# Trigonometric functions

## Transcription

Trigonometric functions

18 Chapter Trigonometric functions Syllabus reference: 4.5 Contents: A B C D cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 E F black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\557IB_STSL-2_18.CDR Monday, 15 February 2010 10:06:40 AM PETER Observing periodic behaviour The sine function The cosine function Modelling using sine and cosine functions Trigonometric equations Using trigonometric models IB_STSL-2ed 558 TRIGONOMETRIC FUNCTIONS (Chapter 18) Periodic phenomena occur all the time in the physical world. For example, in: ² ² ² ² ² ² seasonal variations in our climate variations in average maximum and minimum monthly temperatures the number of daylight hours at a particular location tidal variations in the depth of water in a harbour the phases of the moon animal populations. These phenomena illustrate variable behaviour which is repeated over time. The repetition may be called periodic, oscillatory, or cyclic in different situations. In this chapter we will consider how trigonometric functions can be used to model periodic phenomena. We will then extend our knowledge of the trigonometric functions by considering formulae that connect them. OPENING PROBLEM A Ferris wheel rotates at a constant speed. The wheel’s radius is 10 m and the bottom of the wheel is 2 m above ground level. From a point in front of the wheel, Andrew is watching a green light on the perimeter of the wheel. Andrew notices that the green light moves in a circle. He estimates how high the light is above ground level at two second intervals and draws a scatterplot of his results. Things to think about: ² ² ² ² What does his scatterplot look like? Could a known function be used to model the data? How could this function be used to find the light’s position at any point in time? How could this function be used to find the times when the light is at its maximum and minimum heights? ² What part of the function would indicate the time interval over which one complete cycle occurs? Click on the icon to visit a simulation of the Ferris wheel. You will be able to view the light from: ² in front of the wheel ² above the wheel. ² a side-on position DEMO cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 You can then observe the graph of height above or below the wheel’s axis as the wheel rotates at a constant rate. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\558IB_STSL-2_18.CDR Monday, 15 February 2010 10:06:54 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS A 559 (Chapter 18) OBSERVING PERIODIC BEHAVIOUR Consider the table below which shows the mean monthly maximum temperature for Cape Town, South Africa. Month Jan Temp (o C) 28 Feb Mar 27 25 12 Apr May 22 18 12 Jun Jul 16 On the scatterplot alongside we plot the temperature T on the vertical axis. We assign January as t = 1 month, February as t = 2 months, and so on for the 12 months of the year. Aug 15 16 Sep Oct Nov Dec 18 21 12 24 26 T (°C) 30 20 (10,¡21 Qw ) 10 t (months) JAN 3 6 9 12 JAN The temperature shows a variation from an average of 28o C in January through a range of values across the months. The cycle will repeat itself for the next 12 month period. By the end of the chapter we will be able to establish a function which approximately fits this set of points. T (°C) 30 20 10 t (months) JAN 3 6 9 12 15 JAN 18 21 24 HISTORICAL NOTE direction of rotation + voltage lines of magnetic force 90° - 180° 270° 360° In 1831 Michael Faraday discovered that an electric current was generated by rotating a coil of wire in a magnetic field. The electric current produced showed a voltage which varied between positive and negative values as the coil rotated through 360o . cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 Graphs with this basic shape where the cycle is repeated over and over are called sine waves. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\559IB_STSL-2_18.CDR Monday, 15 February 2010 10:06:57 AM PETER IB_STSL-2ed 560 TRIGONOMETRIC FUNCTIONS (Chapter 18) GATHERING PERIODIC DATA Data on a number of periodic phenomena can be found online or in other publications. For example: ² Maximum and minimum monthly temperatures can be found at http://www.bom.gov.au/silo/ ² Tidal details can be obtained from daily newspapers or internet sites such as http://tidesandcurrents.noaa.gov or http://www.bom.gov.au/oceanography TERMINOLOGY USED TO DESCRIBE PERIODICITY A periodic function is one which repeats itself over and over in a horizontal direction. The period of a periodic function is the length of one repetition or cycle. f (x) is a periodic function with period p , f (x + p) = f (x) for all x, and p is the smallest positive value for this to be true. For example, for the temperature graph on page 559, we see that f (x + 12) = f (x) for all x on the curve. This means that f(0) = f (12) = f(24) = :::: f(1) = f (13) = f(25) = :::: and so on. We say the period of the temperature curve is 12 months. A cycloid is another example of a periodic function. It is the curve traced out by a point on a circle as the circle rolls across a flat surface in a straight line. However, the cycloid function cannot be written as a Cartesian equation in the form y = :::::: or f(x) = :::::: DEMO horizontal flat surface In this course we are mainly concerned with periodic phenomena which show a wave pattern when graphed. the wave principal axis The wave oscillates about a horizontal line called the principal axis or mean line. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 A maximum point occurs at the top of a crest, and a minimum point at the bottom of a trough. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\560IB_STSL-2_18.CDR Wednesday, 24 February 2010 9:23:54 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS (Chapter 18) 561 The amplitude of a periodic function is the distance between a maximum (or minimum) point and the principal axis. amplitude = max ¡ min 2 principal axis y = max + min 2 maximum point amplitude principal axis period minimum point EXERCISE 18A 1 For each set of data below, draw a scatterplot and decide whether or not the data exhibits approximately periodic behaviour. 0 0 a x y b x y 0 4 c x y 0 0 d x y 0 0 1 1 1 1 2 1:4 2 0 3 1 3 1 4 0 5 ¡1 6 ¡1:4 7 ¡1 8 0 9 1 4 4 10 1:4 11 1 12 0 TI-nspire TI-84 Casio 0:5 1:9 2 4:7 1:0 3:5 3 3:4 1:5 4:5 4 1:7 2:0 4:7 2:5 4:3 5 2:1 6 5:2 3:0 3:4 7 8:9 3:5 2:4 8 10:9 9 10:2 10 8:4 12 10:4 2 The following tabled values show the height above the ground of a point on a bicycle wheel as the bicycle is wheeled along a flat surface. Distance travelled (cm) 0 20 40 60 80 100 120 140 160 180 200 Height above ground (cm) 0 6 23 42 57 64 59 43 23 7 1 220 240 260 280 300 320 340 360 380 400 Distance travelled (cm) 5 Height above ground (cm) 27 40 55 63 60 44 24 9 3 cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 a Plot the graph of height against distance. b Is the data periodic? If so, estimate: i the equation of the principal axis ii the maximum value iii the period iv the amplitude. c Is it reasonable to fit a curve to this data, or should we leave it as discrete points? black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\561IB_STSL-2_18.CDR Monday, 15 February 2010 10:09:59 AM PETER IB_STSL-2ed 562 TRIGONOMETRIC FUNCTIONS (Chapter 18) 3 Which of these graphs show periodic behaviour? a b y 1 2 3 4 6 x 5 3 y c y x e 3 4 5 6 7 8 9 12 18 x 4 x 2 -1 9 1 f y 15 y d 1 2 6 2 3 y x 5 10 15 4 -8 -12 2 20 6 x y 1 -4 4 8 4 16 12 x -1 -2 -3 For the given periodic function: a state its amplitude b state its period. c State the coordinates of the first maximum point such that x > 0. d What is the distance between successive maxima? e What is the equation of the principal axis? B THE SINE FUNCTION In previous studies of trigonometry we have only considered right angled triangles, or static situations where an angle µ is fixed. However, when an object moves around a circle, the situation is dynamic. The angle between the radius OP and the horizontal axis continually changes with time. y Consider again the Opening Problem in which a Ferris wheel of radius 10 m revolves at constant speed. P 10 The height of P, the point representing the green light on the wheel relative to the principal axis O, can be determined using right angled triangle trigonometry. h q x 10 h , so h = 10 sin µ: 10 As time goes by, µ changes and so does h. sin µ = cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 So, h is a function of µ, but more importantly h is a function of time t. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\562IB_STSL-2_18.CDR Monday, 15 February 2010 10:07:05 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS (Chapter 18) 563 Suppose the Ferris wheel observed by Andrew takes 100 seconds for a full revolution. The graph below shows the height of the light above or below the principal axis against the time in seconds. 10 DEMO height (metres) 50 time (seconds) 100 -10 We observe that the amplitude is 10 metres and the period is 100 seconds. THE BASIC SINE CURVE Suppose the point P moves around a circle of radius 1 unit centred at the origin O, so the angle OP makes with the horizontal axis is x. In this case the y-coordinate of P will be sin x. If we project the values of sin x from the unit circle onto a set of axes on the right, we obtain the graph of y = sin x. y y 1 y = sin_x 90° 180° 270° 360° x -1 The wave of course can be continued beyond 0o 6 x 6 360o as shown: y 1 90° 180° 270° 360° 450° 540° x y = sin_x -1 We expect the period to be 360o , since the point P returns to its starting point after one full revolution. The maximum value is 1 and the minimum is ¡1. GRAPHING PACKAGE The amplitude is 1. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 Use your graphics calculator or graphing package to obtain the graph of y = sin x to check these features. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\563IB_STSL-2_18.CDR Monday, 15 February 2010 10:07:07 AM PETER IB_STSL-2ed 564 TRIGONOMETRIC FUNCTIONS (Chapter 18) INVESTIGATION 1 THE FAMILY y = a sin x What to do: 1 Use a b c technology to graph on the same set of axes: y = sin x and y = 2 sin x y = sin x and y = 0:5 sin x y = sin x and y = ¡ sin x GRAPHING PACKAGE If using a graphics calculator, make sure that the mode is set in degrees and that your viewing window is appropriate. 2 For each of y = sin x, y = 2 sin x, y = 0:5 sin x, and y = ¡ sin x: a record the maximum and minimum values b state the period and amplitude. 3 Describe the effect of a in the function y = a sin x. 4 State the amplitude of: a y = 3 sin x b y= p 7 sin x c y = ¡2 sin x INVESTIGATION 2 THE FAMILY y = sin bx What to do: 1 Use technology to graph on the same set of axes: b y = sin x and y = sin( 12 x) a y = sin x and y = sin 2x 2 For each of y = sin x, y = sin 2x, and y = sin( 12 x): a record the maximum and minimum values b state the period and amplitude. GRAPHING PACKAGE 3 On the same set of axes, sketch y = sin x and y = sin(¡x). State the period and amplitude of y = sin(¡x): 4 Describe the effect of b in the function y = sin bx. 5 State the period of: b y = sin( 13 x) a y = sin 3x d y = sin(¡2x) e y= c y = sin(1:2x) sin(¡ 12 x) f y = sin bx From the previous investigations you should have observed that: jaj is the modulus of a. It is the size of a, and cannot be negative. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 ² In y = a sin x, jaj determines the amplitude. The graph of y = sin x is vertically stretched if jaj > 1, or compressed if jaj < 1, to obtain the graph of y = a sin x. We call this a vertical dilation. If a < 0 the graph is also reflected in the x-axis. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\564IB_STSL-2_18.CDR Friday, 5 March 2010 9:22:23 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS ² In y = sin bx, b 6= 0, b affects the period and the period is (Chapter 18) 565 360o . jbj The graph of y = sin x is horizontally stretched if 0 < jbj < 1, or compressed if jbj > 1, to obtain the graph of y = sin bx. We call this a horizontal dilation. If b < 0 the graph is also reflected in the x-axis. Self Tutor Example 1 Without using technology, sketch the graphs of: b y = ¡2 sin x for 0o 6 x 6 360o . a y = 2 sin x a The amplitude is 2 and the period is 360o . y 2 270° 90° 360° x 180° y = 2_sin_x -2 We place the 5 points as shown and fit the sine wave to them. b The amplitude is 2, the period is 360o , and y = ¡2 sin x is the reflection of y = 2 sin x in the x-axis. y 2 y = -2_sin_x 270° 90° 360° 180° x y = 2_sin_x -2 Self Tutor Example 2 Without using technology, sketch the graph of y = sin 2x for 0o 6 x 6 360o . The period is 360o 2 = 180o . The maximum values are therefore 180o apart. As sin 2x has half the period of sin x, the first maximum is at 45o not 90o . y 1 270° 90° 360° 180° yellow 95 50 75 25 0 5 95 100 50 75 25 0 5 