Chapter 2

Trigonometry

§ 2.1 The Tangent Ratio

tanopp

Xadj

The Tangent Ratio

TOA

x

Hypotenuse

(h)

Opposite(o)

Adjacent(a)

xHypotenuse

(h)

Opposite(o)

Adjacent(a)

Hypotenuse

OppositeAdjacent

Example #1Determine the tangent ratio for the following.

Hypotenuse

(h)

Opposite(o)

Adjacent(a)

Vtanadj

opp

4

3tan V

Qtanadj

opp

3

4tan Q

G and J Example #2Determine the measures of to the nearest tenth of a degree.

Hypotenuse

(h)

Opposite(o)

Adjacent(a)

Gtan adj

opp

G

5

4tan 1

Jtan adj

opp

J3.51

5

4

G7.38

4

5

J

4

5tan 1

Example #3The latitude of Fort Smith, NWT, is approximately 60o. Determine whether this design for a solar panel is the best for Fort Smith. Justify your answer.

h

ao

Adj

Opp

12

9

x =

36.8699°

The best angle of inclination for the solar panel is the same as the latitude

(60°)

xtan

x

12

9tan 1

x

This is not the best design for the solar panel because it is not equal to the

latitude of Fort Smith (60 °)

Example #4A 10-ft. ladder leans against the side of a building that is 7 ft. tall.What angle, to the nearest degree, does the ladder make with the ground?

h

a

O

107

Adj

Opp

51

7xtan

x

a2 + b2 = c2

a2 + o2 = h2

a2 = 51a2 + 49 = 100

a = √51

a2 + 72 = 102

√51

on your calculator:

x = tan-1(7÷√(51))

<x = 44.43

<x = 44°

§ 2.2 Using the Tangent Ratio to Calculate

Lengths

tanopp

Xadj

The Tangent Ratio

TOA

x

Hypotenuse

(h)

Opposite(o)

Adjacent(a)

xHypotenuse

(h)

Opposite(o)

Adjacent(a)

Hypotenuse

OppositeAdjacent

Example #1Determine the length of AB to the nearest tenth of a centimeter

Hypotenuse

(h)

Opposite(o)

Adjacent(a)

30tanadj

opp

1030tan

AB

ABcm8.5

1010

AB 30tan10

Example #2 (first way)Determine the length of EF to the nearest tenth of a centimeter.

hO

a

20tanadj

opp

EF

5.320tan

cmEF 6.9

EFEF

5.320tan EF20tan20tan

20tan

5.3EF

Example #2 (second way)Determine the length of EF to the nearest tenth of a centimeter.

h

O

a

70tanadj

opp

5.370tan

EF

EFcm6.9

5.35.3

EF 70tan5.3

Angles in a triangle add up to

90°

70°

Example #3A searchlight beam shines vertically on a cloud. At a horizontal distance of 250m from the searchlight, the angle between the ground and the line of sight to the cloud is 75o. Determine the height of the cloud to the nearest metre.

250 m

hO

a

75tanadj

opp

25075tan

opp

oppm933

250250

opp 75tan250

The cloud is 933 metres above the ground

§ 2.4 The Sine and Cosine Ratios

The Sine & Cosine Ratios

x

Hypotenuse

(h)

Opposite(o)

Adjacent(a)

x

Hypotenuse

(h)Opposite

(o)

Adjacent(a)

Hypotenuse

OppositeAdjacent

sinopp

hyp cos

adj

hyp

SOH CAH

Example #1a) In identify the side opposite , the side adjacent to and the hypotenuse.

DEF D D

b) Determine using sin and cos to the nearest hundredthDHypotenus

e

Opposite

Adjacent

Dsinhyp

opp

13

5sin D

62.22D

Dcoshyp

adj

13

12cos D

D

13

5sin 1

62.22D

D

13

12cos 1

Example #2Determine the measures of and to the nearest tenth of a degree.

Hypotenuse

OppositeAdjacentGcoshyp

adj

14

6cos G

G6.64

Hsinhyp

opp

14

6sin H

H4.25

G H

G

14

6cos 1

H

14

6sin 1

Example #3A water bomber is flying at an altitude of 5000ft. The plane’s radar shows that it is 8000 ft. from the target site. What is the angle of elevation of the plane measured from the target site to the nearest degree?

5000 ft

8000 ft

x

hO

a

xsinhyp

opp

8000

5000sin x

x39

x

80000

5000sin 1

§ 2.5 Using Sine and Cosine Ratios To Calculate

Lengths

h

a Ocos(50) = Hyp

Adj

2.5

BC

2.5

50cosBC

5.2 x cos(50) = BC

3.34249557 = BC

Example #1Determine the length of BC to the nearest tenth of a centimetre.

2.52.5

BC = 3.3 cm

What trig ratio uses adjacent and hypotenuse?

h

a

O

sin(55) = Hyp

Opp

DE

8.6

DE

8.655sin

Example #2Determine the length of DE to the nearest tenth of a centimetre.

