https://script.spoken-tutorial.org/index.php?title=Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English-timed&feed=atom&action=historyApplications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English-timed - Revision history2024-03-28T14:34:16ZRevision history for this page on the wikiMediaWiki 1.23.17https://script.spoken-tutorial.org/index.php?title=Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English-timed&diff=54080&oldid=prevPoojaMoolya: Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- ||00:01 ||Welcome to this tutorial on '''Inverse Trigonometric Functions'''. |- ||00:06 ||In this '''tutorial''', we will learn..."2020-10-21T07:18:58Z<p>Created page with "{|border=1 ||'''Time''' ||'''Narration''' |- ||00:01 ||Welcome to this tutorial on '''Inverse Trigonometric Functions'''. |- ||00:06 ||In this '''tutorial''', we will learn..."</p>
<p><b>New page</b></p><div>{|border=1<br />
||'''Time'''<br />
||'''Narration'''<br />
<br />
|-<br />
||00:01<br />
||Welcome to this tutorial on '''Inverse Trigonometric Functions'''.<br />
<br />
|-<br />
||00:06<br />
||In this '''tutorial''', we will learn to use '''GeoGebra''' to<br />
<br />
|-<br />
||00:11<br />
||Plot graphs of '''inverse trigonometric functions''' <br />
<br />
|-<br />
||00:15<br />
||Compare them to graphs of '''trigonometric functions''' <br />
<br />
|-<br />
||00:19<br />
||Create '''check-boxes''' to group and show or hide '''functions'''<br />
<br />
|-<br />
||00:24<br />
||To follow this '''tutorial''', you should be familiar with: <br />
<br />
|-<br />
||00:28<br />
||'''GeoGebra''' interface<br />
<br />
|-<br />
||00:31<br />
||'''Trigonometry'''<br />
<br />
|-<br />
||00:33<br />
||For relevant '''tutorials''', please visit our website.<br />
|-<br />
||00:37<br />
||Here I am using:<br />
<br />
'''Ubuntu Linux OS version 14.04'''<br />
<br />
'''GeoGebra 5.0.388.0 hyphen d'''<br />
|-<br />
||00:51<br />
||I have already opened the '''GeoGebra''' interface.<br />
|-<br />
||00:56<br />
||Now let us change '''x Axis units''' to '''radians'''.<br />
<br />
|-<br />
||01:01<br />
||In '''Graphics''' view, double-click on the '''x axis''' and then on '''Object Properties'''.<br />
|-<br />
||01:08<br />
||In the '''Object Properties''' menu, click on '''Preferences Graphics''' and then on '''xAxis'''.<br />
|-<br />
||01:17<br />
||Check the '''Distance''' option, select '''pi''' divided by 2 and then the '''Ticks first option'''.<br />
|-<br />
||01:28<br />
||Close the''' Preferences''' box.<br />
|-<br />
||01:31<br />
||Units of '''x-axis''' are in '''radians''' with interval of '''pi''' divided by 2 as shown.<br />
<br />
|-<br />
||01:38<br />
||'''GeoGebra''' will convert '''degrees''' of angle '''alpha''' to '''radians'''.<br />
|-<br />
||01:44<br />
||Note that the name appears when you place the mouse over any '''tool icon''' in the '''toolbar'''.<br />
|-<br />
||01:52<br />
||In the '''Graphics toolbar''', click on '''Slider''' and then in the top of '''Graphics''' view.<br />
|-<br />
||02:01<br />
||A '''slider dialog-box''' appears.<br />
|-<br />
||02:04<br />
||By default, '''Number''' radio button is selected.<br />
|-<br />
||02:08<br />
||In the '''Name''' field, select '''theta''' from the '''Symbol menu'''.<br />
|-<br />
||02:14<br />
||Type the '''Min''' value as minus 360 and '''Max''' plus 360 with '''Increment''' 1.<br />
<br />
Click '''OK'''.<br />
|-<br />
||02:27<br />
||This creates a '''number slider theta''' from minus 360 to plus 360.<br />
|-<br />
||02:34<br />
||In the '''input bar''', type '''alpha is equal to theta divided by 180 in parentheses''', and then '''pi'''.<br />
<br />
|-<br />
||02:47<br />
||Note how '''GeoGebra''' inserts a space between the right parenthesis and '''pi''' for multiplication.<br />
<br />
Press '''Enter'''.<br />
|-<br />
||02:57<br />
||Drag '''slider theta''' to minus 360 and then back to 360.<br />
|-<br />
||03:04<br />
