Unit 4 Trigonometry 1

Solving Triangles by Law of Sines

(I) Review of Right Triangle Trigonometry

Labeling Sides of a Right Triangle

State the primary Trigonometric Ratios

soh cah toa

Applying primary trig ratios to determine an unknown side or unknown angle

Example Solve for x

1 A lighthouse is 168 feet up

from the ocean A fishing ship

is 360 feet from the base of

the cliff Determine the angle

of inclination

Reference angle

360 ft

xdeg

168 ft

Objectives

Review of Right Triangle Trigonometry

Applying the Law of Sines to determine an unknown side length or

angle measure

Unit 4 Trigonometry 2

2 A cable 200 m in length is attached to a telephone pole and forms a 65deg angle on the

pole Determine the distance from the base of the pole to where the cable attaches to

the ground Include a sketch

(II) Solving an unknown side length or angle measure in an acute triangle

by applying the law of sines

Law of Sines

sin 119860

119886=

sin 119861

119887 =

sin 119862

119888

or

119886

sin 119860=

119887

119904119894119899 119861 =

119888

119904119894119899119862

A

a

B

b C

c

Sine of the angle

NOTE

bullThe law of sines is applied when

(i) an angle is known

and

(ii) the side opposite that angle is known

Unit 4 Trigonometry 3

Example 1 Determining an unknown side

Determine which ship is closer

to the ship in distress

Example 2 Determining an unknown angle

Determine the angle at the air traffic control tower

20 km

70deg

60deg

x

y

2 km

6 km 10deg

x

Unit 4 Trigonometry 3

Example 1 Determining an unknown side

Determine which ship is closer

to the ship in distress

Example 2 Determining an unknown angle

Determine the angle at the air traffic control tower

20 km

70deg

60deg

x

y

2 km

6 km 10deg

x

Unit 4 Trigonometry 4

Example 3 Applying the law of sines based on a verbal description

A pulley is suspended from the ceiling by two chains One chain 62 m in length

forms an angle of 55deg with the ceiling Determine the length of the other chain

which forms an angle 30deg with the other ceiling

P139 ndash 141 3a c d 4b 5 7 10 13

P143 4 5a 7

Unit 4 Trigonometry 5

Solving Triangles by Law of Cosines

(I) Law of Cosines

Law of Cosines

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862

2119887119888

Applied when there isnrsquot an angle and side opposite known

A

a

B

b C

c

Side Opposite this angle

Other Two sides

Objectives

Introduction of Law of Cosines

Applying the Law of Cosines to Determine an Unknown Side

Applying the Law of Cosines to Determine an Unknown Angle

Unit 4 Trigonometry 6

(II) Applying the Law of Cosines to determine an unknown side

Example Determine the distance between the two satellites

(II) Applying the Law of Cosines to Determine an Unknown Angle

Example

Two flights depart from Deer Lake

One flight heads to St Anthony and

the other towards St Johnrsquos

Determine the angle formed from

the departing flights at Deer Lake

36 km 40 km

25deg

x

374 km

241 km

489 km

x

Unit 4 Trigonometry 5

Solving Triangles by Law of Cosines

(I) Law of Cosines

Law of Cosines

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

Or rearranging for cos of the angle cos 119860 = 1198872+ 1198882minus 1198862

2119887119888

Applied when there isnrsquot an angle and side opposite known

A

a

B

b C

c

Side Opposite this angle

Other Two sides

Objectives

Introduction of Law of Cosines

Applying the Law of Cosines to Determine an Unknown Side

Applying the Law of Cosines to Determine an Unknown Angle

Unit 4 Trigonometry 6

(II) Applying the Law of Cosines to determine an unknown side

Example Determine the distance between the two satellites

(II) Applying the Law of Cosines to Determine an Unknown Angle

Example

Two flights depart from Deer Lake

One flight heads to St Anthony and

the other towards St Johnrsquos

Determine the angle formed from

the departing flights at Deer Lake

36 km 40 km

25deg

x

374 km

241 km

489 km

x

Unit 4 Trigonometry 6

(II) Applying the Law of Cosines to determine an unknown side

Example Determine the distance between the two satellites

(II) Applying the Law of Cosines to Determine an Unknown Angle

Example

Two flights depart from Deer Lake

One flight heads to St Anthony and

the other towards St Johnrsquos

Determine the angle formed from

the departing flights at Deer Lake

36 km 40 km

25deg

x

374 km

241 km

489 km

x

Unit 4 Trigonometry 7

(III) Applying the Law of Cosines to solve a word problem

Example A ship passing an island establishes by sonar a distance of 35 km from

the ship to one end of the island and 51 km to the other end The angle

from the ship contained between the tips of the island is 115deg Determine

the length of the island

P151 ndash 153 2 3 4 5 8 9 12

Unit 4 Trigonometry 8

Applying the Law of SinesCosines to Solve Problems

(I) Remember

Primary Trig Ratios

Consider a reference right angle

triangle

The three primary trigonometry ratios are

Law of Sines

sin 119860

119886=

sin 119861

119887 =

sin 119862

119888

or

119886

sin 119860=

119887

119904119894119899 119861 =

119888

119904119894119899119862

Law of Cosines

NOTE

The law of sines is applied when

(i) an angle is known

and

(ii) the side opposite that angle is known

A

a

B

b C

c Side opposite the angle

Sine of the angle

Objectives

Review of Trig Ratios Law of Sines amp Law of Cosines

Applying Trig to Solve Triangle Problems

Unit 4 Trigonometry 9

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in

more than one triangle to solve a problem

Example A ship navigating the coast knows from a map that the height of the cliff is

