Teaching trigonometry with technology

Transcription

1 Teaching trigonometry with technology Barry Kissane & Marian Kemp Murdoch University, WA In this workshop, we explore some of the ways in which the teaching of trigonometry might be supported by the availability of various forms of technology. We will make use of hand-held technologies (such as Casio graphics calculators or ClassPads) as well as demonstrate some Java applets. We will consider circular measures, graphs of functions, trigonometric identities, equations and statistical modelling and focus on activities that are not possible without the use of technology. Modern technology provides an excellent means of exploring many of the concepts associated with trigonometry, both trigonometric and circular functions. Many of these opportunities for learning were not available before technology development and access within schools we enjoy today. This workshop suggests some of the avenues for exploration. A. Introduction to circular measure Trigon The Flash applet called Round and Round a Circle from the Nrich website gives an excellent introduction to the powerful idea of a unit circle: See (You may be able to save this as a Flash applet to your computer to use offline from the link at depending on your browser and IT expertise.) The unit circle on a graphics calculator A dynamic unit circle can be drawn in parametric mode and traced. Use T from 0 to 4π and pitch = π/30. X in [-3.15,3.15] and Y in [-1.55,1.55] ensures circle is OK and can be traced. Turn off graph Func. Pitch setting ensures extreme values correct. Make sure GC set to radians. Trace the graph to see values for sin T and cos T as y and x values respectively. (A limitation is that T can t be negative.) As well as tracing, particular values can be shown. The example below shows the result of using G- Solve (F5) and then Y-CAL to obtain the sine and cosine of π/4, which can be seen to be the same in this case.

2 A table can be traced too, and will allow negative values for T. Table settings -2π to 2π and step π/60 shown below will ensure results for the main exact values for T (such as π/3, π/4 etc.) will be displayed correctly. In the diagram here, T = π/3 with sin T = ½ and cos T = 3/2 are shown. Reference: Kissane, B. & Kemp, M. 2006, Mathematics with a graphics calculator: Casio fx-9860g AU, Mirabooka: Mathematical Association of Western Australia, p.55) An Internet version for a computer The GeoGebra applet at provides a nice example of a unit circle that would be helpful for classroom display. B. Functions and their graphs Graphing on graphics calculators Graphs of trigonometric and circular functions are readily produced on graphics calculators. Set the calculator to radians to focus on circular functions with radian measure. In Graph mode (MENU 5) Draw any trigonometric graphs on the default Trig screen. Use trace or G-Solve (Root, Max, Min) to explore key points, noting periodicity, amplitude, etc. In Dynamic Graphing mode (MENU 6) Note that some trigonometric functions are B-IN (built in) functions or enter these manually. Use VAR (F4) to first SELect (F1) a dynamic variable, then fix (carefully) the other two parameters. Uses SET (F2) to set range of values (not too many!). Use SPEED (F3) to choose dynamic plotting method and speed. Then EXIT and tap DYNA (F6) to draw the graphs. Press AC/ON to stop graphing, if necessary.

3 Graphing applets on computers Maths Online The Austrian Maths Online website at has some excellent educational materials, with sound pedagogical advice for their use (including commentary for teachers and exercises with solutions for students. Some of the applets in the Gallery relate directly to trigonometry, using Java applets, with powerful connections between the unit circle and various graphs. Importantly, materials from this site can be downloaded for use on computers when Internet access is problematic. (Go to the Welcome page for details of this.) Manipula math The Japanese Manipula Math collection of Java applets at can be used online or can be purchased for individual or class use offline. A free download set of applets includes some concerned with trigonometric graphs. Statistical modelling on a graphics calculator Trigonometry is useful for modelling periodic phenomena. If periodic data are obtained, they can be entered into lists and graphed in Statistics mode of a calculator. One of the regression functions is Sine, which will suit some data quite well. (Students need to know about statistics, of course.) For example, the following data show the predicted times of sunrise in Fremantle on the first day of each month of (Source: Geoscience Australia at Date (day of year) Sunrise time (hr:min) January (1) 5:14 February (32) 5:42 March (61) 6:06 April (92) 6:28 May (122) 6:49 June (153) 7:09 July (183) 7:18 August (214) 7:06 September (245) 6:34 October (275) 5:55 November (305) 5:19 December (335) 5:04 Data are entered as day of the year and hours past midnight. A graph shows a roughly sinusoidal shape. The graph shows that a sine function fits the data well in this case:

