222 60º 60º 60º 1 60º 2 30º 3333 this triangle will provide exact values for sin, cos and tan...

67
2 2 2 2 2 2 60º 60º 60º 60º 60º 60º 1 1 60º 60º 2 2 30º 30º 3 3 This triangle will provide This triangle will provide exact values for exact values for sin, cos and tan 30º and 60º sin, cos and tan 30º and 60º Exact Values Exact Values Some special values of Sin, Cos and Tan are Some special values of Sin, Cos and Tan are useful left as fractions, We call these useful left as fractions, We call these exact values exact values

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Page 1: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

2222

22

60º60º

60º60º60º60º11

60º60º

2230º30º

33

This triangle will provide exact values for This triangle will provide exact values for

sin, cos and tan 30º and 60ºsin, cos and tan 30º and 60º

Exact ValuesExact ValuesSome special values of Sin, Cos and Tan are useful left as Some special values of Sin, Cos and Tan are useful left as fractions, We call these fractions, We call these exact valuesexact values

Page 2: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

xx 0º0º 30º30º 45º45º 60º60º 90º90º

Sin xºSin xº

Cos xºCos xº

Tan xºTan xº

½

½½

3

23

23

31

1160º60º

2230º30º33

Exact ValuesExact Values

0

1

0

1

0

∞∞

Page 3: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Exact ValuesExact Values

11 1145º45º

45º45º

22

Consider the square with sides 1 unitConsider the square with sides 1 unit

1111

We are now in a position to calculate We are now in a position to calculate exact values for sin, cos and tan of 45exact values for sin, cos and tan of 45oo

Page 4: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Exact ValuesExact Values11

45º45º

45º45º

22

11

Tan xºTan xº

Cos xºCos xº

Sin xºSin xº

90º90º60º60º45º45º30º30º0º0ºxx

00

00

11

11

00

21

21

23

23

33

1

21

21

11

Page 5: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Apr 21, 2023Apr 21, 2023

Naming the sides of a TriangleNaming the sides of a Triangle

AA

BB

CC

aa

bb

cc

Page 6: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Apr 21, 2023Apr 21, 2023

Area of ANY TriangleArea of ANY Triangle

AA

BB

CC

aa

bb

cc

The area of ANY triangle can be found The area of ANY triangle can be found by the following formula.by the following formula.

Another versionAnother version

Another versionAnother version

Key feature Key feature

To find the areaTo find the areayou need to knowyou need to know

2 sides and the angle 2 sides and the angle in between (SAS)in between (SAS)

AbcArea sin21

BacArea sin21

CabArea sin21

If you know A, b and cIf you know A, b and cIf you know B, a and cIf you know B, a and cIf you know C, a and bIf you know C, a and b

Remember:Remember:

Sides use lower case Sides use lower case letters of anglesletters of angles

Page 7: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Area of ANY TriangleArea of ANY Triangle

AA

BB

CC

AA

20cm20cmBB

25cm25cm

CCcc

Example : Find the area of the triangle.Example : Find the area of the triangle.

The version we use isThe version we use is

3030oo

120 25 sin 30

2oArea

210 25 0.5 125Area cm

CabArea sin21

aa

bb

Page 8: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Area of ANY TriangleArea of ANY Triangle

DD

EE

FF

10cm10cm

8cm8cm

Example : Find the area of the triangle.Example : Find the area of the triangle.

sin1

Area= df E2

The version we use isThe version we use is

6060oo

18 10 sin 60

2oArea

240 0.866 34.64Area cm

d

e

f

Page 9: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

What Goes In The Box ?What Goes In The Box ?

Calculate the areas of the triangles Calculate the areas of the triangles below:below:

(1)

23o

15cm

12.6cm

(2)

71o

5.7m

6.2m

A =36.9cm2

A =16.7m2

Key Key feature feature

Remember Remember (SAS)(SAS)

Page 10: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Apr 21, 2023Apr 21, 2023

