TEK: P.3E Use trigonometry in mathematical and real-world problems, including directional bearing

Transcription

1 Precalculus Notes 4.8 Applications of Trigonometry Solving Right Triangles TEK: P.3E Use trigonometry in mathematical and real-world problems, including directional bearing Page 1 link: Label the Triangles Based on the Given Angle, θ. θ θ The three main trig functions, defined in terms of opposite, adjacent and hypotenuse are: sinθ = cosθ = tanθ = Basic Solving: Using the above three trig functions, solve the following triangles. Round all angles and sides to three decimal places when necessary. Note: To solve a triangle is to find all its missing parts. A R 48 o C B P 29 o Q

2 Basic Applications Page 2 link: From a point on the ground 100 feet from a flag pole the angle of elevation to the top of the flag pole is 21.8 o To the nearest thousandth of a foot, how tall is the flag pole? A 5.5-foot man standing on top of a 35-foot building looks down at a spot on the street with an angle o of depression of 14. How far away from the building is the spot on the street? How far is the spot on the street from the man on the building? Definitions: Angle of Elevation Angle of Depression

3 Precalculus Notes 4.8 Solving Right Triangles Navigation Page 3 link: When an object in motion, like a ship or an airplane, has its bearing or course given, it is given in terms of direction (north, south, east or west) and the angle given is always measured clockwise from north. Draw angles to approximate the following bearings: 75 o 220 o 110 o 300 o An airplane travels 100 miles on a bearing of 80 miles. 40 o o then changes to a bearing of 130 and travels another a) How far away from its original position is the airplane at the end of this trip? b) What is the bearing from the original position to the final position?

4 Page 4 link: A ship travels 9 hours at 10 knots on a bearing of 150 then changes to a bearing of another 16 hours at 10 knots. o 240 o and travels a) How far away from its original position is the ship at the end of this trip? b) What is the bearing from the original position to the final position? Simple Harmonic Motion A point moving on a number line is in if its directed distance from the origin is given by, where at andw,, R andw> 0. The motion has of which is the number of oscillations per unit of time. Example: A mass oscillating up and down on the bottom of a spring (assuming perfect elasticity and no air friction)can be modeled as simple harmonic motion. If the weight is displaced a maximum of 5 cm. find the equation to model this situation if it take 2 seconds to complete one cycle.

5 More Applications of trigonometry Page 5 link: At Oyster Cove, Maine, the depth of water at the end of a dock varies with the tides. A high tide of 11.3 feet occurs at 4:00 a.m, and a low tide of 0.1 feet occurs at 10:00 a.m. These tide levels then repeat themselves at approximately 12 hour intervals. The data can be represented as in the table below. Time 4 a.m. (t=4) 10 a.m. (t=10) 4 p.m. (t=16) 10 p.m. (t=22) Depth of Water 11.3 ft. 0.1 ft ft. 0.1 ft. Plotting the Data: y x Amplitude: A = = Vertical Shift: 2 k = = 2 Horizontal Stretch: Period is 12 hours, so 2π B B Horizontal Shift: The first low point occurs when x = and the highest data point first occurs when x =. Halfway between these is 2 = Once we have all these values, we can put them into our equation: y =

6 Page 6 link: We can now use this model (function) to make predictions about tide levels at given times. Example 3: Using the tide model of example 2, what is the expected water depth at 2:00 p.m. (t=14)? ( ) y π = 5.6sin = 6 14 feet Example 4: Using the tide model of example 2, what is the expected water depth at midnight (t=0 or t=24)? ( ) y π = 5.6sin = 6 feet Example 5: During what times can we expect the water level to be 10 feet or more? By looking at the graph, we can see that the water levels are 10 feet or higher from approximately t = to t = and again from t = to t =. So, the water is 10 feet deep or greater from approximately and again from Is there a pattern? What is it? What other predictions can be made from this pattern?

7 Alternate page 5 link: Repeat to remember!! More Applications of trigonometry. At Oyster Cove, Maine, the depth of water at the end of a dock varies with the tides. A high tide of 11.3 feet occurs at 4:00 a.m, and a low tide of 0.1 feet occurs at 10:00 a.m. These tide levels then repeat themselves at approximately 12 hour intervals. Use this grid to model the scenario with a sine or cosine wave. Plotting the Data: y x y = Remember to repeat!! What key elements of this lesson need to be remembered in order to repeat this type of sinusoidal modeling?

