Difference between revisions of "Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English"

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|-
 
|-
−
| | '''Slide Number 1'''
+
||'''Slide Number 1'''
  
 
'''Title Slide'''
 
'''Title Slide'''
−
| | Welcome to this tutorial on '''Inverse Trigonometric Functions'''.
+
||Welcome to this tutorial on '''Inverse Trigonometric Functions'''.
 
|-
 
|-
−
| | '''Slide Number 2'''
+
||'''Slide Number 2'''
  
 
'''Learning Objectives'''
 
'''Learning Objectives'''
−
| | In this '''tutorial''', we will learn to use '''GeoGebra''' to
+
||In this '''tutorial''', we will learn to use '''GeoGebra''' to
  
 
Plot graphs of '''inverse trigonometric functions'''  
 
Plot graphs of '''inverse trigonometric functions'''  
Line 20: Line 20:
 
Create '''check-boxes''' to group and show or hide '''functions'''
 
Create '''check-boxes''' to group and show or hide '''functions'''
 
|-
 
|-
−
| | '''Slide Number 3'''
+
||'''Slide Number 3'''
 
'''Pre-requisites'''
 
'''Pre-requisites'''
  
 
'''www.spoken-tutorial.org'''
 
'''www.spoken-tutorial.org'''
−
| | To follow this '''tutorial''', you should be familiar with:
+
||To follow this '''tutorial''', you should be familiar with:
 
'''GeoGebra''' interface
 
'''GeoGebra''' interface
  
Line 31: Line 31:
 
For relevant '''tutorials''', please visit our website.
 
For relevant '''tutorials''', please visit our website.
 
|-
 
|-
−
| | '''Slide Number 4'''
+
||'''Slide Number 4'''
 +
 
 
'''System Requirement'''
 
'''System Requirement'''
−
| | Here I am using:
+
||Here I am using:
 
'''Ubuntu Linux OS version. 14.04'''
 
'''Ubuntu Linux OS version. 14.04'''
  
 
'''GeoGebra 5.0.388.0-d'''
 
'''GeoGebra 5.0.388.0-d'''
 
|-
 
|-
−
| | Show the '''GeoGebra''' window.
+
||Show the '''GeoGebra''' window.
−
| | I have already opened the '''GeoGebra''' interface.
+
||I have already opened the '''GeoGebra''' interface.
 
|-
 
|-
−
| | '''Switching x axis to radians'''
+
||'''Switching x axis to radians'''
  
 
Double click on '''x axis''' in '''Graphics '''view >> '''Object Properties'''
 
Double click on '''x axis''' in '''Graphics '''view >> '''Object Properties'''
−
| | Now let us change '''x Axis units''' to '''radians'''.
+
||Now let us change '''x Axis units''' to '''radians'''.
  
 
In '''Graphics''' view, double-click on the '''x axis'''  and then on '''Object Properties'''.
 
In '''Graphics''' view, double-click on the '''x axis'''  and then on '''Object Properties'''.
 
|-
 
|-
−
| | Click on '''Preferences-Graphics''' >> '''x axis'''.
+
||Click on '''Preferences-Graphics''' >> '''x axis'''.
−
| | In the '''Object Properties''' menu, click on '''Preferences-Graphics''' and then on '''x Axis'''.
+
||In the '''Object Properties''' menu, click on '''Preferences-Graphics''' and then on '''x Axis'''.
 
|-
 
|-
−
| | Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
+
||Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
−
| | Check the '''Distance''' option, select '''pi''' divided by 2 and then the '''Ticks first option'''.
+
||Check the '''Distance''' option, select '''pi''' divided by 2 and then the '''Ticks first option'''.
 
|-
 
|-
−
| | Close the '''Preferences''' box.
+
||Close the '''Preferences''' box.
−
| | Close the''' Preferences''' box.
+
||Close the''' Preferences''' box.
 
|-
 
|-
−
| | Point to '''x-axis'''.
+
||Point to '''x-axis'''.
−
| | Units of '''x-axis''' are in '''radians''' with interval of '''pi''' divided by 2 as shown.
+
||Units of '''x-axis''' are in '''radians''' with interval of '''pi''' divided by 2 as shown.
  
 
'''GeoGebra''' will convert '''degrees''' of angle '''alpha''' to '''radians'''.
 
'''GeoGebra''' will convert '''degrees''' of angle '''alpha''' to '''radians'''.
 
|-
 
|-
−
| | Point to the '''toolbar'''.
+
||Point to the '''toolbar'''.
−
| | Note that the name appears when you place the mouse over any '''tool icon''' in the '''toolbar'''.
+
||Note that the name appears when you place the mouse over any '''tool icon''' in the '''toolbar'''.
 
|-
 
|-
−
| | Click on '''Slider''' tool >> click on '''Graphics''' view.
+
||Click on '''Slider''' tool >> click on '''Graphics''' view.
−
| | In the '''Graphics toolbar''', click on '''Slider''' and then in the top of '''Graphics''' view.
+
||In the '''Graphics toolbar''', click on '''Slider''' and then in the top of '''Graphics''' view.
 
|-
 
|-
−
| | Point to the '''Slider dialog box'''.
+
||Point to the '''Slider dialog box'''.
−
| | A '''slider dialog-box''' appears.
+
||A '''slider dialog-box''' appears.
 
|-
 
|-
−
| | Point to Number radio button.
+
||Point to Number radio button.
−
| | By default, '''Number''' radio-button is selected.
+
||By default, '''Number''' radio-button is selected.
 
|-
 
|-
−
| | Type '''Name''' as '''symbol theta ϴ'''.
+
||Type '''Name''' as '''symbol theta ϴ'''.
−
| | In the '''Name''' field, select '''theta''' from the '''Symbol menu'''.
+
||In the '''Name''' field, select '''theta''' from the '''Symbol menu'''.
 
|-
 
|-
−
| | Point to''' Min, Max''' and '''Increment''' values.
+
||Point to''' Min, Max''' and '''Increment''' values.
  
 
Click '''OK'''.
 
Click '''OK'''.
−
| | Type the '''Min''' value as minus 360 and '''Max''' plus 360 with '''Increment''' 1.
+
||Type the '''Min''' value as minus 360 and '''Max''' plus 360 with '''Increment''' 1.
  
 
Click '''OK'''.
 
Click '''OK'''.
 
|-
 
|-
−
| | Point to slider '''ϴ'''.
+
||Point to slider '''ϴ'''.
−
| | This creates a '''number slider theta''' from minus 360 to plus 360.
+
||This creates a '''number slider theta''' from minus 360 to plus 360.
 
|-
 
|-
−
| | In '''input bar''', type '''α = (ϴ /180) π'''.
+
||In '''input bar''', type '''α = (ϴ /180) π'''.
  
 
Point to space between the right parenthesis and '''pi''' for multiplication.
 
Point to space between the right parenthesis and '''pi''' for multiplication.
  
 
Press '''Enter'''
 
Press '''Enter'''
−
| | In the '''input bar''', type '''alpha  is equal to theta divided by 180 in parentheses''', and then '''pi'''.
+
||In the '''input bar''', type '''alpha  is equal to theta divided by 180 in parentheses''', and then '''pi'''.
  
 
Note how '''GeoGebra''' inserts a space between the right parenthesis and '''pi''' for multiplication.
 
Note how '''GeoGebra''' inserts a space between the right parenthesis and '''pi''' for multiplication.
Line 98: Line 99:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to -360 and then back to 360.
+
||Drag '''slider ϴ''' to -360 and then back to 360.
−
| | Drag '''slider theta''' from minus 360 and then back to 360.
+
||Drag '''slider theta''' from minus 360 and then back to 360.
 
|-
 
|-
−
| | Point to values of '''α''' in '''Algebra''' view.
+
||Point to values of '''α''' in '''Algebra''' view.
−
| | In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to 2 '''pi radians''' as you change '''theta'''.
+
||In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to 2 '''pi radians''' as you change '''theta'''.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to minus 360.
+
||Drag '''slider ϴ''' to minus 360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | '''Sine function'''
+
||'''Sine function'''
  
−
In '''input bar''', type''' f_S: = Function[sin(x), -2π, α]''' >> press '''Enter'''.
+
In '''input bar''', type '''f_S: = Function[sin(x), -2π, α]''' >> press '''Enter'''.
−
| | In the '''input bar''', type the following command:
+
||In the '''input bar''', type the following command:
  
 
'''f underscore S colon is equal to Function with capital F'''
 
'''f underscore S colon is equal to Function with capital F'''
Line 120: Line 121:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Drag the boundary to see '''Algebra''' view properly.
+
||Drag the boundary to see '''Algebra''' view properly.
−
| | Drag the boundary to see '''Algebra''' view properly.
+
||Drag the boundary to see '''Algebra''' view properly.
 
|-
 
|-
−
| | Point to '''fS''' in '''Algebra''' view.  
+
||Point to '''fS''' in '''Algebra''' view.  
−
| | Here, '''fS''' defines the '''sine function''' of '''x'''.  
+
||Here, '''fS''' defines the '''sine function''' of '''x'''.  
  
 
'''x''' is between -2 '''pi''' and '''alpha''' which can take a maximum value of 2 '''pi'''.   
 
'''x''' is between -2 '''pi''' and '''alpha''' which can take a maximum value of 2 '''pi'''.   
Line 130: Line 131:
 
This is called the '''domain''' of the '''function'''.   
 
This is called the '''domain''' of the '''function'''.   
 
|-
 
|-
−
| | Drag the boundary to see '''Graphics''' View properly.
+
||Drag the boundary to see '''Graphics''' View properly.
−
| | Drag the boundary to see '''Graphics''' View properly.
+
||Drag the boundary to see '''Graphics''' View properly.
 
|-
 
|-
−
| | Drag '''slider theta''' from minus 360 to 360.   
+
||Drag '''slider theta''' from minus 360 to 360.   
−
| | Drag '''slider theta''' from minus 360 to 360.  
+
||Drag '''slider theta''' from minus 360 to 360.  
 
|-
 
|-
−
| | Point to '''fS sine function''' graph.   
+
||Point to '''fS sine function''' graph.   
−
| | This graphs the '''sine function''' of '''x'''.   
+
||This graphs the '''sine function''' of '''x'''.   
 
|-
 
|-
−
| | In the toolbar, click on the bottom right triangle of the last button.  
+
||In the toolbar, click on the bottom right triangle of the last button.  
  
 
Point to '''Move Graphics View''' button.  
 
Point to '''Move Graphics View''' button.  
−
| | In the toolbar, click on the bottom right triangle of the last button.  
+
||In the toolbar, click on the bottom right triangle of the last button.  
  
 
Note that this button is called '''Move Graphics View'''.
 
Note that this button is called '''Move Graphics View'''.
 
|-
 
|-
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
+
||Under '''Move Graphics View''', click on '''Zoom Out''' tool.
  
 
Click on '''Graphics view''' to see 2 '''pi radians''' on either side of '''origin'''.
 
Click on '''Graphics view''' to see 2 '''pi radians''' on either side of '''origin'''.
−
| | In the menu that appears, click on '''Zoom Out'''.
+
||In the menu that appears, click on '''Zoom Out'''.
  
 
Click in '''Graphics view''' to see 2 '''pi radians''' on either side of the '''origin'''.
 
Click in '''Graphics view''' to see 2 '''pi radians''' on either side of the '''origin'''.
 
|-
 
|-
−
| | Again, click on '''Move Graphics View''' and drag the background to see the graph properly.   
+
||Again, click on '''Move Graphics View''' and drag the background to see the graph properly.   
−
| | Again, click on '''Move Graphics View''' and drag the background to see the graph properly.  
+
||Again, click on '''Move Graphics View''' and drag the background to see the graph properly.  
 
|-
 
|-
−
| | Drag '''slider ϴ''' back to -360.
+
||Drag '''slider ϴ''' back to -360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | '''Inverse Trigonometric Functions'''
+
||'''Inverse Trigonometric Functions'''
  
 
e.g., If '''sin<sup>-1</sup>z''' (or '''arcsin z''') '''= w''', then '''z = sin w'''
 
e.g., If '''sin<sup>-1</sup>z''' (or '''arcsin z''') '''= w''', then '''z = sin w'''
Line 168: Line 169:
  
 
Change curvature of '''trigonometric function graph'''
 
Change curvature of '''trigonometric function graph'''
−
| | '''Inverse Trigonometric Functions'''
+
||'''Inverse Trigonometric Functions'''
  
 
For example, If '''inverse sine''' of '''z''' (also known as '''arcsin''' of '''z''') is '''w'''.   
 
For example, If '''inverse sine''' of '''z''' (also known as '''arcsin''' of '''z''') is '''w'''.   
Line 184: Line 185:
 
You can pause and refer to the example in the '''additional material''' provided for this '''tutorial'''.   
 
You can pause and refer to the example in the '''additional material''' provided for this '''tutorial'''.   
 
|-
 
|-
−
| |  
+
||
−
| | Let us go back to the '''GeoGebra''' window.   
+
||Let us go back to the '''GeoGebra''' window.   
 
|-
 
|-
−
| | '''Inverse sine function'''
+
||'''Inverse sine function'''
  
 
Type '''i_S: = Function[asin(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
 
Type '''i_S: = Function[asin(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
−
| | In the '''input bar''', type the following command:
+
||In the '''input bar''', type the following command:
  
 
'''i underscore S colon is equal to Function with capital F'''
 
'''i underscore S colon is equal to Function with capital F'''
Line 200: Line 201:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Drag the boundary to see '''Algebra''' view properly.   
+
||Drag the boundary to see '''Algebra''' view properly.   
−
| | Drag the boundary to see '''Algebra''' view properly.  
+
||Drag the boundary to see '''Algebra''' view properly.  
 
|-
 
|-
−
| | Point to''' i<sub>S</sub> function''' graph.
+
||Point to''' i<sub>S</sub> function''' graph.
−
| | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.
+
||This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.
  
 
Note that '''x''' and '''y axes''' are interchanged for this '''inverse sine function'''.   
 
Note that '''x''' and '''y axes''' are interchanged for this '''inverse sine function'''.   
Line 212: Line 213:
 
Observe the graph.
 
Observe the graph.
 
|-
 
|-
−
| | Drag the boundary to see '''Graphics''' view properly.   
+
||Drag the boundary to see '''Graphics''' view properly.   
−
| | Drag the boundary to see '''Graphics''' view properly.  
+
||Drag the boundary to see '''Graphics''' view properly.  
 
|-
 
|-
−
| | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
+
||Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
−
| | In the '''input bar''', type the following command:
+
||In the '''input bar''', type the following command:
  
 
'''P underscore S colon is equal to'''
 
'''P underscore S colon is equal to'''
Line 226: Line 227:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to''' P<sub>S</sub>'''.
+
||Point to''' P<sub>S</sub>'''.
−
| | This creates point '''PS''' on the '''inverse sine''' graph.
+
||This creates point '''PS''' on the '''inverse sine''' graph.
  
 
On the '''sine function''' graph, '''PS''' would be '''alpha comma sine alpha'''.   
 
On the '''sine function''' graph, '''PS''' would be '''alpha comma sine alpha'''.   
 
|-
 
|-
−
| | In '''Algebra''' view, right-click on''' P<sub>S</sub>, check '''Trace On''' option.
+
||In '''Algebra''' view, right-click on''' P<sub>S</sub>, check '''Trace On''' option.
−
| | In '''Algebra''' view, right-click on '''PS''', check the '''Trace On''' option.
+
||In '''Algebra''' view, right-click on '''PS''', check the '''Trace On''' option.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to 360.
+
||Drag '''slider ϴ''' to 360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
 
|-
 
|-
−
| | Point to traces of '''P<sub>S</sub>''', '''i<sub>S</sub>''' and '''F<sub>S</sub>'''.
+
||Point to traces of '''P<sub>S</sub>''', '''i<sub>S</sub>''' and '''F<sub>S</sub>'''.
−
| | Traces appear for the '''inverse sine function''' graph for '''alpha'''.
+
||Traces appear for the '''inverse sine function''' graph for '''alpha'''.
  