17 cm17 cm

95 95 cmcm22

12 cm12 cm

PP

QQ

RR

Finding the Angle given the areaFinding the Angle given the area

Find the size of angle PFind the size of angle P

pp

qq

rr

Area = Area = ½½qrqrsinPsinP

95 = 95 = ½ ½ 1717 × × 1212 sinPsinP

95 = 95 = 102102 sinPsinP

sinP sinP = 95 ÷ 102 = 95 ÷ 102 = 0∙931 = 0∙931

P = sinP = sin-1-1 0∙931 = 68∙6° 0∙931 = 68∙6°

Page 11: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

The side opposite angle The side opposite angle AA is labelled is labelled aa

The Sine RuleThe Sine Rule

Bsinb

Asin

a

AA

BB

CC

aa

bb

cc

The side opposite angle The side opposite angle BB is labelled is labelled bbThe side opposite angle The side opposite angle CC is labelled is labelled cc

Csinc

The RuleThe Rule

Page 12: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Calculating Sides Using The Sine RuleCalculating Sides Using The Sine Rule

Find the length of Find the length of xx in in this triangle.this triangle.

ox

41sin

o34sin

10 Now cross Now cross multiply.multiply.

oox 41sin1034sin

o

ox

34sin

41sin10

mx 74.11559.0

656.010

Example 1Example 1

PP3434oo

4141oo

xx10m10m

QQ

RR

Rr

Qq

Pp

sinsinsin

1010

sin 34sin 34°°

xx

sin 41sin 41°°

Page 13: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Find the length of x in this triangle.

ox

133sin

o37sin

10

oox 133sin1037sin

o

ox

37sin

133sin10

602.0731.010

x = 12.14m= 12.14m

Example 210m133o

37o

xD

E

F

Ff

Ee

Dd

sinsinsin

1010

sin 37sin 37°°

xx

sin 133sin 133°°

Cross Cross multiplymultiply

Page 14: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

The balloon is anchored to the ground as shown in the diagram.

o70sin

25

o

x

35sin

Problem

C

c

B

b

A

a

sinsinsin

xx

sin 35sin 35°°

2525

sin 70sin 70°°

25 m25 m

7575°°xx m

7070°°

Calculate the distance between the anchor points.

A

B Caa

bbcc

3535°°

sin 75sin 75°°

oox 35sin2570sin

oox 70sin35sin25

mx 315

Cross Cross multiplymultiply

Page 15: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Find the unknown side in each of the triangles below:Find the unknown side in each of the triangles below:

(1) 12cm

72o

32o

x(2)

93o

y47o

16mm

(3)

87o

89m

35o

a (4) 143o

g12o

17m

x = 6.7cm y = 21.8mm

a = 51.12m51.12m g = 49.21m

Page 16: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Calculating Angles Using The Sine RuleCalculating Angles Using The Sine Rule

Example 1.

Find the angle Find the angle aaoo

oasin

45o23sin

38

ooa 23sin45sin38

38

23sin45sin

ooa = 0.463

ooa 6.27463.0sin 1

ao

45m

23o

38m

Z

Y

X

Zz

Yy

Xx

sinsinsin

4545

sin sin aaºº

3838

sin 23sin 23ºº

Cross Cross multiplymultiply

Use sinUse sin-1-1

Page 17: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Example 2.

143o

75m

38m

boFind the size of Find the size of the angle bthe angle boo

obsin

38

oob 143sin38sin75

o143sin

75

75

143sin38sin

oob = 0.305

oob 8.17305.0sin 1

Page 18: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Calculate the unknown angle in the following:Calculate the unknown angle in the following:

(1)

14.5m

8.9m

ao

100o(2)

14.7cm

bo

14o

12.9cm

(3)93o

64mm

co

49mm

aaoo = 37.2 = 37.2oo bbo o = 16= 16oo

c = 49.9c = 49.9oo

Page 19: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Sine Rule

Basic Examples

Page 20: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

In the examples which follow use the sine rule to find the required side or angle.

asin A

bsin B

csinCB

5.8

C74° 34°A

Find AB

5.7

9.6

76°

C

B

A

Find angle B

DD

4.64.6

EE80°80°

3.23.2

FF Find angle FFind angle F

Page 21: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

In the examples which follow use the sine rule to find the required side or angle.

asin A

bsin B

csinC

13.8

127°127°

36°36°

P

R

Q

Find PQ

M

12

N

86°86° 7

TFind angle NFind angle N

A

8

B

C

72° 72°

Find ABFind AB

Page 22: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

RR60°60°

SS

78°78°

3.83.8

TTFind RTFind RT

II

27°27°

130°130°4.14.1

GG

HH

Find GIFind GI

AA

99

BB

75°75°

32°32°

CC

Find BCFind BC

88

39°39°

1010

YY

XX

ZZFind angle ZFind angle Z

BB

70°70°

CC

99

1313

AAFind angle AFind angle A

Page 23: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Sine Rule (Bearings)Sine Rule (Bearings)

