s4 3 sine cosine ch12

7/27/2019 S4 3 Sine Cosine Ch12

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14-Aug-13 Created by Mr. Lafferty Maths Dept.

Trigonometry

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Cosine Rule Finding a Length

Sine Rule Finding a length

Mixed Problems

S4 Credit

Sine Rule Finding an Angle

Cosine Rule Finding an Angle

Area of ANY Triangle


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Starter Questions

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2

1. Multiply out the brackets and simplify

5(y -5) - 7(5 - y)

2. True or false the gradient of the line is 5

3y = 5x -

4

3. Factorise x -100

S4 Credit


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www.math

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Success Criteria

1. Know how to use the sinerule to solve REAL LIFEproblems involving lengths.

1. To show how to use thesine rule to solve REALLIFE problems involvingfinding the length of aside of a triangle .

Sine RuleS4 Credit


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C

B

A

14-Aug-13 Created by Mr Lafferty Maths Dept

Sine Rule

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S4 Credit

a

b

c

The Sine Rule can be used with ANY triangleas long as we have been given enough information.

Works for any Triangle

a b c= =

SinA SinB SinC


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Deriving the rule

B

C

A

b

c

a

Consider a general triangle ABC.

The Sine Rule

Draw CP perpendicular to BA

P

CPSinB CP aSinB

a

CP

also SinA CP bSinA

b

aSinB bSinA

aSinBb

SinA

a b

SinA SinB

This can be extended to

a b c

SinA SinB SinC

or equivalentlySinA SinB SinC

a b c


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Calculating SidesUsing The Sine Rule

10m

34o41o

a

Match up corresponding sides and angles:

sin 41o

a

10

sin 34o Now cross multiply.

sin 34 10 sin 41o o

a Solve for a.

10 sin 41

sin 34

o

o

a 10 0.656

11.740.559a m

Example 1 : Find the length of a in this triangle.

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S4 Credit

A

B

C


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Calculating SidesUsing The Sine Rule

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S4 Credit

10m133o

37o

d

sin133o

d

10

sin 37o

sin 37 10sin133o o

d

10sin133

sin 37

o

od

10 0.731

0.602

d

= 12.14m


Now cross multiply.

Solve for d.

Example 2 : Find the length of d in this triangle.

C

D

E


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What goes in the Box ?

Find the unknown side in each of the triangles below:

(1)12cm

72o32oa

(2)

93o

b47o

16mm

A = 6.7cm

B = 21.8mm

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S4 Credit



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Now try MIA Ex 2.1Ch12 (page 247)

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S4 Credit

Sine Rule


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Starter Questions

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1. True or false 9x - 36 = 9(x + 6)(x - 6)

2. Find the gradient and the y - intercept

3 1for the line with equation y = - x +

4 5

3. Solve the equation tanx - 1 = 0

S4 Credit


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www.math

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com Learning Intention Success Criteria

1. Know how to use the sinerule to solve problemsinvolving angles.

1. To show how to use thesine rule to solve problemsinvolving finding an angleof a triangle .

Sine RuleS4 Credit


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Calculating AnglesUsing The Sine Rule

Example 1 :

Find the angle AoAo

45m

23o38m


45

sino

A

38

sin 23o

Now cross multiply:

38sin 45sin 23o oA Solve for sin Ao

45sin 23sin

38

o

oA = 0.463 Use sin-1 0.463 to find Ao

1

sin 0.463 27.6

o o

A

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S4 Credit


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Calculating AnglesUsing The Sine Rule

143o

75m

38m

Bo

38

sino

B

75sin 38sin143o oB

75

sin143o

38sin143sin

75

o

oB = 0.305

1

sin 0.305 17.8

o o

B

Example 2 :

Find the angle Bo


Now cross multiply:

Solve for sin Bo

Use sin-1 0.305 to find Bo

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S4 Credit


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What Goes In The Box ?

Calculate the unknown angle in the following:

(1)

14.5m

8.9mAo

100o (2)

14.7cm

Bo

14o

12.9cm

Ao = 37.2o

Bo

= 16ow

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S4 Credit


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Now try MIA Ex3.1Ch12 (page 249)

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S4 Credit

Sine Rule


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Starter Questions

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2

1. Find the gradient of the line that passes

through the points ( 1,1) and (9,9).

2. Find the gradient and the y - intercept

for the line with equation y = 1 -x

3. Factorise x -64

S4 Credit


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14-Aug-13 Created by Mr. Lafferty Maths Dept.www.math

srevision.com Learning Intention Success Criteria

1. Know when to use the cosinerule to solve problems.

1. To show when to use thecosine rule to solveproblems involving findingthe length of a side of atriangle .

Cosine RuleS4 Credit

2. Solve problems that involvefinding the length of a side.


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C

B

A14-Aug-13 Created by Mr Lafferty Maths Dept

Cosine Rule

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S4 Credit

a

b

c

The Cosine Rule can be used with ANY triangleas long as we have been given enough information.


cos2 2 2a =b +c - 2bc A


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Deriving the rule

A

B

C

a

b

c

Consider a general triangle ABC. Werequire a in terms of b, c and A.

Draw BP perpendicular to AC

b

Px b - x

BP2 = a2 (b x)2

Also: BP2 = c2 x2

a2 (b x)2 = c2 x2

a2 (b2 2bx + x2) = c2 x2

a2 b2 + 2bx x2 = c2 x2

a2

= b2

+ c2

2bx* a2 = b2 + c2 2bcCosA

*Since Cos A = x/c x = cCosA

When A = 90o, CosA = 0 and reduces to a2 = b2 + c2 1

When A > 90o, CosA is positive, a2 > b2 + c2 2

When A < 90o, CosA is negative, a2 > b2 + c2 3

The Cosine Rule

The Cosine Rulegeneralises Pythagoras Theorem andtakes care of the 3 possible cases for Angle A.

a2 > b2 + c2

a2 < b2 + c2

a2 = b2 + c2

A

A

A

1

2

3

Pythagoras + a bit

Pythagoras - a bit

Pythagoras


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a2 = b2 + c2 2bcCosA

Applying the same method asearlier to the other sidesproduce similar formulae for

b and c. namely:b2 = a2 + c2 2acCosB

c2 = a2 + b2 2abCosC

A

B

C

a

b

c

The Cosine Rule

The Cosine rule can be used to find:

1. An unknown side when two sides of the triangle and theincluded angle are given.

2. An unknown angle when 3 sides are given.

Finding an unknown side.


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Cosine Rule

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S4 Credit

How to determine when to use the Cosine Rule.


1. Do you know ALL the lengths.

2. Do you know 2 sides and the angle in between.

SASOR

If YES to any of the questions then Cosine Rule

Otherwise use the Sine Rule

Two questions


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Using The Cosine Rule

Example 1 : Find the unknown side in the triangle below:

L5m

12m

43o

Identify sides a,b,c and angle Ao

a = L b = 5 c = 12 Ao = 43o

Write down the Cosine Rule.a2 = b2 + c2 -2bccosAo

Substitute values to find a2.a2 = 52 + 122 - 2 x 5 x 12 cos 43o

a2 = 25 + 144 - (120 x 0.731 )

a2 = 81.28 Square root to find a.

a = L = 9.02mwww.math

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S4 Credit



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Example 2 :

Find the length of side M.

137o17.5 m

12.2 m

MIdentify the sides and angle.a = M b = 12.2 C = 17.5 Ao = 137o

Write down Cosine Rulea2 = b2 + c2 -2bccosAo

a2 = 12.22 + 17.52 ( 2 x 12.2 x 17.5 x cos 137o )

a2 = 148.84 + 306.25 ( 427 x 0.731 )Notice the two negative signs.

a2 = 455.09 + 312.137

a2 = 767.227

a = M = 27.7m

Using The Cosine Rule

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S4 Credit



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Find the length of the unknown side in the triangles:

(1)

78o

43cm

31cmL

(2)

8m

5.2m

38o

M

L = 47.5cm

M =5.05mwww.math

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S4 Credit


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S4 Credit

Cosine Rule


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Starter Questions

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2

o o

1. If lines have the same gradient

What is special about them.