150 m high From the ships position the angle of elevation to the base of

the lighthouse is 10deg and the angle of elevation to the top of the lighthouse

is 15deg Determine the height of the lighthouse

A

a

B

b C

c

Applied when there isnrsquot an angle AND side opposite known

Side Opposite this angle

Other Two sides

150 m

15deg 10deg

h

Unit 4 Trigonometry 8

Applying the Law of SinesCosines to Solve Problems

(I) Remember

Primary Trig Ratios

Consider a reference right angle

triangle

The three primary trigonometry ratios are

Law of Sines

sin 119860

119886=

sin 119861

119887 =

sin 119862

119888

or

119886

sin 119860=

119887

119904119894119899 119861 =

119888

119904119894119899119862

Law of Cosines

NOTE

The law of sines is applied when

(i) an angle is known

and

(ii) the side opposite that angle is known

A

a

B

b C

c Side opposite the angle

Sine of the angle

Objectives

Review of Trig Ratios Law of Sines amp Law of Cosines

Applying Trig to Solve Triangle Problems

Unit 4 Trigonometry 9

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in

more than one triangle to solve a problem

Example A ship navigating the coast knows from a map that the height of the cliff is

150 m high From the ships position the angle of elevation to the base of

the lighthouse is 10deg and the angle of elevation to the top of the lighthouse

is 15deg Determine the height of the lighthouse

A

a

B

b C

c

Applied when there isnrsquot an angle AND side opposite known

Side Opposite this angle

Other Two sides

150 m

15deg 10deg

h

Unit 4 Trigonometry 9

___2 = ___2 + ___2 ndash 2(___)(___) cos ___

(II) Applying Primary Trig Ratios Law of Sines or Law of Cosines in

more than one triangle to solve a problem

Example A ship navigating the coast knows from a map that the height of the cliff is

150 m high From the ships position the angle of elevation to the base of

the lighthouse is 10deg and the angle of elevation to the top of the lighthouse

is 15deg Determine the height of the lighthouse

A

a

B

b C

c

Applied when there isnrsquot an angle AND side opposite known

Side Opposite this angle

Other Two sides

150 m

15deg 10deg

h

Unit 4 Trigonometry 10

Example Determine the value of x

Example Two ships 2 km apart using sonar have located a sunken ship One ship

emits its sonar at an angle of depression of 30deg and the other ship at 80deg

Determine the distance of each ship from the sunken ship

x 32ordm

55ordm

25ordm

10

9

P161 4 5b 6 7 9 11a 13

Unit 4 Trigonometry 11

TRIGONOMETRY TEST REVIEW SHEET

Formulae

CSin

c

BSin

b

ASin

a

c

CSin

b

BSin

a

ASin

AbcCoscba 2222 bc

acbACos

2

222

1 Which represents the correct trigonometric equation for the diagram below

(A) 17

25cosx

(B) x

1725cos

(C) 17

25sinx

(D) x

1725sin

2 For which triangle would you use the Law of Sines to determine the missing side of x

(A) (B)

(C) (D)

105ordm

35ordm

7

x

105ordm 7

x

10

25

35ordm

x

10

x

7

20

15

x

25ordm

17

Unit 4 Trigonometry 12

3 Solve for x

10

20

30 SinSin

x

(A) 1000 (B) 1012 (C) 5926 (D) 6392

4 Solve for x

40

2510

SinxSin

(A) 588deg (B) 1611deg (C) 1511deg (D) 7902

5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm

(A) 624 (B) 436 (C) 342 (D) 450

6 Solve for x 60

162536 xCos

(A) 46deg (B) 54deg (C) 41deg (D) 60deg

7 Which represents the appropriate equation to solve for x

(A)

55sin

28

45sin

x

(B)

45sin

28

55sin

x

(C)

45sin

28

80sin

x

(D)

55sin

28

80sin

x

x

28

55deg 45deg

Unit 4 Trigonometry 13

8 Which represents the appropriate equation to solve for x

(A) 144

496481 xCos (B)

112

814964 xCos

(C) 126

644981 xCos (D)

126

644981 xCos

9 Determine the value of x

(a) (b)

