Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English-timed
From Script | Spoken-Tutorial
Time | Narration |
00:01 | Welcome to this tutorial on Inverse Trigonometric Functions. |
00:06 | In this tutorial, we will learn to use GeoGebra to |
00:11 | Plot graphs of inverse trigonometric functions |
00:15 | Compare them to graphs of trigonometric functions |
00:19 | Create check-boxes to group and show or hide functions |
00:24 | To follow this tutorial, you should be familiar with: |
00:28 | GeoGebra interface |
00:31 | Trigonometry |
00:33 | For relevant tutorials, please visit our website. |
00:37 | Here I am using:
Ubuntu Linux OS version 14.04 GeoGebra 5.0.388.0 hyphen d |
00:51 | I have already opened the GeoGebra interface. |
00:56 | Now let us change x Axis units to radians. |
01:01 | In Graphics view, double-click on the x axis and then on Object Properties. |
01:08 | In the Object Properties menu, click on Preferences Graphics and then on xAxis. |
01:17 | Check the Distance option, select pi divided by 2 and then the Ticks first option. |
01:28 | Close the Preferences box. |
01:31 | Units of x-axis are in radians with interval of pi divided by 2 as shown. |
01:38 | GeoGebra will convert degrees of angle alpha to radians. |
01:44 | Note that the name appears when you place the mouse over any tool icon in the toolbar. |
01:52 | In the Graphics toolbar, click on Slider and then in the top of Graphics view. |
02:01 | A slider dialog-box appears. |
02:04 | By default, Number radio button is selected. |
02:08 | In the Name field, select theta from the Symbol menu. |
02:14 | Type the Min value as minus 360 and Max plus 360 with Increment 1.
Click OK. |
02:27 | This creates a number slider theta from minus 360 to plus 360. |
02:34 | In the input bar, type alpha is equal to theta divided by 180 in parentheses, and then pi. |
02:47 | Note how GeoGebra inserts a space between the right parenthesis and pi for multiplication.
Press Enter. |
02:57 | Drag slider theta to minus 360 and then back to 360. |
03:04 | In Algebra view, observe how alpha changes from minus 2pi to 2pi radians as you change theta. |
03:13 | Drag slider theta back to minus 360. |
03:18 | In the input bar, type the following command: |
03:22 | f underscore S colon is equal to Function with capital F |
03:29 | Type the following words in square brackets.
sin, x in parentheses, comma minus 2 pi comma alpha. |
03:42 | Press Enter. |
03:44 | Drag the boundary to see Algebra view properly. |
03:49 | Here, fS defines the sine function of x. |
03:54 | x is between -2 pi and alpha which can take a maximum value of 2pi. |
04:03 | This is called the domain of the function. |
04:08 | Drag the boundary to see Graphics View properly. |
04:13 | Drag slider theta from minus 360 to 360. |
04:19 | This graphs the sine function of x. |
04:23 | In the toolbar, click on the bottom right triangle of the last button. |
04:29 | Note that this button is called Move Graphics View. |
04:34 | In the menu that appears, click on Zoom Out. |
04:39 | Click in Graphics view to see 2 pi radians on either side of the origin. |
04:47 | Again, click on Move Graphics View and drag the background to see the graph properly. |
04:56 | Drag slider theta back to minus 360. |
05:01 | Inverse Trigonometric Functions |
05:04 | For example, If inverse sine of z also known as arcsin of z is w.
Then, z is sin w. |
05:15 | w can have multiple values.
So a principal value has to be defined and the domain has to be restricted. |
05:23 | To get the inverse function graph, interchange x and y axes. |
05:29 | Next, change curvature of trigonometric function graph. |
05:34 | You can pause and refer to the example in the additional material provided for this tutorial. |
05:41 | Let us go back to the GeoGebra window. |
05:45 | In the input bar, type the following command:
i underscore S colon is equal to Function with capital F |
05:56 | Type the following words in square brackets.
a sin, x in parentheses, comma minus 1 comma 1 |
06:06 | Press Enter. |
06:08 | Drag the boundary to see Algebra view properly. |
06:12 | This graphs the inverse sine (or arc sine) function of x. |
06:18 | Note that x and y axes are interchanged for this inverse sine function. |
06:25 | Its domain set of x values lies between minus 1 and 1.
