ; a = SH; and b = GH

The approximate size of GH is 18,29


2918

31389181390

31389181390

9620

116327

1

,

,(

,

,

))(,(

)((

HSG

SinHSG

HSSinG

SinHSSinG

SH

HGSinSGHHSSinG

GH

HSSinG

SH

HGSinS

15 LO3 Rules for Triangles (3)

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