Warm-up 10/29

4.8 Trigonometric Applications and Models 2014 Objectives: Use right triangles to solve real-life problems. Use directional bearings to solve real-life problems. Use harmonic motion to solve real-life problems.

4.8 Applications and Models Terminology Angle of depression – angle from the horizontal downward to an object. Angle of elevation – angle from the horizontal upward to an object. Horizontal Observer Angle of elevation Object Horizontal Observer Angle of depression Object Talk about radian mode vs degree mode Talk about finding more than one answer on the unit circle Precalculus 4.8 Applications and Models

Example Solve the right triangle for all missing sides and angles.

You Try Solve the right triangle for all missing sides and angles.

Example – Solving Rt. Triangles At a point 200 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 35°, and the angle of elevation to the top of the smokestack is 53°. Find the height of the smokestack. Place triangle based on properties of inverse function Precalculus 4.8 Applications and Models

You try: A swimming pool is 20 meters long and 12 meters wide. The bottom of the pool is slanted so that the water depth is 1.3 meters at the shallow end and 4 meters at the deep end, as shown. Find the angle of depression of the bottom of the pool. 7.69°

Trigonometry and Bearings In surveying and navigation, directions are generally given in terms of bearings. 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

Trig and Bearings You try. Draw a bearing of: N800W S300E N S E W N S

Trig and Bearings You try. Draw a bearing of: N800W S300E

Example – Finding Directions Using Bearings A hiker travels at 4 miles per hour at a heading of S 35° E from a ranger station. After 3 hours how far south and how far east is the hiker from the station? Place triangle based on properties of inverse function Precalculus 4.8 Applications and Models

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

Two lookout towers are 50 kilometers apart Two lookout towers are 50 kilometers apart. Tower A is due west of tower B. A roadway connects the two towers. A dinosaur is spotted from each of the towers. The bearing of the dinosaur from A is N 43o E. The bearing of the dinosaur from tower B is N 58o W. Find the distance of the dinosaur to the roadway that connects the two towers. h 43o 58o 47o 32o A B x 50– x

h 47o 32o A B x 50– x 19.741 km

Two lookout towers spot a fire at the same time 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 45cos(330) – 72sin(630) b – d A 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) 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

4.8 Applications and Models Homework 4.8 p 326 1, 5, 9, 17-37 Odd Quiz tomorrow on sections 4.5,4.6, and 4.7 Precalculus 4.8 Applications and Models

4.8 Trigonometric Applications and Models Day 2 Objectives: Use harmonic motion to solve real-life problems.

4.8 Applications and Models HWQ 11/14 A plane is 160 miles north and 85 miles east of an airport. The pilot wants to fly directly to the airport. What bearing should be taken? Precalculus 4.8 Applications and Models

4.8 Applications and Models Terminology Harmonic Motion – Simple vibration, oscillation, rotation, or wave motion. It can be described using the sine and cosine functions. Displacement – Distance from equilibrium. Precalculus 4.8 Applications and Models

Simple Harmonic Motion A point that moves on a coordinate line is in simple harmonic motion if its distance d from the origin at time t is given by where a and ω are real numbers (ω>0) and frequency is number of cycles per unit of time. Precalculus 4.8 Applications and Models

Simple Harmonic Motion Precalculus 4.8 Applications and Models

Example – Simple Harmonic Motion Given this equation for simple harmonic motion Find: Maximum displacement Frequency Value of d at t=4 The least positive value of t when d=0 Place triangle based on properties of inverse function Precalculus 4.8 Applications and Models

You Try – Simple Harmonic Motion A mass attached to a spring vibrates up and down in simple harmonic motion according to the equation Find: Maximum displacement Frequency Value of d at 2 lvalues of t for which d=0

Example – Simple Harmonic Motion A weight attached to the end of a spring is pulled down 5 cm below its equilibrium point and released. It takes 4 seconds to complete one cycle of moving from 5 cm below the equilibrium point to 5 cm above the equilibrium point and then returning to its low point. Find the sinusoidal function that best represents the motion of the moving weight. Find the position of the weight 9 seconds after it is released.

You Try – Simple Harmonic Motion A buoy oscillates in simple harmonic motion as waves go past. At a given time it is noted that the buoy moves a total of 6 feet from its low point to its high point, returning to its high point every 15 seconds. Write a sinusoidal function that describes the motion of the buoy if it is at the high point at t=0. Find the position of the buoy 10 seconds after it is released.

Homework 4.8 Applications and Models Worksheet (Bearings and Harmonic Motion) Test next Tuesday.