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Warm-up 10/28 Evaluate the following. Give exact values when possible: 1 arcsin 2 1 arccos 2 1 arctan 2 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. Terminology • Angle of elevation – angle from the horizontal upward to an object. • Angle of depression – angle from the horizontal downward to an object. Object Horizontal Angle of elevation Observer Angle of depression Observer Horizontal Object Precalculus 4.8 Applications and Models 3 Example • 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. Precalculus 4.8 Applications and Models 6 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 35E N N S 70W 35° W E W E 70° S S Trig and Bearings • You try. Draw a bearing of: N800W S300E N N W E S W E S Trig and Bearings • You try. Draw a bearing of: N800W S300E 800 300 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? Precalculus 4.8 Applications and Models 11 20sin(36o) 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 a θ b o) 20cos(36 a 20 o 20sin(36 ) a b cos(36o ) 20 o 20cos(36 ) b sin(36o ) 40 nmfor 2 20 nmph hrso ) 40 20cos(36 20sin(36o ) tan( ) 20cos(36o ) 40 o 20sin(36 ) 1 tan o 20cos(36 ) 40 11.819o Bearing: N 78.181o W 20sin(36o) 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 θ b o) 20cos(36 a 20 o 20sin(36 ) a b cos(36o ) 20 o 20cos(36 ) b sin(36o ) 40 nmfor 2 20 nmph hrso ) 40 Bearing: N 78.181o W 20cos(36 20sin(36 ) 20sin(36 ) o o 20 cos(36 ) 40 2 20 cos(36 ) 40 d 2 2 o o 2 2 d 57.397 nm d 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 A h tan(47 ) x x tan(47o ) h o 32o x 50– x h tan(32 ) 50 x 50 x tan(32o ) h o B h 47o A 32o x x tan(47o ) h 50 tan(32o ) o tan(47 )h o o tan(47 ) tan(32 ) 19.741 h 19.741 50– x B 50 x tan(32o ) h x tan(47o ) 50 x tan(32o ) x tan(47o ) 50 tan(32o ) x tan(32o ) x tan(47o ) x tan(32o ) 50 tan(32o ) x tan(47o ) tan(32o ) 50 tan(32o ) 50 tan(32o ) x km tan(47o ) tan(32o ) 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 19 4.8 Trigonometric Applications and Models Day 2 Objectives: •Use harmonic motion to solve real-life problems. 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 22 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 d a sin t a amplitude or d a cos t 2 period frequency 2 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 23 Simple Harmonic Motion 10 cm 10 cm 0 cm 0 cm 10 cm 10 cm Precalculus 4.8 Applications and Models 24 Example – Simple Harmonic Motion Given this equation for simple harmonic motion 3 d 6 cos t 4 Find: a) Maximum displacement 6 b) Frequency 83 cycle per unit of time c) Value of d at t=4 6 d) The least positive value of t when d=0 Precalculus 4.8 Applications and Models 2 t 3 25 You Try – Simple Harmonic Motion A mass attached to a spring vibrates up and down in simple harmonic motion according to the equation d 4sin 2 t Find: a) Maximum displacement 4 1 b) Frequency 4 cycle per unit of time c) Value of d at t 2 0 d) 2 values of t for which d=0 0 and 2 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. f t 5cos t 2 • Find the position of the weight 9 seconds after it is released. 0 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 high point to its low 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. 2 f t 3cos t 15 • Find the position of the buoy 10 seconds after it is released. 3 2 Homework: 4.8 Applications and Models Worksheet (Bearings and Harmonic Motion) Test next Tuesday.