||In '''Algebra''' view, observe how '''alpha''' changes from minus '''2pi''' to '''2pi radians''' as you change '''theta'''.<br />
|-<br />
||03:13<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||03:18<br />
||In the '''input bar''', type the following command:<br />
<br />
|-<br />
||03:22<br />
||'''f underscore S colon is equal to Function with capital F'''<br />
<br />
|-<br />
||03:29<br />
||Type the following words in square brackets.<br />
<br />
'''sin, x in parentheses, comma minus 2 pi comma alpha'''.<br />
<br />
|-<br />
||03:42<br />
||Press '''Enter'''.<br />
|-<br />
||03:44<br />
||Drag the boundary to see '''Algebra''' view properly.<br />
|-<br />
|| 03:49<br />
||Here, '''fS''' defines the '''sine function''' of '''x'''. <br />
<br />
|-<br />
||03:54<br />
||'''x''' is between '''-2 pi''' and '''alpha''' which can take a maximum value of '''2pi'''. <br />
<br />
|-<br />
||04:03<br />
||This is called the '''domain''' of the '''function'''. <br />
|-<br />
||04:08<br />
||Drag the boundary to see '''Graphics''' View properly.<br />
|-<br />
|| 04:13<br />
||Drag '''slider theta''' from minus 360 to 360. <br />
|-<br />
||04:19<br />
||This graphs the '''sine function''' of '''x'''. <br />
|-<br />
||04:23<br />
||In the toolbar, click on the bottom right triangle of the last button. <br />
<br />
|-<br />
||04:29<br />
||Note that this button is called '''Move Graphics View'''.<br />
|-<br />
||04:34<br />
||In the menu that appears, click on '''Zoom Out'''.<br />
<br />
|-<br />
||04:39<br />
||Click in '''Graphics view''' to see '''2 pi radians''' on either side of the '''origin'''.<br />
|-<br />
|| 04:47<br />
||Again, click on '''Move Graphics View''' and drag the background to see the graph properly. <br />
|-<br />
||04:56<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||05:01<br />
||'''Inverse Trigonometric Functions'''<br />
<br />
|-<br />
||05:04<br />
||For example, If '''inverse sine''' of '''z''' also known as '''arcsin''' of '''z''' is '''w'''. <br />
<br />
Then, '''z''' is '''sin w'''.<br />
<br />
|-<br />
||05:15<br />
||'''w''' can have multiple values. <br />
<br />
So a '''principal value''' has to be defined and the '''domain''' has to be restricted. <br />
<br />
|-<br />
||05:23<br />
||To get the '''inverse function''' graph, interchange '''x''' and '''y''' axes.<br />
<br />
|-<br />
||05:29<br />
||Next, change curvature of '''trigonometric function graph'''.<br />
<br />
|-<br />
||05:34<br />
||You can pause and refer to the example in the '''additional material''' provided for this '''tutorial'''. <br />
<br />
|-<br />
||05:41<br />
||Let us go back to the '''GeoGebra''' window. <br />
<br />
|-<br />
||05:45<br />
||In the '''input bar''', type the following command:<br />
<br />
'''i underscore S colon is equal to Function with capital F'''<br />
<br />
|-<br />
||05:56<br />
||Type the following words in square brackets.<br />
<br />
'''a sin, x in parentheses, comma minus 1 comma 1'''<br />
<br />
|-<br />
||06:06<br />
||Press '''Enter'''.<br />
<br />
|-<br />
||06:08<br />
||Drag the boundary to see '''Algebra''' view properly. <br />
<br />
|-<br />
||06:12<br />
||This graphs the '''inverse sine''' (or '''arc sine''') function of '''x'''.<br />
<br />
|-<br />
||06:18<br />
||Note that '''x''' and '''y axes''' are interchanged for this '''inverse sine function'''. <br />
<br />
|-<br />
||06:25<br />
||Its '''domain''' set of '''x''' values lies between minus 1 and 1.<br />
<br />
Observe the graph.<br />
|-<br />
|| 06:34<br />
||Drag the boundary to see '''Graphics''' view properly. <br />
|-<br />
||06:39<br />
||In the '''input bar''', type the following command:<br />
<br />
'''P underscore S colon is equal to''' Type the following words in parentheses '''sin alpha in parentheses comma alpha'''. Press '''Enter'''.<br />
<br />
|-<br />
||06:58<br />
||This creates point '''PS''' on the '''inverse sine''' graph.<br />
<br />
|-<br />
||07:04<br />
||On the '''sine function''' graph, '''PS''' would be '''alpha comma sine alpha'''. <br />