Observe the graph. |
06:34 | Drag the boundary to see Graphics view properly. |
06:39 | In the input bar, type the following command:
P underscore S colon is equal to Type the following words in parentheses sin alpha in parentheses comma alpha. Press Enter. |
06:58 | This creates point PS on the inverse sine graph. |
07:04 | On the sine function graph, PS would be alpha comma sine alpha. |
07:10 | In Algebra view, right-click on PS, check the Trace On option. |
07:17 | Drag slider theta to 360. |
07:21 | Traces appear for the inverse sine function graph for alpha. |
07:26s | fs also appears in Graphics view. |
07:30 | Compare iS and traces of PS. |
07:35 | Note that the domain for the graph that PS traces is not restricted from minus 1 to 1. |
07:43 | Drag slider theta back to minus 360. |
07:47 | Click and drag the background in Graphics view to erase traces of PS. |
07:53 | In Algebra view, uncheck fS, iS, and PS to hide them. |
08:02 | Cosine and Inverse Cosine Functions |
08:06 | Follow the steps shown for SINE to graph the cosine function fC. |
08:12 | Its domain should be from minus 2 pi to alpha. |
08:17 | Graph the inverse cosine function iC" in the domain from minus 1 to 1. |
08:24 | Create a point PC whose co-ordinates are cos alpha comma alpha. |
08:30 | The domain of the inverse cosine graph that PC traces will go beyond minus 1 and 1. |
08:37 | The cosine and inverse cosine functions should look like this. |
08:46 | In Algebra view, uncheck fC, iC and PC and move the background to erase traces of PC. |
08:58 | Drag slider theta back to minus 360. |
09:03 | Tangent and Inverse Tangent Functions |
09:07 | Now graph the tangent function fT. |
09:11 | Its domain should also be from minus 2 pi to alpha. |
09:16 | We will look at the graph for the inverse tangent function iT. |
09:21 | Its domain will be from minus infinity to infinity. |
09:26 | Create a point PT whose co-ordinates are tan alpha comma alpha. |
09:32 | The domain of the inverse tangent graph that PT traces will go beyond minus 1 and 1. |
09:39 | Let us look at the inverse tangent function graph in the domain from minus 1 to 1. |
09:45 | To type infinity, click in the input bar and on symbol alpha appearing at the right end of the bar. |
09:53 | In the symbol menu, click on the infinity symbol in the third row and third column from the right. |
10:01 | In the input bar, type the following command:
i underscore T colon is equal to Function with capital F |
10:12 | Type the following words in square brackets atan, x in parentheses, comma minus infinity comma infinity. Press Enter. |
10:26 | This graphs the inverse tangent function of x. |
10:31 | x lies between minus infinity and infinity.
Observe the graph. |
10:39 | Drag slider theta to 360. |
10:43 | Compare traces of PT and iT. |
10:48 | Drag slider theta back to minus 360. |
10:53 | Drag the background slightly to the erase the traces of PT |
10:58 | In Algebra view, uncheck fT and PT. |
11:04 | In Algebra view, check fS, iS, and PS to show them again. |
11:14 | Let us create check boxes to make it easier to group and see different functions at a time. |
11:21 | Under Slider, click on Check-box. |
11:25 | Click on the top of the grid in Graphics view. |
11:29 | ACheck-Box to Show/Hide Objects dialog-box appears. |
11:35 | In the Caption field, type SIN. |
11:38 | Click on Objects drop-down menu to select f_{S}, i_{S} and P_{S}, one by one, click Apply. |
11:50 | A check-box SIN is created in Graphics view. |
11:55 | It gives us the option to display or hide sine, arcsine graphs and point P_{S}. |
12:02 | Again, click on Check Box. |
12:05 | Click on the top of the grid in Graphics view. |
12:09 | A Checkbox to Show/Hide Objects dialog-box appears. |
12:14 | In the Caption field, type TAN. |
12:16 | Click on Objects drop-down menu to select f_{T}, i_{T} and P_{T}, one by one, click Apply. |
12:29 | A checkbox TAN is created in Graphics view. |
12:33 | It gives us the option to display or hide tangent, arctangent graphs and point P_{T}. |
12:40 | In the toolbar, click on the first Move button and uncheck all boxes. |
12:48 | Check the SIN box. |
12:51 | Drag slider theta to 360. |
12:55 | Observe fS, iS and traces of PS appear in Graphics view. |
13:03 | Uncheck the SIN box. |
13:06 | Click on and move Graphics view slightly to erase traces of PS. |
13:12 | Drag slider theta back to minus 360. |
13:17 | Check the TAN box. |
13:19 | Drag slider theta to 360. |
13:23 | Observe fT, iT and traces of PT appear in Graphics view. |
13:31 | Drag slider theta back to minus 360. |
13:35 | Check the SIN box. |
13:37 | Drag slider theta to 360. |
13:41 | Observe the functions appearing in Graphics view. |
13:45 | Let us summarize. |
13:47 | In this tutorial, we have learnt how to use GeoGebra to: |
13:52 | Graph trigonometric functions |
13:54 | Graph inverse trigonometric functions |
13:57 | Create checkboxes to group and show/hide functions |
14:02 | As an assignment, |
14:04 | Plot graphs of, Secant and arcsecant, Cosecant and arccosecant, Cotangent and arccotangent |
14:17 | For hints, you can refer to the additional material provided. |
14:22 | The video at the following link summarizes the Spoken Tutorial project.
Please download and watch it. |
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14:56 | This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |