Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 5 Trigonometric Functions 5.8 Part 2 Applications of Trigonometric Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Solve a right triangle. Solve problems involving bearings. Model simple harmonic motion.

Trigonometry and Bearings

Bearings 2 Types True Bearings Compass Bearings

True Bearings In true bearing, all bearings are measured from the north in a clockwise direction. A three-digit integer is used to represent the integral part of true bearing. The true bearing of B from A is 035. The true bearing of A from B is 215.

Compass Bearings In compass bearing, all bearings are measured from the north or from the south in an acute angle to the east or to the west. The compass bearing of B from A is N35E. The compass bearing of A from B is S35W.

Compass Bearings In navigation and surveying problems, the term bearing is used to specify the location of one point relative to another. The compass bearing from point O to point P is the acute angle, measured in degrees, between ray OP and a north-south line. The north-south line and the east-west line intersect at right angles. Each bearing has three parts: a letter (N or S), the measure of an acute angle, and a letter (E or W).

Compass Bearings (continued) If the acute angle is measured from the north side of the north-south line, then we write N first. Second, we write the measure of the acute angle. If the acute angle is measured on the east side of the north-south line, then we write E last. N 40° E

Compass Bearings (continued) If the acute angle is measured from the north side of the north-south line, then we write N first. Second, we write the measure of the acute angle. If the acute angle is measured on the west side of the north-south line, then we write W last. N 65° W

Compass Bearings (continued) If the acute angle is measured from the south side of the north-south line, then we write S first. Second, we write the measure of the acute angle. If the acute angle is measured on the east side of the north-south line, then we write E last. S 70° E

Compass Bearing - Examples Name the Bearing

Your Turn Name the Bearing N 13° 2. 26° 4. N S 45° 3. S 72° 1.

Examples: A bearing measures the acute angle that a path or line of sight makes with a fixed north-south line. N S 70° E W N S 35° E W

Your Turn: Draw a bearing of: N800W S300E N S E W N S E W

Example: Understanding Bearings Use the figure to find the bearing from O to D. Point D is located to the south and to the east of the north-south line. The bearing from O to D is S25°E.

Your Turn Use the figure to find the bearing from O to C.

Bearing Problems Caution A correctly labeled sketch is crucial when solving bearing applications. Some of the necessary information is often not directly stated in the problem and can be determined only from the sketch.

20 nmph for 2 hrs 40 nm Bearing: N 78.181o W Example – Finding Directions Using Bearings A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N 54o W. Find the ship’s bearing and distance from the port of departure at 3 P.M. d 20sin(36o) a 20 nmph for 1 hr 54o 78.181o 20 nm 36o θ 20cos(36o) b 20 nmph for 2 hrs 40 nm Bearing: N 78.181o W

A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N 54o W. Find the ship’s bearing and distance from the port of departure at 3 P.M. d a 20 nm 54o 20sin(36o) 78.181o 20 nmph for 1 hr 36o θ 20cos(36o) b 20 nmph for 2 hrs 40 nm Bearing: N 78.181o W

Example: Two lookout towers spot a fire at the same time Example: Two lookout towers spot a fire at the same time. Tower B is Northeast of Tower A. The bearing of the fire from tower A is N 33o E and is calculated to be 45 km from the tower. The bearing of the fire from tower B is N 63o W and is calculated to be 72 km from the tower. Find the distance between the two towers and the bearing from tower A to tower B. a c 63o 72 d B b 45 33o s b – d 45cos(330) – 72sin(630) A a + c 45sin(330) + 72sin(630)

B A 72 45 s a c 63o d b 88.805 km 33o b – d 45cos(330) – 72sin(630) 45sin(330) + 72sin(630) a + c

B A 72 45 s a c 63o d b 88.805 km 33o b – d 45cos(330) – 72sin(630) θ 45sin(330) + 72sin(630) a + c

Your Turn: Radar stations A and B are on an east-west line, 3.7 km apart. Station A detects a plane at C, on a bearing of 61°. Station B simultaneously detects the same plane, on a bearing of 331°. Find the distance from A to C.

Your Turn: The bearing from A to C is S 52° E. The bearing from A to B is N 84° E. The bearing from B to C is S 38° W. A plane flying at 250 mph takes 2.4 hours to go from A to B. Find the distance from A to C.