95 50 75 25 0 5 95 100 50 75 25 0 5 100 magenta 100 y = sin_2x -1 cyan x black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\565IB_STSL-2_18.CDR Monday, 15 February 2010 10:14:43 AM PETER IB_STSL-2ed 566 TRIGONOMETRIC FUNCTIONS (Chapter 18) EXERCISE 18B.1 1 Without using technology, sketch the graphs of the following for 0o 6 x 6 360o : a y = 3 sin x b y = ¡3 sin x c y= 3 2 d y = ¡ 32 sin x sin x 2 Without using technology, sketch the graphs of the following for 0o 6 x 6 540o : ¡ ¢ a y = sin 3x b y = sin x2 c y = sin(¡2x) 3 State the period of: a y = sin 4x b y = sin(¡4x) c y = sin ¡x¢ d y = sin(0:6x) 3 4 Find b given that the function y = sin bx, b > 0 has period: a 900o b 120o c 2160o INVESTIGATION 3 THE FAMILY y = sin x +c What to do: GRAPHING PACKAGE 1 Use technology to graph on the same set of axes: a y = sin x and y = sin x + 3 b y = sin x and y = sin x ¡ 2 each of y = sin x, y = sin x + 3 and y = sin x ¡ 2: record the maximum and minimum values calculate the equation of the principal axis state the period and amplitude. 2 For a b c 3 Explain the connection between the graphs of y = sin x and y = sin x + c. From Investigation 3 we observe that the graph of y = sin x + c has the same shape as the graph of y = sin x, but it is shifted up or down depending on the sign of c. We say: y = sin x + c is a vertical translation of y = sin x through c units. Its principal axis has equation y = c. Self Tutor Example 3 On the same set of axes graph for 0 6 x 6 720o : y = sin x and y = sin x ¡ 1 y 1 180° -1 -1 360° 540° -1 720° -1 -1 cyan principal axis magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 y = sin_x - 1 75 25 0 5 95 100 50 75 25 0 5 -2 x y = sin_x black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\566IB_STSL-2_18.CDR Monday, 15 February 2010 10:20:19 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS 567 (Chapter 18) THE GENERAL SINE FUNCTION y = a sin bx + c is called the general sine function. affects amplitude affects period affects principal axis The principal axis of the general sine function is y = c. The period of the general sine function is 360o . jbj Consider y = 2 sin 3x + 1. So, starting with y = sin x we would: ² double the amplitude to produce y = 2 sin x, then ² divide the period by 3 to produce y = 2 sin 3x, then ² translate upwards 1 unit to produce y = 2 sin 3x + 1. EXERCISE 18B.2 1 Find a formula in the form y = a sin bx + c which produces the following graphs: a 4 y b 3 2 90° -2 -4 c 4 270° 4 (270°, 1) 90° 270° -4 f 2 2 g 4 360° 270° (45°, 1) 90° 180° 270° 360° x h 4 180° 90° (180°, 1) x 90° 180° 270° 360° y 2 360° x 270° y -2 y -2 y -1 (180°, -4) 2 (270°, -3) 1 x -4 360° x -4 y 180° 270° -2 (90°, -3) 90° (90°, -1) 2 360° x 180° 180° -2 d y -2 90° -4 -2 4 y 2 360° x -3 2 e 4 180° -2 -4 (45°, -1) 90° x 180° 270° 360° -4 cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 (135°, -5) black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\567IB_STSL-2_18.CDR Tuesday, 16 February 2010 10:24:52 AM PETER IB_STSL-2ed 568 TRIGONOMETRIC FUNCTIONS (Chapter 18) 2 Without using technology, sketch the graphs of the following for 0o 6 x 6 360o : a y = sin x + 1 b y = sin x ¡ 2 c y = 1 ¡ sin x GRAPH e y = sin 3x + 1 d y = 2 sin x ¡ 1 f y = 1 ¡ sin 2x PAPER Click on the icon to obtain printable graph paper for the above question. C THE COSINE FUNCTION DEMO We return to the Ferris wheel to see the cosine function being generated. y Click on the icon to inspect a simulation of the view from above the wheel. 10 q The graph being generated over time is a cosine function. d This is no surprise as cos µ = and so 10 d = 10 cos µ. x d The graph alongside shows the horizontal displacement of the light on the Ferris wheel over time. The cosine curve y = cos x, like the sine curve y = sin x, has a period of 360o , an amplitude of 1, and its range is ¡1 6 y 6 1. Use your graphics calculator or graphing package to check these features. horizontal displacement (m) 10 50 100 time (s) -10 GRAPHING PACKAGE TI-nspire TI-84 Casio INVESTIGATION 4 COSINE FUNCTIONS In this investigation we consider families of cosine functions in the same way we did for sine functions. If you are using your graphics calculator, make sure it is set to degrees mode. GRAPHING PACKAGE What to do: 1 Use technology to graph on the same set of axes: a y = cos x and y = 2 cos x c y = cos x and y = ¡ cos x b y = cos x and y = 0:5 cos x 2 For each of y = cos x, y = 2 cos x, y = 0:5 cos x, y = ¡ cos x: a record the maximum and minimum values b state the period and amplitude. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 3 Describe the effect of a in the function y = a cos x. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\568IB_STSL-2_18.CDR Friday, 5 March 2010 9:24:47 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS (Chapter 18) 4 State the amplitude of: a y = 3 cos x b y= p 7 cos x 569 c y = ¡2 cos x 5 Use technology to graph on the same set of axes: b y = cos x and cos( 12 x) a y = cos x and y = cos 2x 6 For each of y = cos x, y = cos 2x, y = cos( 12 x), y = cos(¡2x): a record the maximum and minimum values b state the period and amplitude. 7 Describe the effect of b in the function y = cos bx. 8 State the period of: a y = cos 3x b y = cos( 13 x) c y = cos(1:2x) d y = cos(¡x) e y = cos(¡3x) f y = cos bx 9 Use technology to graph on the same set of axes: a y = cos x and y = cos x + 3 10 For a b c b y = cos x and y = cos x ¡ 2 each of y = cos x, y = cos x + 3 and y = cos x ¡ 2: record the maximum and minimum values find the equation of the principal axis state the period and amplitude. 11 Explain the connection between the graphs of y = cos x and y = cos x + c. You should have observed that: ² in y = a cos x, jaj determines the amplitude ² in y = cos bx, b affects the period and the period is 360o jbj ² in y = sin x + c, c affects the principal axis. Self Tutor Example 4 Without using technology, sketch the graph of y = 2 cos 2x for 0o 6 x 6 360o . a = 2, so the amplitude is j2j = 2. y y = 2_cos_2x 2 b = 2, so the period is 360o 360o = = 180o . jbj 2 1 90° 180° 270° 360° x -1 cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 -2 black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\569IB_STSL-2_18.CDR Monday, 15 February 2010 10:33:31 AM PETER IB_STSL-2ed 570 TRIGONOMETRIC FUNCTIONS (Chapter 18) EXERCISE 18C y 1 Given the graph of y = cos x, sketch the graphs of: 1 -180° 180° -360° GRAPH PAPER 360° x -1 a y = cos x + 2 d y= 3 2 cos x ¡ ¢ g y = cos x2 2 3 b y = cos x ¡ 1 c y= e y = ¡ cos x f y = cos 2x cos x h y = 3 cos 2x 2 Without graphing them, state the periods of: ¡ ¢ a y = cos 3x b y = cos x3 c y = cos ¡x¢ 2 3 The general cosine function is y = a cos bx + c. State the geometrical significance of a, b and c. 4 For the following graphs, find the cosine function representing them: a b y c y y 3 2 360° 5 180° x x 1 -2 360° D 720° x -5 MODELLING USING SINE AND COSINE FUNCTIONS When patterns of variation can be identified and quantified in terms of a formula or equation, predictions may be made about behaviour in the future. Examples of this include tidal movement which can be predicted many months ahead, and the date of a future full moon. In this section we use sine and cosine functions to model certain biological and physical phenomena that are periodic in nature. MEAN MONTHLY TEMPERATURE cyan magenta yellow Jul 16 95 50 15 75 25 0 16 Aug Sep Oct Nov Dec 18 21 12 24 26 100 Jun 5 95 22 18 12 100 May 50 Apr 75 0 27 25 12 5 Mar 95 50 28 75 0 5 95 100 50 75 25 0 5 Temp ( C) Feb 100 Jan o 25 Month 25 Consider again the mean monthly maximum temperature for Cape Town: black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\570IB_STSL-2_18.CDR Tuesday, 16 February 2010 10:26:07 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS 571 (Chapter 18) The graph over a two year period is shown below: 40 T (°C) 30 20 10 0 t (months) Jan Mar May Nov Sep Jul Jan Mar May Jul Sep Nov Since the maximum is at the start of the year, we attempt to model this data using the general cosine function y = a cos bx + c, or in this case T = a cos bt + c. 360o = 12 months. jbj 360o b= = 30o per month. 12 The period is 12 months, so ) The amplitude = max ¡ min 28 ¡ 15 ¼ ¼ 6:5, so a ¼ 6:5 : 2 2 The principal axis is midway between the maximum and minimum, 28 + 15 ¼ 21:5 . so c ¼ 2 So, the model is T ¼ 6:5 cos(30t) + 21:5 and is superimposed on the graph which follows: 40 T (°C) 30 20 10 0 t (months) Jan Mar May Nov Sep Jul Jan Mar May Jul Sep Nov EXERCISE 18D 1 At the Mawson base in Antarctica, the mean monthly temperatures for the last 30 years are as follows: Month Jan Feb Mar Apr May Jun July Aug Sept Oct Nov Dec Temperature (o C) 0 ¡4 ¡10 ¡15 ¡16 ¡17 ¡18 ¡19 ¡17 ¡13 ¡6 ¡1 Find a cosine model for this data without using technology. cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 2 Some of the largest tides in the world are observed in Canada’s Bay of Fundy. The difference between high and low tides is 14 metres and the average time difference between high tides is about 12:4 hours. a Find a sine model for the height of the tide H in terms of the time t hours after mean tide. b Sketch the graph of the model over one period. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\571IB_STSL-2_18.CDR Tuesday, 16 March 2010 3:02:39 PM PETER IB_STSL-2ed 572 TRIGONOMETRIC FUNCTIONS (Chapter 18) 3 Revisit the Opening Problem on page 558. The wheel takes 100 seconds to complete one revolution. Find the cosine model which gives the height of the light above the ground at any point in time. Assume that at time t = 0, the light is at its lowest point. 10 m 2m 4 Geraldine rides her bicycle along a flat road at night. Ian can clearly see a reflector on the spoke of her wheel rotating. He measures the height of the reflector above the ground at different times using a video. 0:1 0:2 0:3 0:4 0:5 0:6 0:7 0:8 0:9 Time (t s) Height (H cm) 19 17 38 62 68 50 24 15 31 a b c d Find a sine model for H in terms of t. Sketch the graph of the model over one period. Find the diameter of Geraldine’s wheel. How fast was Geraldine travelling? E reflector TRIGONOMETRIC EQUATIONS Linear equations such as 2x + 3 = 11 have exactly one solution. Quadratic equations have up to two real solutions. Trigonometric equations generally have infinitely many solutions unless a restrictive domain such as 0o 6 x 6 540o is given. We will examine solving trigonometric equations using: ² pre-prepared graphs ² technology. For the Ferris wheel Opening Problem the model is H = ¡10 cos(3:6t) + 12. t¡=¡50 We can easily check this by substituting t = 0, 25, 50, 75 H(0) = ¡10 cos 0o + 12 = ¡10 + 12 = 2 X 10 m t¡=¡75 H(25) = ¡10 cos 90o + 12 = 12 X t¡=¡25 10 m H(50) = ¡10 cos 180o + 12 = 22 X H(75) = ¡10 cos 270o + 12 = 12 X 2m However, we may be interested in the times when the light is some other height above the ground, for example 16 m. We would then need to solve the equation cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 ¡10 cos(3:6t) + 12 = 16. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\572IB_STSL-2_18.CDR Monday, 15 February 2010 12:03:56 PM PETER IB_STSL-2ed 573 TRIGONOMETRIC FUNCTIONS (Chapter 18) GRAPHICAL SOLUTION OF TRIGONOMETRIC EQUATIONS Sometimes simple trigonometric graphs are available on grid paper. In such cases we can estimate solutions straight from the graph. For example, we could use a graph to find approximate solutions for trigonometric equations such as cos x = 0:4 for 0o 6 x 6 540o . y 1 A y = 0.4 C B 180° 540° 360° 90° x 450° 270° -1 y = cos_x y = 0:4 meets y = cos x at A, B and C. Hence, to the nearest degree, x ¼ 66o , 294o or 426o . So, the solutions of cos x = 0:4 for 0o 6 x 6 540o are 66o , 294o and 426o . EXERCISE 18E.1 1 y y = sin_x 1 0.5 x 100° 200° 300° 400° 500° 600° 700° 800° -0.5 -1 Use the graph of y = sin x to find, correct to the nearest 5o , the solutions of: cyan magenta yellow 95 100 50 75 b sin x = ¡0:4 for 180o 6 x 6 360o . 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 a sin x = 0:3 for 0o 6 x 6 720o black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\573IB_STSL-2_18.CDR Tuesday, 16 February 2010 10:34:58 AM PETER IB_STSL-2ed 574 TRIGONOMETRIC FUNCTIONS (Chapter 18) 2 y y = cos_x 1 0.5 x 100° 200° 300° 400° 500° 600° 700° 800° -0.5 -1 Use the graph of y = cos x to find, correct to the nearest 5o , the solutions of: a cos x = 0:6, 0o 6 x 6 720o 3 b cos x = ¡0:3, 180o 6 x 6 540o . y y = sin_2x 1 0.5 x 100° 200° 300° 400° 500° 600° 700° 800° -0.5 -1 Use the graph of y = sin 2x to find, correct to the nearest 5o , the solutions of: a sin 2x = 0:7, 0o 6 x 6 540o b sin 2x = ¡0:3, 0o 6 x 6 540o . SOLVING TRIGONOMETRIC EQUATIONS USING TECHNOLOGY Trigonometric equations may be solved using either a graphing package or a graphics calculator. GRAPHING PACKAGE When using a graphics calculator make sure that the mode is set to degrees. Self Tutor Example 5 Solve