**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.

**Horizontal** Observer Angle of depression Object Object Angle of elevation Observer Horizontal Terminology • Angle of elevation – angle from the horizontal upward to an object. • Angle of depression – angle from the horizontal downward to an object. 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. 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°

**N** W E 70° S 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 35° W E S

**Trig and Bearings** • You try. Draw a bearing of: N800W S300E N N W W E E S 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? 4.8 Applications and Models

**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 54o 20 nmph for 1 hr 78.181o 20 nm 36o θ b 20cos(36o) 40 nm 20 nmph for 2 hrs 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 54o 20 nm 78.181o 20sin(36o) 20 nmph for 1 hr 36o θ b 20cos(36o) 40 nm 20 nmph for 2 hrs Bearing: N 78.181o W

**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. 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 s 33o 45cos(330) – 72sin(630) b – d A a + c 45sin(330) + 72sin(630)

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

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

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

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

**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? 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. 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. 4.8 Applications and Models

**10 cm** 10 cm 0 cm 0 cm 10 cm 10 cm Simple Harmonic Motion 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 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.