TrigonometryTrigonometry

Page 24: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

In the examples which follow use the sine ruleIn the examples which follow use the sine rule to find the required side or angle.to find the required side or angle.

asin A

bsin B

csinC

Consider two radar stations Alpha and Beta. Consider two radar stations Alpha and Beta. Alpha is 140 miles west of Beta.Alpha is 140 miles west of Beta. The bearing of an aero plane from Alpha is 032° The bearing of an aero plane from Alpha is 032° and from Beta it is 316°. and from Beta it is 316°. How far is the aeroplane from each of the radar stations? How far is the aeroplane from each of the radar stations?

1.1.

Page 25: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

2.2. Two ports Dundee and Stonehaven are 54 miles apart with Two ports Dundee and Stonehaven are 54 miles apart with Dundee approximately due south of Stonehaven. Dundee approximately due south of Stonehaven. The bearing of a ship at sea from Stonehaven is 098° The bearing of a ship at sea from Stonehaven is 098° and from Dundee it is 048°.and from Dundee it is 048°.

How far is the ship from Dundee? How far is the ship from Dundee?

3.3. A ship is sailing north. At noon its bearing from a lighthouse A ship is sailing north. At noon its bearing from a lighthouse is 240°. Five hours later the ship is 84km further north and is 240°. Five hours later the ship is 84km further north and its new bearing from the lighthouse is 290°. its new bearing from the lighthouse is 290°. How far is the ship from the lighthouse now? How far is the ship from the lighthouse now?

Page 26: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

4. Two ports P and Q are 35 miles apart with P due east of Q. The bearings of a ship from P and Q are 190° and 126° respectively.

a) What is the bearing of port Q from the ship?

b) How far is the ship from port P

5. The bearing of an aeroplane from Aberdeen is 068° and from Dundee it is 030°. Dundee is 69 miles south of Aberdeen. How far is the aero plane from Aberdeen?

Page 27: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

6.6. Two oil rigs are 80 miles apart with rigTwo oil rigs are 80 miles apart with rig B being 80 miles east of rig A. B being 80 miles east of rig A.

The bearing of a ship from rig B is 194° The bearing of a ship from rig B is 194° and from A the bearing is 140°. and from A the bearing is 140°. Find the distance of the ship from rig A. Find the distance of the ship from rig A.

7.7. At noon a ship lies on a bearing 070° from Dundee. At noon a ship lies on a bearing 070° from Dundee. The ship sails a distance of 45 miles south and at 3pm its The ship sails a distance of 45 miles south and at 3pm its new bearing from Dundee is 130°. new bearing from Dundee is 130°.

How far is the ship from Dundee now? How far is the ship from Dundee now?

If the ship maintains the same speed, If the ship maintains the same speed, how long will it take to return to Dundee?how long will it take to return to Dundee?

Page 28: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

8.8. An aeroplane leaves Leuchers airport and flies 80 miles north.An aeroplane leaves Leuchers airport and flies 80 miles north. It then changes direction and flies 120 miles east. It then changes direction and flies 120 miles east. The plane now turns onto a bearing of 126° The plane now turns onto a bearing of 126°

and flies a further 164 miles. and flies a further 164 miles.

Calculate the bearing and distance of Calculate the bearing and distance of the plane from Leuchers. the plane from Leuchers.

Page 29: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

QuestionQuestion SolutionSolution

11 122.4 miles ; 103.8 miles122.4 miles ; 103.8 miles

22 69.8 miles69.8 miles

33 95.0 kilometers95.0 kilometers

44 Bearing 306° ; 22.9 milesBearing 306° ; 22.9 miles

55 56.0 miles56.0 miles

66 114.9 miles114.9 miles

77 48.8 miles ; 3 hours 15 minutes48.8 miles ; 3 hours 15 minutes

88 236.2 miles ; bearing 269°236.2 miles ; bearing 269°

Note: the last question requires the cosine rule as well as the sine rule.

Page 30: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

C

B

AApr 21, 2023Apr 21, 2023

Cosine RuleCosine Rule

a

b

c

The Cosine Rule can be used with ANY triangle The Cosine Rule can be used with ANY triangle

as long as we have been as long as we have been givengiven enough information enough information.