(c) (d)

x

9

8 7

x

12

x

65ordm 45deg

x 8

34ordm 42ordm 12

10

x 30ordm

55ordm

25ordm

12

9

11 50deg

9 32ordm

9

Unit 4 Trigonometry 14

10 Find the distance between the two police officers

11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an

angle of 62deg with the ceiling Determine the angle the second chain makes with the

ceiling if it has a length of 10 m

12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away

from the store If G = 63deg and H = 67deg then how far apart are George and Henry

before traveling to the store

S

G H

Unit 4 Trigonometry 15

13 Determine the distance d between the two satellites

14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one

end of the island and 9 km to the other end of the island The angle formed at the ship

from the sonar is 74deg Determine the length of the island

15 What is the angle between the two planes as recorded at the radar

d

40 km 42 km

35deg

80 km

100 km

120 km

SOLUTIONS

1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C

9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km

13 2473 km 14 1026 km 15 8282 km

Unit 4 Trigonometry 12

3 Solve for x

10

20

30 SinSin

x

(A) 1000 (B) 1012 (C) 5926 (D) 6392

4 Solve for x

40

2510

SinxSin

(A) 588deg (B) 1611deg (C) 1511deg (D) 7902

5 Solve for x x2 = 22 + 52 ndash 20 Cos 60ordm

(A) 624 (B) 436 (C) 342 (D) 450

6 Solve for x 60

162536 xCos

(A) 46deg (B) 54deg (C) 41deg (D) 60deg

7 Which represents the appropriate equation to solve for x

(A)

55sin

28

45sin

x

(B)

45sin

28

55sin

x

(C)

45sin

28

80sin

x

(D)

55sin

28

80sin

x

x

28

55deg 45deg

Unit 4 Trigonometry 13

8 Which represents the appropriate equation to solve for x

(A) 144

496481 xCos (B)

112

814964 xCos

(C) 126

644981 xCos (D)

126

644981 xCos

9 Determine the value of x

(a) (b)

(c) (d)

x

9

8 7

x

12

x

65ordm 45deg

x 8

34ordm 42ordm 12

10

x 30ordm

55ordm

25ordm

12

9

11 50deg

9 32ordm

9

Unit 4 Trigonometry 14

10 Find the distance between the two police officers

11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an

angle of 62deg with the ceiling Determine the angle the second chain makes with the

ceiling if it has a length of 10 m

12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away

from the store If G = 63deg and H = 67deg then how far apart are George and Henry

before traveling to the store

S

G H

Unit 4 Trigonometry 15

13 Determine the distance d between the two satellites

14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one

end of the island and 9 km to the other end of the island The angle formed at the ship

from the sonar is 74deg Determine the length of the island

15 What is the angle between the two planes as recorded at the radar

d

40 km 42 km

35deg

80 km

100 km

120 km

SOLUTIONS

1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C

9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km

13 2473 km 14 1026 km 15 8282 km

Unit 4 Trigonometry 13

8 Which represents the appropriate equation to solve for x

(A) 144

496481 xCos (B)

112

814964 xCos

(C) 126

644981 xCos (D)

126

644981 xCos

9 Determine the value of x

(a) (b)

(c) (d)

x

9

8 7

x

12

x

65ordm 45deg

x 8

34ordm 42ordm 12

10

x 30ordm

55ordm

25ordm

12

9

11 50deg

9 32ordm

9

Unit 4 Trigonometry 14

10 Find the distance between the two police officers

11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an

angle of 62deg with the ceiling Determine the angle the second chain makes with the

ceiling if it has a length of 10 m

12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away

from the store If G = 63deg and H = 67deg then how far apart are George and Henry

before traveling to the store

S

G H

Unit 4 Trigonometry 15

13 Determine the distance d between the two satellites

14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one

end of the island and 9 km to the other end of the island The angle formed at the ship

from the sonar is 74deg Determine the length of the island

15 What is the angle between the two planes as recorded at the radar

d

40 km 42 km

35deg

80 km

100 km

120 km

SOLUTIONS

1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C

9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km

13 2473 km 14 1026 km 15 8282 km

Unit 4 Trigonometry 14

10 Find the distance between the two police officers

11 A pulley is suspended from the ceiling by two chains One chain 7 m in length forms an

angle of 62deg with the ceiling Determine the angle the second chain makes with the

ceiling if it has a length of 10 m

12 George (G) and Henry (H) are getting ready to walk to the store (S) and George is 56 km away

from the store If G = 63deg and H = 67deg then how far apart are George and Henry

before traveling to the store

S

G H

Unit 4 Trigonometry 15

13 Determine the distance d between the two satellites

14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one

end of the island and 9 km to the other end of the island The angle formed at the ship

from the sonar is 74deg Determine the length of the island

15 What is the angle between the two planes as recorded at the radar

d

40 km 42 km

35deg

80 km

100 km

120 km

SOLUTIONS

1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C

9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km

13 2473 km 14 1026 km 15 8282 km

Unit 4 Trigonometry 15

13 Determine the distance d between the two satellites

14 A ship passing an island establishes by sonar a distance of 8 km from the ship to one

end of the island and 9 km to the other end of the island The angle formed at the ship

from the sonar is 74deg Determine the length of the island

15 What is the angle between the two planes as recorded at the radar

d

40 km 42 km

35deg

80 km

100 km

120 km

SOLUTIONS

1 B 2 A 3 576 4 149deg 5 B 6 C 7 D 8 C

9(a) 1839deg (b) 1095 (c) 1194 (d) 3449deg 10 1887 m 11 3817deg 12 466 km

13 2473 km 14 1026 km 15 8282 km