 
'''fs''' also appears in '''Graphics''' view.
 
'''fs''' also appears in '''Graphics''' view.
Line 246: Line 247:
 
Note that the '''domain''' for the graph that '''PS''' traces is not restricted from -1 to 1.   
 
Note that the '''domain''' for the graph that '''PS''' traces is not restricted from -1 to 1.   
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||Drag '''slider theta''' back to -360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Click and drag the background in '''Graphics''' view to erase traces of '''P<sub>S</sub>'''.
+
||Click and drag the background in '''Graphics''' view to erase traces of '''P<sub>S</sub>'''.
−
| | Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.
+
||Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.
 
|-
 
|-
−
| | In '''Algebra''' view, uncheck '''f<sub>S</sub>''', '''i<sub>S</sub>''' and '''P<sub>S</sub>'''.
+
||In '''Algebra''' view, uncheck '''f<sub>S</sub>''', '''i<sub>S</sub>''' and '''P<sub>S</sub>'''.
−
| | In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
+
||In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
 
|-
 
|-
−
| | '''Cosine and Inverse Cosine Functions'''
+
||'''Cosine and Inverse Cosine Functions'''
  
 
'''Cosine function f<sub>C</sub>''' in '''domain [-2π, α]'''
 
'''Cosine function f<sub>C</sub>''' in '''domain [-2π, α]'''
Line 262: Line 263:
  
 
'''P<sub>C</sub> (cos(α),α)'''
 
'''P<sub>C</sub> (cos(α),α)'''
−
| | '''Cosine and Inverse Cosine Functions'''
+
||'''Cosine and Inverse Cosine Functions'''
  
 
Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.
 
Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.
Line 274: Line 275:
 
The '''domain''' of the inverse cosine''' graph that '''PC''' traces will go beyond -1 and 1.   
 
The '''domain''' of the inverse cosine''' graph that '''PC''' traces will go beyond -1 and 1.   
 
|-
 
|-
−
| | Point to '''f<sub>C</sub>, i<sub>C</sub> and traces of P<sub>C</sub> in '''Graphics''' view.   
+
||Point to '''f<sub>C</sub>, i<sub>C</sub> and traces of P<sub>C</sub> in '''Graphics''' view.   
−
| | The '''cosine''' and '''inverse cosine functions''' should look like this.   
+
||The '''cosine''' and '''inverse cosine functions''' should look like this.   
 
|-
 
|-
−
| | In '''Algebra view, uncheck '''f<sub>C</sub>, i<sub>C</sub> and P<sub>C</sub> and move the background to erase traces of P<sub>C</sub>.   
+
||In '''Algebra view, uncheck '''f<sub>C</sub>, i<sub>C</sub> and P<sub>C</sub> and move the background to erase traces of P<sub>C</sub>.   
−
| | In '''Algebra view, uncheck '''fC, iC and PC and move the background to erase traces of PC.
+
||In '''Algebra view, uncheck '''fC, iC and PC and move the background to erase traces of PC.
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||Drag '''slider theta''' back to -360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | '''Tangent and Inverse Tangent Functions'''
+
||'''Tangent and Inverse Tangent Functions'''
 
'''Tangent function f<sub>T</sub>''' in '''domain [-2π, α]'''
 
'''Tangent function f<sub>T</sub>''' in '''domain [-2π, α]'''
  
Line 289: Line 290:
  
 
'''P<sub>T</sub> (tan(α),α)'''
 
'''P<sub>T</sub> (tan(α),α)'''
−
| | '''Tangent and Inverse Tangent Functions'''
+
||'''Tangent and Inverse Tangent Functions'''
 +
 
 
Now graph the '''tangent function fT'''.
 