DEDE

What trig ratio uses opposite and hypotenuse?

cmDE 3.8

8.6)55sin( DE)55sin()55sin(

)55sin(

8.6DE

Example #3A surveyor made the measurements shown in the diagram. How could the surveyor determine the distance from the transit to the survey pole to the nearest hundredth of a meter?

h

a

O

cos(67.3) = Hyp

AdjH

86.20

H

86.203.67cos HH

What trig ratio uses adjacent and hypotenuse?

mH 05.54

86.20)3.67cos( H)3.67cos()3.67cos(

)3.67cos(

86.20H

§ 2.6 Applying the Trigonometric Ratios

Example #1Solve . Give the measures to the nearest tenth.

h

ao

Adj

Opp0.6

0.10xtan

a2 + b2 = c2

6.02 + 10.02 = c2

136 = c2

√136 = c

36 + 100 = c2 on your calculator:

x = tan-1(10÷6)<x = 59.0°

Solve means find the measures of all the sides and

angles.

11.7 cm

11.7 = c

59.0°

z

Angles in a triangle add up to

90°

59.0° + 90° + z = 180°149.0° + z = 180°-149.0° -149.0°

z = 31.0°

31.0°

Example #2Solve this triangle. Give the measures to the nearest tenth where necessary.

h

a

o

Adj

Opp0.5

Opp)65tan(

a2 + b2 = c2

10.72 + 5.02 = c2

139.49 = c2

√139.49 = c

114.49 + 25 = c2

Solve means find the measures of all the sides and angles.

10.7 cm

11.8 = c

11.8 cmz

Angles in a triangle add up to

90°

25° + 90° + z = 180°115° + z = 180°-115° -115°

z = 65°

65°

5.0 x tan(65) = Opp

0.50.5

Opp = 10.7 cm

0.5)65tan(Opp

Example #3A helicopter leaves its base, and flies 35km due west to pick up a sick person. It then flies 58km due north to a hospital.

a) When the helicopter is at the hospital, how far is it from its base to the nearest kilometre?

35 Km

58 Km

x

a2 + b2 = c2

582 + 352 = x2

4589 = x2

√4589 = x

3364 + 1225 = x2

67.7421= x68 = x

Example #3A helicopter leaves its base, and flies 35km due west to pick up a sick person. It then flies 58km due north to a hospital.

b) When the helicopter is at the hospital, what is the measure of the angle between the path it took due north and the path it will take to return directly to its base? Write the angel to the nearest degree.

35 Km

58 Km

68 Km

xh

O

a

Adj

Oppxtan

Hyp

Oppxsin

Hyp

Adjxcos

58

35

68

35

68

58

on your calculator:

x = sin-1(35÷68)<x = 31°

x = cos-1(58÷68)<x = 31°

x = tan-1(35÷58)<x = 31°

§ 2.7 Solving Problems Involving More than One

Right Triangle

7.5)26cos(

x

ha

O

ha

O

5.7 cm

Example #1Calculate the length of CD to the nearest tenth of a centimetre.

x

What trig ratio uses opposite and hypotenuse?

sin(47) = Hyp

Opp

h

2.4

hh

cmh 7.5

2.4)47sin( h)47sin()47sin(

)47sin(

2.4h

h

2.4)47sin(

What trig ratio uses adjacent and hypotenuse?

cos(26) = Hyp

Adj

7.5

x

5.7 x cos(26) = x

7.57.5

x = 5.1 cm

Example #2From the top of a 20-m high building, a surveyor measured the angle of elevation of the top of another building and the angle of depression of the base of the building. The surveyor sketched this plan of her measurements. Determine the height of the taller building to the nearest tenth of a metre.

20 ha

o

What trig ratio uses opposite and adjacent?tan(15) =

Adj

Oppa

20

ma 6.74

)15tan(

20a

a

20)15tan(

74.6 cm

h

a

o tan(30) = Adj

Opp6.74

Opp

mOpp 1.43

Opp )30tan(6.74

6.74)30tan(Opp

Height = 20 + 43.1 =

63.1 m

Example #3In the given diagram find . Round your answer to the nearest tenth.

HJK

x

a2 + b2 = c2

102 + 32 = c2

109 = c2

√109 = c

100 + 9 = c2

√109

h

ao

What trig ratio uses opposite and adjacent?

tan(x) = Adj

Opp5

109

4.64HJK

x

5

109tan 1

5

109)tan( x

Example #4From the top of a 90-ft. observation tower, a fire ranger observes one fire due west of the tower at an angle of depression of 5o, and another fire due south of the tower at an angle of depression of 2o. How far apart are the fires to the nearest foot?

ha

o

What trig ratio uses opposite and adjacent?

tan(85) = Adj

Opp90

Opp

Opp7.1028

Opp )85tan(90

90)85tan(Opp

1028.7 ft

ha

o

tan(88) = Adj

Opp90

Opp

Opp3.2577

Opp )88tan(90

90)88tan(Opp

2577.3 ft

a2 + b2 = c2

1028.72 + 2577.32 = c2

7700698.98 = c2

√ 7700698.98 = c

1058223.69 + 6642475.29= c2

2775 ft = c