|-<br />
||07:10<br />
||In '''Algebra''' view, right-click on '''PS''', check the '''Trace On''' option.<br />
|-<br />
||07:17<br />
||Drag '''slider theta''' to 360.<br />
|-<br />
||07:21<br />
||Traces appear for the '''inverse sine function''' graph for '''alpha'''.<br />
<br />
|-<br />
||07:26s<br />
||'''fs''' also appears in '''Graphics''' view.<br />
<br />
|-<br />
||07:30<br />
||Compare '''iS''' and traces of '''PS'''.<br />
<br />
|-<br />
||07:35<br />
||Note that the '''domain''' for the graph that '''PS''' traces is not restricted from minus 1 to 1. <br />
|-<br />
||07:43<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||07:47<br />
||Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.<br />
|-<br />
||07:53<br />
||In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.<br />
<br />
|-<br />
||08:02<br />
||'''Cosine and Inverse Cosine Functions'''<br />
<br />
|-<br />
||08:06<br />
||Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.<br />
<br />
|-<br />
||08:12<br />
||Its '''domain''' should be from '''minus 2 pi''' to '''alpha'''. <br />
<br />
|-<br />
||08:17<br />
||Graph the '''inverse cosine function iC" in the '''domain''' from minus 1 to 1. <br />
<br />
|-<br />
||08:24<br />
||Create a point '''PC''' whose '''co-ordinates''' are '''cos alpha comma alpha'''. <br />
<br />
|-<br />
||08:30<br />
||The '''domain''' of the inverse cosine''' graph that '''PC''' traces will go beyond minus 1 and 1. <br />
|-<br />
||08:37<br />
||The '''cosine''' and '''inverse cosine functions''' should look like this. <br />
|-<br />
||08:46<br />
||In '''Algebra view, uncheck '''fC, iC and PC and move the background to erase traces of PC.<br />
|-<br />
||08:58<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||09:03<br />
||'''Tangent and Inverse Tangent Functions'''<br />
<br />
|-<br />
||09:07<br />
||Now graph the '''tangent function fT'''.<br />
<br />
|-<br />
||09:11<br />
||Its domain should also be from '''minus 2 pi''' to '''alpha'''. <br />
<br />
|-<br />
||09:16<br />
||We will look at the graph for the '''inverse tangent function iT'''.<br />
<br />
|-<br />
||09:21<br />
||Its domain will be from minus infinity to infinity. <br />
<br />
|-<br />
||09:26<br />
||Create a point '''PT''' whose '''co-ordinates''' are '''tan alpha comma alpha'''. <br />
<br />
|-<br />
||09:32<br />
||The '''domain''' of the '''inverse tangent''' graph that '''PT''' traces will go beyond minus 1 and 1. <br />
|-<br />
||09:39<br />
||Let us look at the '''inverse tangent function''' graph in the domain from minus 1 to 1. <br />
|-<br />
||09:45<br />
||To type infinity, click in the '''input bar''' and on '''symbol alpha''' appearing at the right end of the bar. <br />
<br />
|-<br />
||09:53<br />
||In the '''symbol menu''', click on the '''infinity symbol''' in the third row and third column from the right. <br />
<br />
|-<br />
||10:01<br />
||In the '''input bar''', type the following command:<br />
<br />
'''i underscore T colon is equal to Function with capital F'''<br />
<br />
|-<br />
||10:12<br />
||Type the following words in square brackets '''atan, x in parentheses, comma minus infinity comma infinity'''. Press '''Enter'''.<br />
|-<br />
||10:26<br />
||This graphs the '''inverse tangent''' function of '''x'''.<br />
<br />
|-<br />
||10:31<br />
||'''x''' lies between minus '''infinity''' and '''infinity'''.<br />
<br />
Observe the graph.<br />
|-<br />
||10:39<br />
||Drag '''slider theta''' to 360.<br />
|-<br />
||10:43<br />
||Compare traces of '''PT''' and '''iT'''.<br />
|-<br />
||10:48<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||10:53<br />
||Drag the background slightly to the erase the traces of '''PT'''<br />
|-<br />
||10:58<br />
||In '''Algebra''' view, uncheck '''fT''' and '''PT'''.<br />
|-<br />
||11:04<br />
||In '''Algebra''' view, check '''fS''', '''iS''', and '''PS''' to show them again.<br />
|-<br />
||11:14<br />