sin x = 0:3 for 0o 6 x 6 360o . cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 We graph the functions Y1 = sin X and Y2 = 0:3 on the same set of axes. We need to use view-window settings just larger than the domain. In this case, Xmin = ¡30o , Xmax = 390o , Xscale = 30o : black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\574IB_STSL-2_18.CDR Tuesday, 16 February 2010 10:31:00 AM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS (Chapter 18) 575 The grid facility on the graphics calculator can also be helpful, particularly when a sketch is required. Using the appropriate function on the calculator gives the solutions x ¼ 17:5o , 163o . EXERCISE 18E.2 1 Solve each of the following for 0o 6 x 6 360o : a sin x = 0:414 b sin x = ¡0:673 ³x´ = 0:606 e cos 2 d sin(2x) = 0:162 F c cos x = 1:289 µ ¶ 2x f sin = 0:9367 3 USING TRIGONOMETRIC MODELS In this section we apply our knowledge of trigonometric functions and equations to different modelling situations. Self Tutor Example 6 The height h(t) metres of the tide above mean sea level on January 24th at Cape Town is modelled approximately by h(t) = 3 sin (30t) where t is the number of hours after midnight. a Graph y = h(t) for 0 6 t 6 24. b When is high tide and what is the maximum height? c What is the height at 2 pm? d If a ship can cross the harbour provided the tide is at least 2 m above mean sea level, when is crossing possible on January 24? a h(t) = 3 sin (30t) has period = h(t) 3 360 = 12 hours and h(0) = 0 30 A B 6 3 9 12 15 18 21 24 t noon -3 cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 b High tide is at 3 am and 3 pm. The maximum height is 3 m above the mean as seen at points A and B. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\575IB_STSL-2_18.CDR Monday, 15 February 2010 1:49:58 PM PETER IB_STSL-2ed 576 TRIGONOMETRIC FUNCTIONS (Chapter 18) c At 2 pm, t = 14 and h(14) = 3 sin (30 £ 14) ¼ 2:60 m So, the tide is 2:6 m above the mean. d h(t) 3 2 A t1 3 (14, 2.60) B t2 6 12 t3 9 15 h=2 t4 18 21 24 t -3 We need to solve h(t) = 2, so 3 sin (30t) = 2: Using a graphics calculator with Y1 = 3 sin (30X) and Y2 = 2 we obtain t1 = 1:39, t2 = 4:61, t3 = 13:39, t4 = 16:61 Now 1:39 hours = 1 hour 23 minutes, and so on. So, the ship can cross between 1:23 am and 4:37 am or 1:23 pm and 4:37 pm. EXERCISE 18F 1 The population of grasshoppers after t weeks where 0 6 t 6 12 is estimated by P (t) = 7500 + 3000 sin (90t). a b c d Find: i the initial estimate ii the estimate after 5 weeks. What is the greatest population size over this interval and when does it occur? When is the population: i 9000 ii 6000? During what time interval(s) does the population size exceed 10 000? 2 The model for the height of a light on a Ferris wheel is H(t) = 20 ¡ 19 sin (120t), where H is the height in metres above the ground, and t is in minutes. a Where is the light at time t = 0? b At what time is the light at its lowest in the first revolution of the wheel? c How long does the wheel take to complete one revolution? d Sketch the graph of the function H(t) over one revolution. 3 The population of water buffalo is given by P (t) = 400 + 250 sin (90t) where t is the number of years since the first estimate was made. a What was the initial estimate? b What was the population size after: i 6 months ii two years? c Find P (1). What is the significance of this value? cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 d Find the smallest population size and when it first occurred. e Find the first time when the herd exceeded 500. black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\576IB_STSL-2_18.CDR Monday, 15 February 2010 1:50:44 PM PETER IB_STSL-2ed TRIGONOMETRIC FUNCTIONS (Chapter 18) 577 4 Over a 28 day period, the cost per litre of petrol was modelled by C(t) = 9:2 cos(25:7t) + 107:8 cents L¡1 . a True or false? i “The cost per litre oscillates about 107:8 cents with maximum price $1:17.” ii “Every 14 days, the cycle repeats itself.” b What was the cost of petrol on day 4, to the nearest tenth of a cent per litre? c On what days was the petrol priced at $1:10 per litre? d What was the minimum cost per litre and when did it occur? REVIEW SET 18A 1 For each set of data below, draw a scatterplot and state if the data exhibits approximately periodic behaviour. a x 0 1 2 3 4 5 6 7 8 9 10 11 12 y 2:7 0:8 ¡1:7 ¡3 ¡2:1 0:3 2:5 2:9 1:3 ¡1:3 ¡2:9 ¡2:5 ¡0:3 b x y 0 5 1 3:5 2 6 3 ¡1:5 4 4 5 ¡2:5 6 ¡0:8 7 0:9 8 2:6 9 4:3 2 Draw the graph of y = 4 sin x for 0o 6 x 6 360o . 3 State the minimum and maximum values of: a y = 1 + sin x b y = ¡2 cos(3x) c y = 5 sin x ¡ 3 d y= 1 3 cos x + 1 4 y y = cos_x 1 0.5 x 100° 200° 300° 400° 500° 600° 700° 800° -0.5 -1 Use the graph of y = cos x to find, correct to the nearest 5o , solutions of: a cos x = ¡0:4, 0o 6 x 6 720o b cos x = 0:9, 0o 6 x 6 540o 5 Use technology to solve for 0o 6 x 6 360o : cyan magenta yellow 95 p 2 100 50 75 25 0 5 95 100 50 75 b 2 sin(3x) = 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 a cos x = 0:3 black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\577IB_STSL-2_18.cdr Friday, 5 March 2010 9:26:13 AM PETER c 43 + 8 sin x = 50:1 IB_STSL-2ed 578 TRIGONOMETRIC FUNCTIONS (Chapter 18) 6 An ecologist studying a species of water beetle estimates the population of a colony over an eight week period. If t is the number of weeks after the initial estimate is made, then the population in thousands can be modelled by P (t) = 5 + 2 sin(60t) where 0 6 t 6 8. a What was the initial population? b What were the smallest and largest populations? c During what time interval(s) did the population exceed 6000? REVIEW SET 18B 1 Which of the following graphs display periodic behaviour? a b y y 6 4 -720° -360 -720 -360° 360° 360 x 720° 720 2 x 360° 720° 2 Draw the graph of y = sin 3x for 0o 6 x 6 360o . 3 State the period of: a y = 4 sin( x5 ) c y= 4 cos( x2 ) b y = ¡2 cos(4x) +4 d y= 1 2 sin(3x) 4 Consider y = sin( x3 ) on the domain ¡360o 6 x 6 360o . Use the graph to solve, correct to the nearest 5o : a sin( x3 ) = ¡0:9 b sin( x3 ) = 14 y x y = sin_&^* 3 1 0.5 -300° -300 -200° -200 -100° -100 100° 100 200° 200 300° 300 x -0.5 cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 -1 black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\578IB_STSL-2_18.cdr Tuesday, 16 February 2010 10:32:29 AM PETER IB_STSL-2ed 579 TRIGONOMETRIC FUNCTIONS (Chapter 18) 5 On the same set of axes, for the domain 0o 6 x 6 360o , sketch: b y = cos x and y = ¡ 12 cos x a y = cos x and y = cos x ¡ 3 d y = cos x and y = 2 cos( x2 ) + 3 c y = sin x and y = 3 sin(2x) 6 A robot on Mars records the temperature every Mars day. A summary series, showing every one hundredth Mars day, is shown in the table below. Number of Mars days 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 Temp. (o C) ¡43 ¡15 ¡5 ¡21 ¡59 ¡79 ¡68 ¡50 ¡27 ¡8 ¡15 ¡70 ¡78 ¡68 a Find the maximum and minimum temperatures recorded by the robot. b Find a sine model for the temperature T in terms of the number of Mars days d. c Use this information to estimate the length of a Mars year. REVIEW SET 18C 1 For each of the following functions, find the: i amplitude ii vertical translation. a y = 2 sin x + 1 b f (x) = ¡3 cos x + 1 2 c g(x) = sin(4x) ¡ 5 a Without using technology, draw the graph of f(x) = sin( x3 ) + 2. 2 b For what values of k will f (x) = k have solutions? 3 Find the cosine function represented in the following graphs: a b y 4 y 3 360° x 1 x 90° 4 Use technology to solve for 0o 6 x 6 1000o : b cos( x2 ) = ¡0:458 a sin x = 0:382 5 For the following graphs, state the: i equation of the principal axis a ii amplitude b y 360° 720° 4 x 2 -1 magenta yellow 95 100 50 75 25 0 5 95 100 50 -2 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 -2 -3 cyan y 6 2 1 -360° iii period black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\579IB_STSL-2_18.cdr Friday, 5 March 2010 9:27:45 AM PETER 180° 360° x IB_STSL-2ed 580 TRIGONOMETRIC FUNCTIONS (Chapter 18) cyan magenta yellow 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 95 100 50 75 25 0 5 6 In an industrial city, the amount of pollution in the air becomes greater during the working week when factories are operating, and lessens over the weekend. The number of milligrams of pollutants in a cubic metre of air is given by P (t) = 40 ¡ 12 sin(51:4t) where t is the number of days after midnight on Saturday night. a What is the minimum level of pollution? b At what time during the week does this minimum level occur? black Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_18\580IB_STSL-2_18.cdr Monday, 15 February 2010 2:34:19 PM PETER IB_STSL-2ed