4 Although the model is a consequence of choosing to use only some of the data (one day per month), it provides a reasonably good fit and the corresponding formula can be used to predict tolerably well the sunrise at any date during the year. One way of doing this is to tabulate the trigonometric function: Other real data (such as tidal data, or daily hours of sunshine over the year) can also be used to show the cyclical nature of circular functions and their appropriateness for modeling periodic phenomena. The Geoscience website above is a good source of data for Australian locations. C. Trigonometric identities and equations Graphics calculators Identities Identities can be checked in either Graph or Table mode by entering each side of the identity as a separate function, Y1(x) and Y2(x). If a relation is an identity, graphs of Y1 and Y2 will overlap, so that only one graph is shown: Similarly, table values for Y1 and Y2 will coincide for an identity; otherwise they won t. This is not a proof of course, but a sort of verification and also a way of clarifying the nature of an identity (distinguishing it from an equation). A nice verification of sin 2 x + cos 2 x = 1, graph Y1(x) = sin 2 x and then Y2(x) = cos 2 x. Trace the graph, occasionally flipping up and down between the two graphs using the up/down cursor! Notice that the sum of the y-coordinates appears to be constant, regardless of where the cursor is. Then Graph Y3(x) = Y1 + Y2. The result is a constant, 1. Turning off graphs of Y1 and Y2 will show only the sum of 1 for all values of x.

5 Equations To solve equations numerically, graph each side separately, and then use G-Solve (ISCT) to get points of intersection. The periodic nature of solutions is clarified, providing a motivation for a general solution. For some equations, it s a good idea to use Zoom Auto (F5) to ensure that all the graphs are visible on the screen. (Not for tan graphs!) For example, there are several sample equations shown in the recent NSW syllabus documents for senior secondary school: 3 + 2cos x = 15 cos x 2 sin x = cos x 2sin x = tan x sec 2 x = 3 2 2cos 2t = 0 sin 2x = x/2 Here is an example of equation solving, using the first sample equation above: 3 + 2cos x = 15 cos x Here are the first three solutions for x > 0: EQUA mode allows for solutions directly, but it s hard to see the periodic nature of the solutions. Note in the examples of equations above, that the equation sin 2x = x/2 highlights the need to use radians. To solve sec 2 x = 3, students can also graph Y1(x) = sec 2 x and then use G-Solve Y-Cal to find the points for which Y=3. Parametric equations The parametric equations for a unit circle were used earlier: {Xt1(T) = cos T, Yt1(T) = sin T}. Other shapes, such as Lissajous figures can be explored using appropriate parametric equations; an example is shown below.

6 CAS Calculators On a CAS calculator, similar kinds of activities can be undertaken. An important difference is that the algebraic capabilities of CAS calculators allow for identities to be examined symbolically, as shown below on the Casio ClassPad, using commands to expand and collect trigonometric terms. These are key components of standard by-hand approaches to constructing proofs of identities. As the second screen below shows, commands texpand and tcollect are both regarded in the ClassPad environment as kinds of transformations of algebraic (in this case, trigonometric) expressions. A difference for equations on a CAS calculator is that exact solutions to equations can be found and, in the case of trigonometric equations, general solutions might also be found. The screen below shows an example for the Casio ClassPad: The first screen indicates that there is an infinite number of solutions to the equation, sin x = 3/2. For convenience, these are shown on two separate lines in the screen dump above, representing the two families of solutions, x = 2πk + π/3 and x = 2πk + 2π/3. (Notice that the ClassPad has used a different symbol (constn(1) and constn(2) respectively) for the parameter k for each of the two parts of the general solution). The second screen above shows an important relationship between equations and identities: that an identity such as sin 2x = 2 sin x cos x might be regarded as an equation for which there are an infinite number of solutions, represented on the ClassPad with a solution set of {x=x}. Computers Manipula Math The website at provides many single-purpose Java applets related to aspect of trigonometry, some of which provide fresh ways of thinking about the concepts involved. One example is the applet concerning a proof of the Sine rule at