||Let us create check boxes to make it easier to group and see different functions at a time.<br />
|-<br />
||11:21<br />
||Under '''Slider''', click on '''Check-box'''.<br />
<br />
|-<br />
||11:25<br />
||Click on the top of the grid in '''Graphics''' view.<br />
|-<br />
||11:29<br />
||A'''Check-Box to Show/Hide Objects dialog-box''' appears.<br />
|-<br />
||11:35<br />
||In the '''Caption''' field, type '''SIN.'''<br />
|-<br />
||11:38<br />
||Click on '''Objects''' drop-down menu to select '''f<sub>S</sub>, i<sub>S</sub>''' and '''P<sub>S</sub>''', one by one, click '''Apply'''.<br />
|-<br />
||11:50<br />
||A '''check-box''' '''SIN''' is created in '''Graphics''' view.<br />
<br />
|-<br />
||11:55<br />
||It gives us the option to display or hide '''sine, arcsine''' graphs and point '''P<sub>S</sub>'''.<br />
|-<br />
||12:02<br />
||Again, click on '''Check Box'''.<br />
<br />
|-<br />
||12:05<br />
||Click on the top of the grid in '''Graphics''' view.<br />
|-<br />
||12:09<br />
||A '''Checkbox to Show/Hide Objects''' dialog-box appears.<br />
|-<br />
||12:14<br />
||In the '''Caption''' field, type '''TAN'''.<br />
|-<br />
||12:16<br />
||Click on '''Objects''' drop-down menu to select '''f<sub>T</sub>, i<sub>T</sub>''' and '''P<sub>T</sub>''', one by one, click '''Apply'''.<br />
|-<br />
||12:29<br />
||A '''checkbox''' '''TAN''' is created in '''Graphics''' view.<br />
<br />
|-<br />
||12:33<br />
||It gives us the option to display or hide '''tangent, arctangent''' graphs and point '''P<sub>T</sub>'''.<br />
|-<br />
||12:40<br />
||In the '''toolbar''', click on the first '''Move''' button and uncheck all boxes.<br />
|-<br />
||12:48<br />
||Check the '''SIN''' box.<br />
|-<br />
||12:51<br />
||Drag '''slider theta''' to 360.<br />
|-<br />
||12:55<br />
||Observe '''fS, iS''' and traces of '''PS''' appear in '''Graphics''' view.<br />
|-<br />
||13:03<br />
||Uncheck the '''SIN''' box.<br />
|-<br />
||13:06<br />
||Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.<br />
|-<br />
||13:12<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||13:17<br />
||Check the '''TAN''' box.<br />
|-<br />
||13:19<br />
||Drag '''slider theta''' to 360.<br />
|-<br />
||13:23<br />
||Observe '''fT, iT''' and traces of '''PT''' appear in '''Graphics''' view.<br />
|-<br />
||13:31<br />
||Drag '''slider theta''' back to minus 360.<br />
|-<br />
||13:35<br />
||Check the '''SIN''' box.<br />
|-<br />
||13:37<br />
||Drag '''slider theta''' to 360.<br />
|-<br />
||13:41<br />
||Observe the '''functions''' appearing in '''Graphics''' view.<br />
|-<br />
||13:45<br />
||Let us summarize.<br />
|-<br />
||13:47<br />
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:<br />
<br />
|-<br />
||13:52<br />
||Graph '''trigonometric functions'''<br />
<br />
|-<br />
||13:54<br />
||Graph '''inverse trigonometric functions'''<br />
<br />
|-<br />
||13:57<br />
||Create '''checkboxes''' to group and show/hide '''functions'''<br />
|-<br />
||14:02<br />
||As an assignment,<br />
<br />
|-<br />
||14:04<br />
||Plot graphs of, '''Secant''' and '''arcsecant''', '''Cosecant''' and '''arccosecant''', '''Cotangent''' and '''arccotangent'''<br />
<br />
|-<br />
||14:17<br />
||For hints, you can refer to the '''additional material''' provided. <br />
|-<br />
||14:22<br />
||The video at the following link summarizes the '''Spoken Tutorial project'''.<br />
<br />
Please download and watch it.<br />
|-<br />
||14:30<br />
||The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.<br />
<br />
For more details, please write to us.<br />
|-<br />
||14:40<br />
||Please post your timed queries on this forum.<br />
|-<br />
||14:44<br />
||The '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.<br />
<br />
More information on this mission is available at this link.<br />
|-<br />
||14:56<br />
||This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.<br />
<br />
Thank you for joining.<br />
|-<br />
|}</div>PoojaMoolya