8 Create! With your partner, using the previous problem as a guide, create a fictional tides problem that addresses algebraic, numeric, and graphical sinusoidal modeling and a series of 4 questions (asked and answered) that the captain of a ship might need to consider. Scenario: y x Algebraic sinusoidal model: Questions: 1) 2) 3)

9 4)

10 PreCalculus Out of Class Learning OoCL Name Date Period Use SOH CAH TOA to solve the right triangle problems. Remember that navigational bearing is measured clockwise from North. Sketch the scenarios before solving the problems. 1) The angle of elevation to the top of an obelisk from a point on the ground 300 feet away from the base of the obelisk is 60 o. Find the height of the obelisk. 2) The angle of depression from an observation platform to a chupacabra burrow hole is 22 o. The diagonal distance to the hole from the platform is 480 feet. Assuming the ground is level, how far is the burrow from the base of the platform? 3) A guy wire connects the top of an antenna to a point on level ground 5 feet from the base of the antenna. The angle of depression from the top of the tower to the point is 80 o. What is the length of the guy wire and the height of the antenna?

11 4) An on ramp accessing a freeway overpass is 470 feet long and rises 32 feet. What is the average angle of elevation of the ramp? 5) From the top of a 100 foot building a man observes a car moving toward the building. o If the angle of depression from the top of the building to the car changes from 15 to 33 o during the period of observation, how far does the car travel?

12 6) a) What is the value of a? b) What is the value of k? c) What is the value of ω?

13 7) a) Given that the period is 12 months. Find b. b) Assuming that the high and low temperatures in the table determine the range of the sinusoid, find a and k using only the high and low. c) Find a value of h that will put the minimum at t=1 and the maximum at t=7. d) Use your sinusoid model to predict dates in the year when the mean temperature in Charleston will be 70 o. (Assume that t=0 represents January.) (Use your calculator.)

14 8) A helicopter flying over relatively level ground has an altitude of 800 feet. The pilot spots a heffalump which is 2800 feet away (diagonal distance) from the helicopter. What is the angle of depression of the helicopter to the heffalump? 9) A woosle on a spring oscillates up and down and completes one cycle in 0.5 seconds. It s maximum displacement is 3 cm. Write an equation that models this motion.

15 10) A submarine embarks on an unbelievably secret mission from undisclosed location #1 with initial bearing 335 o and travels for 5 hours at a speed 25 nautical miles per o hour to undisclosed location #2. The sub then takes a bearing of 245 and continues 5 more hours at 30 nautical miles per hour and stops at undisclosed location #3. Find the distance and bearing of the sub from undisclosed location #1 to undisclosed location # 3.

16 PreCalculus Out of Class Learning OoCL Name Date Period Use SOH CAH TOA to solve the right triangle problems. Remember that navigational bearing is measured clockwise from North. Sketch the scenarios before solving the problems. 4.8 Extra Practice (from Pre- Cal Book p.432) 11. Antenna Height A guy wire attached to the top of the KSAM radio antenna is anchored at a point on the ground 10 meters from the antenna s base. If the wire makes an angle of 55 degrees with level ground, how high is the KSAM antenna? 17. Navigation The Coast Guard cutter Angelica travels at 30 knots from its home port of Corpus Christi on a course of 95 degrees for 2 hours and then changes to a course of 185 degrees for 2 hours. Find the distance and the bearing from the Corpus Christi port to the cutter. 19. Land Measure The angle of depression is 19 degrees from a point 7256 ft above sea level on the north rim of the Grand Canyon level to a point 6159 ft above sea level on the south rim. How wide is the canyon at that point?

17 24. Recreational Flying A hot-air balloon over Park City, Utah, is 760 feet above the ground. The angle of depression from the balloon to an observer is 5.25 degrees. Assuming the ground is relatively flat, how far is the observer from a point on the ground directly under the balloon? 32. Ferris Wheel Motion Jacob and Emily ride a Ferris wheel at a carnival in Billings, Montana. The wheel has a 16 meter diameter and turns at 3 rpm with its lowest point 1 meter above the ground. Assume that Jacob and Emily s height h above the ground is a sinusoidal function of time t (in seconds), where t=0 represents the lowest point of the wheel. (a) Write an equation for h. (b) Use h to estimate Jacob and Emily s height above the ground at t=4 and t=10.