"MHF 4U Unit 3 â€“Trigonometric Functionsâ€“ Outline - DOC"
Abbey Park High School Unit 3 – Trigonometric Functions U3 – Days 8 & 9 – Applications of Trig Functions Applications of trigonometry (Source: wikipedia) “Trigonometry has an enormous variety of applications. The ones mentioned explicitly in textbooks and courses on trigonometry are its uses in practical endeavors such as navigation, land surveying, building, and the like. It is also used extensively in a number of academic fields, primarily mathematics, science and engineering. Among the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics. Among the scientific fields that make use of trigonometry are these: acoustics, architecture, astronomy (and hence navigation, on the oceans, in aircraft, and in space; in this connection, see great circle distance), biology, cartography, chemistry, civil engineering, computer graphics, geophysics, crystallography, economics (in particular in analysis of financial markets), electrical engineering, electronics, land surveying and geodesy, many physical sciences, mechanical engineering, machining, medical imaging (CAT scans and ultrasound), meteorology, music theory, number theory (and hence cryptography), oceanography, optics, pharmacology, phonetics, probability theory, psychology, seismology, statistics, and visual perception. The fact that these fields make use of trigonometry does not mean knowledge of trigonometry is needed in order to learn anything about them. It does mean that some things in these fields cannot be understood without trigonometry. For example, a professor of music may perhaps know nothing of mathematics, but would probably know that Pythagoras was the earliest known contributor to the mathematical theory of music. In some of the fields of endeavour listed above it is easy to imagine how trigonometry could be used. For example, in navigation and land surveying, the occasions for the use of trigonometry are in at least some cases simple enough that they can be described in a beginning trigonometry textbook. In the case of music theory, the application of trigonometry is related to work begun by Pythagoras, who observed that the sounds made by plucking two strings of different lengths are consonant if both lengths are small integer multiples of a common length. The resemblance between the shape of a vibrating string and the graph of the sine function is no mere coincidence. In oceanography, the resemblance between the shapes of some waves and the graph of the sine function is also not coincidental. In some other fields, among them climatology, biology, and economics, there are seasonal periodicities. The study of these often involves the periodic nature of the sine and cosine functions.” Source: wikipedia.com Advanced Functions: MHF4U Page 1 of 4 Abbey Park High School Unit 3 – Trigonometric Functions Application Questions 1. Point (-17, -20) is on the terminal arm of angle in standard position. a. Identify the six trig ratios for . b. Determine the measure of angle in radians (to the nearest hundredth). c. What is angle in degrees (to the nearest tenth)? 2. A water wheel of radius 1m sits in a stream as shown. a. Draw, for one complete revolution of the wheel, a sequence of right angle triangles to represent the position of a point on the water wheel for every rotation of 6 b. Make a table with intervals of to show the displacement from the surface of the stream 6 of the indicated point as it rotates from 0 to 2 . c. Use the table to graph displaced from surface versus angle of rotation d. Describe the graph and write and equation that models the situation. 3. A buoy rises and falls as it rides the waves. The equation h(t ) cos t models the displacement of 5 the buoy in metres at t seconds. a. Graph the displacement from 0 to 20s in 2.5s intervals. b. Determine the period of the function from the graph. c. Determine the period of the function algebraically from the equation d. What is the displacement at 35S? e. At what time, to the nearest second, does the displacement first reach -0.8m? 4. A spring bounces up and down according to the model d (t ) 0.5 cos 2t , where d is the displacement in centimetres from the rest position and t is the time in seconds. The model does not consider the effects of gravity. a. Make a table for 0 t 9 using 0.5s intervals. Use the table to graph d(t) vs. t. b. Explain why the function models periodic behaviour. c. What is the relationship between the amplitude of the function and the displacement of the spring in its rest position? 