Sec 9.5. Applications of Trigonometry to Navigation and Surveying

Sec 9.5 Applications of Trigonometry to Navigation and Surveying

Which direction? In basic Trig standard position:

Which direction? Navigation used by ships, planes etc.

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. Bearing of B from A 060 Bearing of Afrom B 240 The course of a ship or plane is the, measured clockwise, from the north direction to the direction of the ship or plane.

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

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? Law of Cosines 45 x 2 2 2 N N 50 60 30 OPP ADJ ADJ 2 ADJ ADJ cos 60 1 2 1 2

Law of Cosines 45 50 N 60 30 60 x N 2 2 2 OPP ADJ ADJ 2 ADJ ADJ cos x 2 2 2 1 2 1 2 30 45 2 30 45 cos110 x 62.0 nautical miles

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.

In surveying, a compass reading is usually given as an acute from the north-south line toward the east or west.

Navigation

NE Sandy and NE 40 th 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

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!

S Takser Hill Road G 195 ft Q 90 90 32º 260 ft 385 ft 32 80 R 68º

Takser Hill Road G 195 ft Q 90 90 32º 260 ft 385 ft area of GQS=35,561.49 area of GRS=49,289.63 32 TOTAL: 84,851.12 80 R 68º S

A plot of land is taxed according to its area. Sketch the plot of land described, then find its area. 2 From a granite post, proceed 195 ft east along Tasker Hill Road, o 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. N 195 ft 1 2 K 195 260 sin122 21,500 ft 32 2 90 2 2 2 x 195 260 2 260 ft 195 260 cos122 x 399 ft 399 x 24.5 ft N 68 sin sin122 55.5 195 399 24.5 55.5 1 k absin C 385 ft 1 K 399 385 sin 55.5 63,300 2 2 Total Area 84, 800 ft 2 ft

500 m NE; 300 m E; 200 m S15 E; Back Presentation 45 15. N K K d d N 500 m 1 2 2 2 N 300 m d 401 1 300 200 sin105 2 2 29,000 m 15 200 m 300 200 2 300 200 cos105 401 m 90 K K sin sin105 200 401 28.8 m 135 28.8 106.2 2 1 500 401 sin106.2 2 2 96,000 m 2 Area 125,000 m

260 m SW; 240 m S; 280 m N40 E; Back 16. Presentation N 1 45 260 m K 260 182 sin 53 2 53 sin sin 40 2 K 18,900 m d 182 240 182 82 58 58.0 240 m 280 m 40 d d 2 2 2 240 280 2 240 280 cos 40 182 m K K 1 240 280 sin 40 2 2 21,600 m 2 Area 40,500 m

Presentation o Proceed S78 W for 250 m. Then S15 E for 180 m. Then N78 E. Then N30 E to the starting point. sin sin 87 180 300.32 36.8 2 2 2 d d 3 3 15 N 78 180 250 2 180 250 cos87 300.32 m 87 K1 180 m 250 m d 3 N 36.8 78 d1 K 2 48 132 1 11.2 30 N 78 30 d 2 N d1 300.32 sin11.2 sin132 d 78.49 m K1 K2 29,500 m 2

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 Surveying The 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

50 60 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?