2. Factorise x + 4x -12

3. Explain why the missing angles

are 90 and 36

S4 Credit


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1. Know when to use the cosinerule to solve REAL LIFEproblems.

1. To show when to use thecosine rule to solve REALLIFE problems involvingfinding an angle of atriangle .

Cosine RuleS4 Credit

2. Solve REAL LIFE problemsthat involve finding an angle

of a triangle.


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C

B

A14-Aug-13 Created by Mr Lafferty Maths Dept

Cosine Rule

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S4 Credit

a

b

c

The Cosine Rule can be used with ANY triangleas long as we have been given enough information.


cos2 2 2a =b +c - 2bc A


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Finding AnglesUsing The Cosine Rule

Consider the Cosine Rule again: a2 = b2 + c2 -2bccosAo

We are going to change the subject of the formula to cos Ao

Turn the formula around:b2

+ c2

2bc cos Ao

= a2

Take b2 and c2 across.-2bc cos Ao = a2 b2 c2

Divide by 2 bc.2 2 2

cos

2

o a b cA

bc

Divide top and bottom by -12 2 2

cos2

o b c aA

bc

You now have a formula for

finding an angle if you know all

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S4 Credit



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Ao

16cm

9cm 11cm

Write down the formula for cos Ao

2 2 2

cos2

o b c a

Abc

Label and identify Ao and a , b and c.Ao = ? a = 11 b = 9 c = 16

Substitute values into the formula.

2 2 2

9 16 11cos2 9 16

oA

Calculate cos Ao .Cos Ao = 0.75

Use cos-1 0.75 to find AoAo = 41.4o

Example 1 : Calculate the

unknown angle Ao .


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S4 Credit



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Example 2: Find the unknown

Angle yo in the triangle:

26cm

15cm 13cmyo

Write down the formula.

2 2 2

cos2

o b c aA

bc

Identify the sides and angle.Ao = yo a = 26 b = 15 c = 13

2 2 2

15 13 26cos

2 15 13

oA

Find the value of cosAo

cosAo = - 0.723The negative tells youthe angle is obtuse.

Ao = yo = 136.3owww.math

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S4 Credit




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Calculate the unknown angles in the triangles below:

(1)

10m

7m5m AoBo

(2)

12.7cm

7.9cm 8.3cm

Ao =111.8o Bo = 37.3o

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S4 Credit


33/50


Now try MIA Ex 5.1Ch12 (page 256)

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S4 Credit

Cosine Rule


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Starter Questions

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2( 3) (4 ) 3 2x x x

1. True or false

2. Find the equaton of the line passing

through the points ( 3,2) and (10, 9) .

3. Solve the equation sin x - 0.5 = 0

S4 Credit


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1. Know the formula for thearea of any triangle.1. To explain how to use theArea formula for ANYtriangle.

S4 Credit


2. Use formula to find area ofany triangle given two lengthand angle in between.


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Labelling Triangles

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S4 Credit

A

B

C

A

a

B

b

Cc

Small letters a, b, c refer to distancesCapital letters A, B, C refer to angles

In Mathematics we have a convention for labelling triangles.


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F

E

D

F

E

D


Labelling Triangles

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S4 Credit

d

e

f

Have a go at labelling the following triangle.


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General Formula forArea of ANY Triangle

Consider the triangle below:

Ao Bo

Co

ab

c

h

Area = x base x height

1

2A c h

What does the sine of Ao equal

sino h

A

b

Change the subject to h.

h = b sinAo

Substitute into the area formula

1sin

2

oA c b A

1sin

2

oA bc Aw

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S4 Credit


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S4 Credit

A

B

C

A

a

B

b

Cc

The area of ANY triangle can be foundby the following formula.

sin1

Area = ab C2

sin1

Area = ac B

2

sin1

Area = bc A2

Another version

Another version

Key feature

To find the areayou need to knowing2 sides and the angle

in between (SAS)


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S4 Credit

A

B

C

A

20cm

B

25cm

Cc

Example : Find the area of the triangle.

sin C1Area = ab2

The version we use is

30o1

20 25 sin 302

oArea

210 25 0.5 125Area cm


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S4 Credit

D

E

F

10cm

8cm

Example : Find the area of the triangle.

sin1Area= df E2

The version we use is

60o

18 10 sin 60

2

oArea

240 0.866 34.64Area cm


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Calculate the areas of the triangles below:

(1)

23o

15cm

12.6cm

(2)

71o5.7m

6.2m

A = 36.9cm2

A = 16.7m2www.mathsrevision.com

S4 Credit

Key feature

Remember

(SAS)


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S4 Credit



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Starter Questions


2

1. A washing machine is reduced by 10%

in a sale. It's sale price is 360.

What was the original price.

2. Factorise x -7x +12

3. Find the missing angles.

S4 Credit

61o


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14-Aug-13 Created by Mr. Lafferty Maths Dept.www.mathsrevision.com Learning Intention Success Criteria

1. Be able to recognise thecorrect trigonometricformula to use to solve aproblem involving triangles.

1. To use our knowledgegained so far to solvevarious trigonometryproblems.

Mixed problemsS4 Credit


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The Sine Rule a b cSinA SinB SinC

Application Problems

25o

15 m AD

The angle of elevation of thetop of a building measured

from point A is 25o. At pointD which is 15m closer to the

building, the angle ofelevation is 35o Calculate the

height of the building.

T

B

Angle TDA =

145o

Angle DTA =

10o

o o

15

25 10

TD

Sin Sin

o15 2536.5

10

SinTD m

Sin

35o

36.5

o3536.5

TBSin

o36.5 25 0. 93TB Sin m

180 35 = 145o

180 170 = 10o


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The Sine Rule a b cSinA SinB SinC

A

The angle of elevation of the top of a column measured from point A, is 20o.

The angle of elevation of the top of the statue is 25o. Find the height of thestatue when the measurements are taken 50 m from its base

50 m

Angle BCA =

70o

Angle ACT = Angle ATC =

110o

65oo 5020Cos

AC

o

50

20

53.21 (2 )

ACCos

m dp

o o

53.21

5 65

TC

Sin Sin

o

53.21 5(1 )

655.1

SinTC m dp

Sin

B

T

C

180 110 = 70o 180 70 = 110o 180 115 = 65o

20o25o

5o


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A fishing boat leaves a harbour (H) and travels due East for 40 miles to amarker buoy (B). At B the boat turns left and sails for 24 miles to alighthouse (L). It then returns to harbour, a distance of 57 miles.

(a) Make a sketch of the journey.

(b) Find the bearing of the lighthouse from the harbour. (nearest degree)

The Cosine Rule

Application Problems

2 2 2

2

b c aCosA

bc

H40 miles

24 miles

B

L

57 miles

A

2 2 257 40 24

2 57 40CosA

x x

A 20.4o

90 0 020.4 7 oBearing

Th C i R l 2 b2 2 2b C sA


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2 2 2

2

b c aCosA

bc

The Cosine Rule a2 = b2 + c2 2bcCosA

An AWACS aircraft takes off from RAFWaddington (W) on a navigation

exercise. It flies 530 miles North toa point (P) as shown, It then turnsleft and flies to a point (Q), 670miles away. Finally it flies back tobase, a distance of 520 miles.

Find the bearing of Q from point P.

2 2 2530 670 520

2 530 670CosP

x x

48.7oP

180 22948.7 oBearing

P

670 miles

W

530 miles

Not to Scale

Q

520 miles


50/50


Now try MIAEx 7.1 & 7.2Ch12 (page 262)


S4 Credit

Mixed Problems

s4 3 sine cosine ch12

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