5. The average monthly temperature, T, in degrees Celsius in the Kawartha Lakes was modelled by T (t ) 22 cos t 10 where t represents the number of months. For t=0, the month is January; 6 for t=1, the month is February; and so on. a. What is the period? Explain the period in the context of the problem. b. What is the maximum temperature? c. What is the minimum temperature? d. What is the range of temperatures for this model? Advanced Functions: MHF4U Page 2 of 4 Abbey Park High School Unit 3 – Trigonometric Functions 6. A gear of radius 1m turns in a counterclockwise direction and drives a larger gear of radius 3m. Both gears have their central axis along the horizontal. a. Which direction is the larger gear turning? b. If the period of the smaller gear is 2s, what is the period of the larger gear? c. Make a table in convenient intervals for each gear, to show the vertical displacement, d, of the point where the two gears first touched. Begin the table at 0s and end it at 12s. d. Graph vertical displacement versus time. e. What is the displacement of the point on the large wheel when the drive wheel first has a displacement of -0.5m? f. What is the displacement of the drive when the large wheel first has a displacement of 2m? g. What is the displacement of the point on the large wheel at 5 min? 7. The graph of y 2 sin 3 is shifted to the right units and down 2 units. Write the new equation. 2 2 8. The graph of y 3 cos is shifted to the left units and up 1 unit. Write the new equation. 2 3 9. Each person’s blood pressure is different, but there is a range of blood pressure values that is 5 considered healthy. The function P(t ) 20 cos t 100 models the blood pressure of a person at 3 rest, where “P” is the blood pressure in millimetres of mercury, and “t” is the time in seconds. a. What is the period of the function? What does the period represent for an individual? b. How many times does this person’s heart beat each minute? c. Sketch the graph of y P(t ) for 0 t 6 . d. What is the range of the function? Explain the meaning of the range in terms of a person’s blood pressure. 10. The average monthly temperature in a region of Australia is modelled by the function T (t ) 9 23 cos t where T is the temperature in degrees Celsius and t is the month of the year 6 (t=0 is January, t=1 is February, t=2 is March, etc.). a. Build a table of Temperatures for 0 t 23 b. Graph the function. c. Explain how to use the axis of the curve and the amplitude to determine the maximum and minimum values of the function. d. Determine the period of the function from the graph. Verify your answer algebraically. Advanced Functions: MHF4U Page 3 of 4 Abbey Park High School Unit 3 – Trigonometric Functions 11. A ship that is docked in port rises and falls with the waves. The model h(t ) 5 sin( )t describes 5 the vertical movement of the ship, h, in metres at t seconds. a. What is the vertical position of the ship at 22 s (to the nearest hundredth)? b. What is the period of the function? What does this mean in this case? c. Determine all times within the first minute that the vertical position of the ship is -0.9m (to the nearest tenth of a second). 12. A skyscraper sways 55 cm back and forth from “the vertical” during high winds. At t=5s, the building is 55 cm to the right of vertical. The building sways back to the vertical, and at t = 35 s, the building sways 55 cm to the left of the vertical. Write an equation that models the motion of the building in terms of time. 13. The Double Scoop Ice Cream Company tracked its mean monthly production of ice cream in 2006. Month J F M A M J J A S O N D J 2005 168 181 219 222 246 276 264 252 219 204 181 174 0 a. Explain why it is reasonable to expect ice cream production to be periodic. b. Determine a trigonometric model that best represents the data. 14. The table shows the average monthly high temperature for one year in Kapuskasing. Time J F M A M J J A S O N D (months) Temp -18.6 -16.3 -9.1 0.4 8.5 13.8 17.0 15.4 10.3 4.4 -4.3 -14.8 o ( C) a. Draw a scatter plot of the data and the curve of best fit. Let January be month 0. b. What type of model describes the graph? c. Write an equation to model the situation. Describe the constants and the variables in the context of this problem. d. What is the average monthly temperature for the 38th month? 15. The daily maximum temperature in Kenora for each month is shown in the table below. Time J F M A M J J A S O N D (months) Temp -13.1 -9.0 -1.1 8.5 16.8 21.6 24.7 22.9 16.3 9.3 -1.2 -10.2 o ( C) a. Draw a scatter plot of the data and the curve of best fit. Let January be month 0. b. What is the amplitude of the function? c. Use the cosine function to write an equation to model the situation. d. What is the phase shift? e. Use the equation to predict the daily maximum temperature for the 38 th month. Advanced Functions: MHF4U Page 4 of 4