|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:|
|00:33||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.
|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.
|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.
|03:44||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
|06:08||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 fS, iS and PS, 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 PS.|
|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 fT, iT and PT, 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 PT.|
|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.|
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