Abccba cos2222

Bcaacb cos2222

c2 a2 b2 2abcos C

Given Given angle Aangle AGiven Given angle Bangle BGiven Given angle Cangle C

Page 31: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Using The Cosine RuleUsing The Cosine RuleExample 1 : Find the unknown side in the triangle below:Example 1 : Find the unknown side in the triangle below:

Identify sides Identify sides a, b, ca, b, c and and angle angle AAoo

aa = = xx bb = = 55 cc = =1212 AAo o == 4343ooWrite down the Cosine Rule Write down the Cosine Rule

for for aa

Substitute valuesSubstitute valuesxx22 = = 5522 ++ 121222 - 2 - 2 xx 5 5 xx 12 cos 43 12 cos 43oo

xx22 = = 81.2881.28 Square root to find “Square root to find “xx”.”.

xx = 9.02m = 9.02m

xx5m5m

12m12m

4343oo

AA BB

CC

aa22 = = bb22 + + cc22 - 2 - 2bcbc cos cos AA

Page 32: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

pp22 = = qq22 + + rr22 – 2 – 2pqpq cos P cos P

Example 2Example 2 : :

Find the length of side QRFind the length of side QR

Identify the sides and angle.Identify the sides and angle.p = yp = y rr = 12.2 = 12.2 qq = 17.5 = 17.5 PP = 137= 137oo

Write down Cosine RuleWrite down Cosine Rule for for pp

yy22 = 12.2 = 12.222 + 17.5 + 17.522 – 2 – 2 xx 12.2 12.2 xx 17.5 17.5 xx cos 137 cos 137oo

yy22 = 767.227 = 767.227

yy = 27.7m = 27.7m

Using The Cosine RuleUsing The Cosine Rule

137137oo 17.5 m17.5 m12.2 m12.2 m

yy

PP

QQ RR

SubstituteSubstitute

Page 33: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Find the length of the unknown side in the triangles:Find the length of the unknown side in the triangles:

(1)(1)7878oo

43cm43cm

31cm31cmpp

AA

BB

CC

a

b

c

a2 = b2 + c2 – 2bc cosA

p2 = 432 + 312 – 2 × 43 × 31 × cos78°

p2 = 2255∙7

p = 47∙5 cm

Page 34: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Find the length of the unknown side in the triangles:Find the length of the unknown side in the triangles:

(2)(2)

8m8m

55∙∙2m2m

3838oo

mmm = 5m = 5∙∙05m05m

112112ºº17 mm17 mm28 mm28 mm

kk

k = 37k = 37∙∙8 mm8 mm

(3)(3)

Page 35: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine RuleCosine Rule Basic ExamplesBasic Examples 1. Finding a side. 1. Finding a side.

a2 b2 c2 2bc cos A

5.6cm5.6cm

39°39°

CC

BB4.7cm4.7cmAA

Find aFind a

2.4cm2.4cm36°36°

CC

BB

2.72.7cmcm

AA Find bFind b

104°104°7.4cm7.4cm

CC

AA

3.8cm3.8cm

Find aFind a BB

CC Find cFind c

6cm6cm

BB

32°32°

AA

6cm6cm

CC

1.1.2.2.

3.3.4.4.

Page 36: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

BB

6.6cm6.6cm 44°44°

CC

BB

8.7cm8.7cm

AAFind cFind c

8cm8cm

67°67°

BB AAFind a.Find a.

6cm6cm

CC

PP

44

RR

44

QQ

134°134°

Find qFind q

3cm3cm71°71°

NN

PP

5cm5cm

MM

Find mFind m40°40°11.4cm11.4cm

TT

UU

8.8cm8.8cmFind vFind v

VV

Find aFind a

4.5cm4.5cm

22°22°

AA

CC

1.7cm1.7cm 5.5.

6.6. 7.7.

9.9.

1010..

8.8.

Page 37: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule

The Cosine Rule formula can be rearranged to The Cosine Rule formula can be rearranged to allow us to find the size of an angleallow us to find the size of an angle

bcacb

A2

cos222

This formula is cyclic, depending on the angle This formula is cyclic, depending on the angle to be foundto be found

acbca

B2

cos222

ab

cbaC

2cos

222

Page 38: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Label and identify angles and sides

D = xo d = 11 e = 9 f = 16

Substitute values into the formula.

cos x = 0.75Use cos-1 0∙75 to find x

x = 41∙4o

Example 1 : Calculate the

unknown angle, xo .

Finding Angles Using The Cosine Rule

D E

F

Write the formula for cos Dcos D =e 2 + f 2 - d 2

2ef

cos x =92 + 162 - 112

2 x 9 x 16

de

f

Page 39: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Example 2: Find the unknown angle in the triangle:Example 2: Find the unknown angle in the triangle:

Write down the formula for cos BWrite down the formula for cos B

Label and identify the Label and identify the sides and angle.sides and angle.

B = yB = yoo

aa = 13 = 13bb = 26 = 26

cc = 15 = 15

The negative tells you the angle is obtuse.The negative tells you the angle is obtuse.

yy = 136.3 = 136.3oo

AA

BB

CC

cos B =cos B = a a 22 ++ c c 22 -- b b 22

22acac

Substitute valuesSubstitute values

cos cos yy == - 0- 0∙∙723723

cos cos yy = =131322 ++ 151522 -- 262622

2 2 x x 13 13 x x 1515

Use cosUse cos-1-1 -0 -0∙∙723 to find 723 to find yy

Page 40: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Calculate the unknown angles in the triangles below:Calculate the unknown angles in the triangles below:

(2)(2)

10m10m

7m7m5m5m

bboo

aaoo

(1)(1)

12.7cm12.7cm

7.9cm7.9cm 8.3cm8.3cm

bboo =111.8 =111.8oo

aaoo = 37.3 = 37.3oo

A

B

C

B

C D

Page 41: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Basic ExamplesBasic Examples 2. Finding an angle. 2. Finding an angle.

cos A b2 c2 a2

2bc

5.6 5.6 cmcm

4.1 4.1 cmcm

CC

BB4.7cm4.7cmAA

Find AFind A

5.25.2cmcm

3.23.2cmcm

CC

AA

3.6cm3.6cmBB

CC

6cm6cm

BB

5cm5cm

AA

6cm6cm

2.4cm2.4cm

1.9cm1.9cm

CC

BB

2.72.7cmcm

AA

3.6cm3.6cm

4.7cm4.7cmAA

4cm4cm

CC 10.5cm10.5cm

7.47.4cmcm

CC

AA

5.85.8cmcm

BB

Find BFind B

Find CFind C

Find BFind BFind CFind C Find AFind A

BB

1.1. 2.2.3.3.

4.4.5.5. 6.6.

Page 42: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

MM

4.5cm4.5cm

2.82.8cmcm

LL

4.5cm4.5cmKK

6.6cm6.6cm

6.9cm6.9cm

ZZ

XX

8.7cm8.7cm

YY

8cm8cm7cm7cm

BB AA

Find the Find the largest angle.largest angle.

6cm6cm

CC

PP

44

RR

44

QQ

7.6cm7.6cm

Find all the anglesFind all the angles

3cm3cm

4.5cm4.5cm

NN

QQ

5cm5cm

MM

Find the Find the smallest anglesmallest angle

9.6cm

11.9cmT

U

8.8cm

Find the largest angle

V

7.7.

Find LFind L

Find XFind X

8.8. 9.9.

10.10.

11.11.12.12.

Page 43: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

1.1. Do you know the length of ALL the sides? Do you know the length of ALL the sides?

Cosine Rule or Sine RuleCosine Rule or Sine Rule

How to determine which rule to useHow to determine which rule to use

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

SASSASOROR

If YES to either of the questions then Cosine RuleIf YES to either of the questions then Cosine Rule

Otherwise use the Sine RuleOtherwise use the Sine Rule

Two questionsTwo questions

Page 44: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Calculate the size of Calculate the size of xx in each of these diagrams in each of these diagrams

Page 45: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

The Sine Rule a b cSinA SinB SinC

Application Problems

25o

15 m AD

The angle of elevation of the top of a building

measured from point A is 25o. At point D which is

15m closer to the building, the angle of elevation is

35o Calculate the height of the building.

T

B

Angle TDA =

145o

Angle DTA =

10o

o o

1525 10

TDSin Sin

o15 2536.5

10Sin

TD mSin

35o

36.5

o3536.5TB

Sin

o36.5 25 0. 93TB Sin m

180 – 35 = 145o

180 – 170 = 10o

Page 46: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

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 the statue when the measurements are taken 50 m from its base

50 m

Angle BCA =

70o

Angle ACT = Angle ATC =

110o

65o

53.21 m

B

T

C

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

20o25o

5o

oo 65sin

21.53

5sin

TC

o

o

65sin

5sin21.53 TC

=5.1 m=5.1 m

AC50

20cos o

o20cos

50 AC

m21.53

Page 47: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (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

2b c a

CosAbc

H40 miles

24 miles

B

L

57 miles

A

o20.4A

Bearing Bearing = 90 – 20 = = 90 – 20 = 070070°°

2020°°

NN

40572244057

cos222

A

Page 48: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

2 2 2

2b c a

CosAbc

The Cosine Rule a2 = b2 + c2 – 2bcCosA

An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles.

Find the bearing of Q from point P.

P

670 miles

W

530 miles

Not to Scale

Q

520 miles6705302520670530

cos222

P

PP = 48.7 = 48.7° (49°)° (49°)

Bearing Bearing = 180 + = 180 + 49 = 49 = 229°229°

Page 49: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

COSINE RULE

1. Find the length of the third side of triangle ABC when

i) b = 2, c = 5, A = 60°

ii) a = 2, b = 5, A = 65°

iii) a = 2, c = 5, B = 115°

iv) b = 6, c = 8, A = 50°

2. Find QR in ∆PQR in which PR = 4, PQ = 3, P = 18°

3. Find XY in ∆ XYZ in which YZ = 25, XZ = 30, Z = 162°

4. In ∆ ABC, a = 7, b = 4, C = 53°. Calculate c.

Page 50: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

5. In ∆ ABC, b = 4·2, c = 6·5, A = 24°. Calculate a.

6. In ∆ ABC, a = 1·64, c = 1·64, B = 110°. Calculate b.

7. In ∆ ABC, a = 18·5, b = 22·6, C = 72·3°. Calculate c.

8. In ∆ABC, a = 100, b = 120, C = 15°. Calculate c.

9. In ∆ABC, b = 80, c = 100, A = 123°. Calculate a.

Page 51: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

10. A town B is 20 km due north of town A and a town C is 15 km north-west of A. Calculate the distance between B and C.

11. Two ships leave port together. One sails on a course of 045° at 9 km/h and the other on a course of 090° at 12 km/h.

After 2h 30 min, how far apart will they be?

12. From a point O, the point P is 3 km distant on a bearing of 040° and the point Q is 5 km distant on a bearing of 123°.

What is the distance between P and Q ?

Page 52: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

COSINE RULE (1) Solutions

1. Find the length of the third side of triangle ABC when

i) a2 = 22 + 52 225cos60° = 4 + 25 200·5 = 19

a = 4·36

ii) c2 = 22 + 52 225cos65° = 4 + 25 200·4226

= 20·54763 c = 4·533

iii) b2 = 22 + 52 225cos115° = 4 + 25

225(0·42262 )

= 29 + 8·45237 = 37·45237

b = 6·12

iv) a2 = 62 + 82 268cos50° = 36 + 64 960·64279

= 100 661·7076 = 38·29239

a = 6·188

Page 53: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

2. QR2 = 42 + 32 243cos18° = 16 + 9 240·951057 = 2·17464

QR = 1·4747

3. XY2 = 252 + 302 22530cos162° = 625 + 900 1500(0·951057 )

= 1525 + 1426·5848

= 2951·5848

XY = 54·33

4. c2 = 72 + 42 274cos53°

= 31·2984

c = 5·5945

Page 54: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

5. a2 = 4·22 + 6·52 24·26·5cos24°

= 17·64 + 42·25 554·60·91355 = 10·0104

a = 3·164

6. b2 = 1·642 + 1·642 21·641·64cos110°

= 22·6896 22·6896(0·34202 ) =

7·2190

b = 2·6868

7. c2 = 18·52 + 22·62 218·522·6cos72·3° =

598·778

c = 24·50

8. c2 = 1002 + 1202 2100120cos15° =

1217·780

c = 34·90

Page 55: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

9. a2 = 802 + 1002 280100cos123° = 25114·225 a = 158·47

10. BC2 = 152 + 202 21520cos45° = 225 + 400 6000·7071 = 200·7359

BC = 14·17 km

45°

West

15 km

20 km

North

C

A

B

11. QR2 = 302 + 22·52 23022·5cos45°

= 900 + 506·25 954·59 = 451·6558

QR = 21·25 km NorthQ

P

22·5 km

30 km

45°R

12. PQ2 = 32 + 52 235cos83° = 9 + 25 300·12187

= 30·344PQ = 5·51 km

Page 56: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule

Bearings problems

Page 57: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule- Bearings:-Two Ships.

