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Warm-up 10/29

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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.

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Terminology Angle of elevation – angle from the horizontal upward to an object. Angle of depression – angle from the horizontal downward to an object. Precalculus4.8 Applications and Models3 Horizontal Observer Angle of elevation Object Horizontal Observer Angle of depression Object

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

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

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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. Precalculus4.8 Applications and Models6

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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°

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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 35° EW N S 70° EW

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Trig and Bearings You try. Draw a bearing of: N80 0 W S30 0 E N S EW N S EW

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Trig and Bearings You try. Draw a bearing of: N80 0 W S30 0 E

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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? Precalculus4.8 Applications and Models11

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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 54 o W. Find the ship’s bearing and distance from the port of departure at 3 P.M. 54 o 20 nmph for 2 hrs 40 nm 20 nmph for 1 hr 20 nm a b 20sin(36 o ) 36 o θ d 20cos(36 o ) o Bearing: N o W Example – Finding Directions Using Bearings

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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 54 o W. Find the ship’s bearing and distance from the port of departure at 3 P.M. 54 o 20 nmph for 2 hrs 40 nm 20 nmph for 1 hr 20 nm a b 20sin(36 o ) 36 o θ d 20cos(36 o ) o Bearing: N o W

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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 43 o E. The bearing of the dinosaur from tower B is N 58 o W. Find the distance of the dinosaur to the roadway that connects the two towers. AB 47 o 32 o x50– x h 43 o 58 o

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AB 47 o 32 o x50– x h km

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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 33 o E and is calculated to be 45 km from the tower. The bearing of the fire from tower B is N 63 o 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 B 33 o 63 o a b c d s a + c b – d 45sin(33 0 ) + 72sin(63 0 ) 45cos(33 0 ) – 72sin(63 0 )

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A B 33 o 63 o a b c d s a + c b – d 45sin(33 0 ) + 72sin(63 0 ) 45cos(33 0 ) – 72sin(63 0 ) km

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A B 33 o 63 o a b c d s a + c b – d 45sin(33 0 ) + 72sin(63 0 ) 45cos(33 0 ) – 72sin(63 0 ) θ km

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Homework 4.8 p 326 1, 5, 9, Odd Quiz tomorrow on sections 4.5,4.6, and 4.7 Precalculus4.8 Applications and Models19

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

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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? Precalculus4.8 Applications and Models21

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Terminology Harmonic Motion – Simple vibration, oscillation, rotation, or wave motion. It can be described using the sine and cosine functions. Displacement – Distance from equilibrium. Precalculus4.8 Applications and Models22

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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. Precalculus4.8 Applications and Models23

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Simple Harmonic Motion Precalculus4.8 Applications and Models24 10 cm 0 cm 10 cm

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Example – Simple Harmonic Motion Given this equation for simple harmonic motion Find: a)Maximum displacement b)Frequency c)Value of d at t=4 d)The least positive value of t when d=0 Precalculus4.8 Applications and Models25

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

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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.

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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.

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

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