Sec 9.5 Applications of Trigonometry to

Navigation and Surveying

Which direction?• In basic Trig… standard position:

Start on the x-axis.Counter clockwise

Which direction?

• Navigation… used by ships, planes etc.

Start on the y-axis.ClockwiseGiven using 3 digits

9.5 Applications of Trigonometry to Navigation and Surveying

Objective

.To use trigonometry to solve navigation and surveying problems

The course of a ship or plane is the , measured clockwise,

from the north direction to the direction of the ship or plane.

9.5 Applications of Trigonometry to Navigation and Surveying

Objective

.To use trigonometry to solve navigation and surveying problems

The course of a ship or plane is the , measured clockwise,

from the north direction to the direction of the ship or plane.

Objective

.To use trigonometry to solve navigation and surveying problems

The course of a ship or plane is the , measured clockwise,

from the north direction to the direction of the ship or plane.

Bearing of from 240A B Bearing of from 060B A

Example 1. A ship proceeds on a course of 300º for 2 hours at a speed of 15 knots (1 knot = 1 nautical mile per hour). Then it changes course to 230º, continuing at 15 knots for 3 more hours. At that time, how far is the ship from its starting point?

Always measure clockwise

N

30

N

45

x

606050

Law of

Cosines

2 2 2

1 2 1 22 cosOPP ADJ ADJ ADJ ADJ

A ship proceeds on a course of 300 for 2 hours at a

speed of 15 knots (1 knot = 1 nautical mile per hour).

Then it changes course to 230 , continuing at 15 knots

for 3 more hours. At that time, how far

is the ship from

the starting point?

N

30

N

45

x

606050

Law of

Cosines

2 2 2

1 2 1 22 cosOPP ADJ ADJ ADJ ADJ

2 2 230 45 2 30 45 cos110

62.0 nautical miles

x

x

Which direction?• In surveying, a compass reading is usually

given as an acute angle from the north-south line toward the east or west line.

a) Start on the y-axis.

b) Clockwisec) The angle is

always acute.

In surveying, a compass reading

is usually given as an acute

from the north-south line toward

the east or west.

Navigation Surveying

NE Sandy and NE 40th meet at approx 58 degree angle.

Give directions from the Formation Area to the disband Area.a) Using

navigation system.

b) Using the survey method.

Basic Hints and rules• Make a diagram… give yourself drawing

space all around the diagram.• Although drawing to scale might be hard,

come as close to a scale as possible.• Write all the given information on your

diagram.

Basic Hints and rules• Include a lightly drawn x and y axes at each

point.• Find as many angles and sides as you can.• Apply as many geometry rules as you can.

Camping:

Give direction from the camp site to each of 4 points of interest:

a) The river

b) The lake

c) To the tower

d) The hill

Generate clarifying

questions!

Example 2. Very often a plot of land is taxed according to its area. Sketch the plot of land described. Then find its area.

From a granite post, proceed 195 ft east along Tasker Hill Road, then along a bearing of S32ºE for 260 ft, then along a bearing of S68ºW for 385 ft, and finally along a line back to the granite post.

With these types of problems, a careful diagram is essential. Next slide will demonstrate all the steps.Make sure to have your geometric tools and math wits about you!

GTakser Hill Road

195 ft Q

32º260 ft

R68º

385 ft

S

90°90°

32°80°

GTakser Hill Road

195 ft Q

32º260 ft

R68º

385 ft

S

90°90°

32°

80°

area of ∆ GQS=35,561.49area of ∆ GRS=49,289.63

TOTAL: 84,851.12

A plot of land is taxed according to its area. Sketch the plot

of land described, then find its area.

o

From a granite post, proceed 195 ft east along Tasker Hill Road,

then along a bearing of S32 E for 260 ft, then along a bearing of

S68 W for 385 ft, and finally along a line back to the granite post.

195 ftN

260 ft32

N

68

385 ft

2

2 2 2

1195 260 sin122 21,500 ft

2

195 260 2 195 260 cos122

399 ft

K

x

x

399 ft

2

2

sin sin122

195 399

24.5 55.5

1399 385 sin 55.5 63,300

84,

ft

8

2

0 ft 0

K

Total Area

x 24.5

55.5

90

1sin

2k ab C

500 m NE; 300 m E; 200 m S15 ; BackE

15. N

500 m

300 m

200 m

15

401d

2 2 2300 200 2 300 200 cos105

401 m

d

d

1

2

1300 200 sin105

2

29,000 m

K

K

N

N45

sin sin105

200 401

28.8

135 28.8 106.2m

2

2

1500 401 sin106.2

2

96,000 m

K

K

2125,000 mArea

90

Presentation

45

16.

260 m

182d

2 2 2240 280 2 240 280 cos 40

182 m

d

d

2

1240 280 sin 40

2

21,600 m

K

K

N

2

1260 182 sin 53

2

18,900 m

K

K

240,500 mArea

240 m40

280 m

sin sin 40

240 182

58.0

53

5882

Presentation260 m SW; 240 m S; 280 m N40 E; Back

oProceed S78 W for 250 m.

Then S15 E for 180 m.

Then N78 E.

Then N30 E to the

starting point.

N

250 m78

N

15

180 m

N78

1d

N

2d30

78

871K

30

48

3d

2 2 23

3

180 250 2 180 250 cos87

300.32 m

d

d

sin sin87

180 300.32

36.8

11.2

132

36.8

1

1

300.32

sin11.2 sin13278.49 m

d

d

2K

21 2 29,500 mK K

Presentation

• click here for a sample test

Homework:

• Sec 9.5 written exercises

• 7-13 odds; 15, 16, 17

In class Exit Slip:• Page 353 Problem

#17.• Only full solutions

will be considered.

If you were absent, see Navi for make up Exit Slip.

Applications of Trig to Navigation and SurveyingThe course of a ship or plane is the angle, measured clockwise, from the north direction to the direction of the ship or plane.

As shown, the compass bearing of one location from another is measured in the same way. Note that compass bearings and courses are given with three digits, such as 060º rather than 60º

G

Example 1. A ship proceeds on a course of 300º for 2 hours at a speed of 15 knots (1 knot = 1 nautical mile per hour). Then it changes course to 230º, continuing at 15 knots for 3 more hours. At that time, how far is the ship from its starting point?

5060

60

Example 1. A ship proceeds on a course of 300º for 2 hours at a speed of 15 knots (1 knot = 1 nautical mile per hour). Then it changes course to 230º, continuing at 15 knots for 3 more hours. At that time, how far is the ship from its starting point?

5060

60

Example 2. Very often a plot of land is taxed according to its area. Sketch the plot of land described. Then find its area.

From a granite post, proceed 195 ft east along Tasker Hill Road, then along a bearing of S32ºE for 260 ft, then along a bearing of S68ºW for 385 ft, and finally along a line back to the granite post.

399

24.5

55.5

Example 2. Very often a plot of land is taxed according to its area. Sketch the plot of land described. Then find its area.

From a granite post, proceed 195 ft east along Tasker Hill Road, then along a bearing of S32ºE for 260 ft, then along a bearing of S68ºW for 385 ft, and finally along a line back to the granite post.

399

24.5

55.5