In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the

distance between the two ships.

1. Ship1 [ 74km, 053° ] ; Ship2 [ 104km, 112° ]

Port

Ship1

Ship2

North

Abccba cos2222

5353°°

112112°°

5959°°

Page 58: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule- Bearings:-Two Ships.

In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the distance between

the two ships.

2. Ship1 [ 56km,021° ] ; Ship2 [ 66km,090° ]

Port

Ship1

Ship2

North

Page 59: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule- Bearings:-Two Ships.

In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the distance between

the two ships.

3. Ship1 [ 83km,060° ] ; Ship2 [ 80km,159° ]

Port

Ship1

Ship2

North

Page 60: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule- Bearings:-Two Ships.

In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the distance between

the two ships.

4. Ship1 [ 50km,090° ] ; Ship2 [ 62km,142° ]

Port Ship1

Ship2

North

Page 61: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule- Bearings:-Two Ships.

In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the distance between

the two ships.

5. Ship1 [ 80km,146° ] ; Ship2 [ 70km,190° ]

Port

Ship1Ship2

North

Page 62: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Cosine Rule- Bearings:-Two Ships.

In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the distance between

the two ships.

6. Ship1 [ 47km,180° ] ; Ship2 [ 54km,235° ]

Port

Ship1

Ship2

North

Page 63: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Bearings Problems

1. Dundee is 84 km due south of Aberdeen. A ship at sea is on a bearing of 078° from Aberdeen and 048° from Dundee. How far is the ship from Dundee and from Aberdeen ?

2. An aeroplane is 240 km from an airport on a bearing of 100° while a helicopter is 165 km from the airport on a bearing of 212°. How far apart are the aircraft ?

Page 64: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Bearings Problems

3. A ship sails 80 km on a bearing of 060° from its home port. It then sails 93 km on a bearing of 134°. How far is it now from its home port ?

4. Glasgow airport is 73 km from Edinburgh airport and lies to the west of Edinburgh airport. The bearing of an aeroplane from Glasgow airport is 040° while its bearing from Edinburgh airport is 300°. How far is the aeroplane from each airport ?

Page 65: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Bearings Problems

5. A ship sails 74 km south from its port to a lighthouse. It then sails 85 km on a bearing of 160°. How far is the ship from the port ?

6. From a port P, ship A is 144 km distant on a bearing of 036° and ship B is 97 km distant on a bearing of 114°. What is the distance between the two ships ?

Page 66: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Bearings Problems

7. A ship sails 93 km on a bearing 054° and then another 108 km on a bearing of 110°. How far is it now from its starting point ?

8. Two radar stations Alpha and Beta pick up signals from an incoming aircraft. Alpha is 40 km east of Beta and picks up the signals on a bearing of 300°. Beta picks up the signals on a bearing of 070°. How far is the aircraft from Alpha and from Beta ?

Page 67: 222 60º 60º 60º 1 60º 2 30º 3333 This triangle will provide exact values for sin, cos and tan 30º and 60º Exact Values Some special values of Sin,

Bearings Problems

9. Aeroplane P is 200 km from an airport on a bearing of 208° while Aeroplane P is 200 km from an airport on a bearing of 208° while aeroplane Q is 170 km from the same airport on a bearing of 094°. aeroplane Q is 170 km from the same airport on a bearing of 094°.

How far apart are the two aeroplanes ?How far apart are the two aeroplanes ?

10. The bearings and distances from Aberdeen of three oil platforms are :10. The bearings and distances from Aberdeen of three oil platforms are :

a) 028° 116 kma) 028° 116 km b) 081° 104 kmb) 081° 104 km c) 138° 97 kmc) 138° 97 km

A supply boat leaves Aberdeen, visits each platform and thenA supply boat leaves Aberdeen, visits each platform and then returns to Aberdeen. returns to Aberdeen.

Find the total length of the journey.Find the total length of the journey.