Now graph the '''tangent function fT'''.
  
Line 302: Line 304:
 
The '''domain''' of the '''inverse tangent''' graph that '''PT''' traces will go beyond -1 and 1.   
 
The '''domain''' of the '''inverse tangent''' graph that '''PT''' traces will go beyond -1 and 1.   
 
|-
 
|-
−
| |  
+
||
−
| | Let us look at the '''inverse tangent function''' graph in the domain from -1 to 1.   
+
||Let us look at the '''inverse tangent function''' graph in the domain from -1 to 1.   
 
|-
 
|-
−
| | '''Inverse tangent function'''
+
||'''Inverse tangent function'''
  
 
Type '''i_T: = Function[atan(x), -∞, ∞]''' in '''input bar''' >> press '''Enter'''.
 
Type '''i_T: = Function[atan(x), -∞, ∞]''' in '''input bar''' >> press '''Enter'''.
−
| |  
+
||
 
To type infinity, click in the '''input bar''' and on '''symbol alpha''' appearing at the right end of the bar.   
 
To type infinity, click in the '''input bar''' and on '''symbol alpha''' appearing at the right end of the bar.   
  
Line 323: Line 325:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to '''i<sub>T</sub> function''' graph.
+
||Point to '''i<sub>T</sub> function''' graph.
−
| | This graphs the '''inverse tangent''' function of '''x'''.
+
||This graphs the '''inverse tangent''' function of '''x'''.
  
 
'''x''' lies between minus '''infinity''' and '''infinity'''.
 
'''x''' lies between minus '''infinity''' and '''infinity'''.
Line 330: Line 332:
 
Observe the graph.
 
Observe the graph.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to 360.
+
||Drag '''slider ϴ''' to 360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
 
|-
 
|-
−
| | Point to traces of '''P<sub>T</sub>''' and '''i<sub>T</sub>'''.
+
||Point to traces of '''P<sub>T</sub>''' and '''i<sub>T</sub>'''.
−
| | Compare traces of '''PT''' and '''iT'''.
+
||Compare traces of '''PT''' and '''iT'''.
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||Drag '''slider theta''' back to -360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | In '''Algebra''' view, uncheck '''f<sub>T</sub>''' and '''P<sub>T</sub>'''.
+
||In '''Algebra''' view, uncheck '''f<sub>T</sub>''' and '''P<sub>T</sub>'''.
−
| | In '''Algebra''' view, uncheck '''fT''' and '''PT'''.
+
||In '''Algebra''' view, uncheck '''fT''' and '''PT'''.
 
|-
 
|-
−
| | In '''Algebra''' view, check '''f<sub>S</sub>, f<sub>C</sub>, i<sub>S</sub>, i<sub>C</sub>, P<sub>S</sub>''' and '''P<sub>C</sub>''' to show them again.
+
||In '''Algebra''' view, check '''f<sub>S</sub>, f<sub>C</sub>, i<sub>S</sub>, i<sub>C</sub>, P<sub>S</sub>''' and '''P<sub>C</sub>''' to show them again.
−
| | In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
+
||In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
 
|-
 
|-
−
| | '''Check boxes'''
+
||'''Check boxes'''
  
 
Under '''Slider''', click on check box tool.
 
Under '''Slider''', click on check box tool.
  
 
Click on the top of the grid in '''Graphics''' view.
 
Click on the top of the grid in '''Graphics''' view.
−
| | Under '''Slider''', click on '''Check-box'''.
+
||Under '''Slider''', click on '''Check-box'''.
  
 
Click on the top of the grid in '''Graphics''' view.
 
Click on the top of the grid in '''Graphics''' view.
 
|-
 
|-
−
| | Point to the '''dialog box'''.
+
||Point to the '''dialog box'''.
−
| | '''Check-Box to Show/Hide Objects dialog-box''' appears.
+
||'''Check-Box to Show/Hide Objects dialog-box''' appears.
 
|-
 
|-
−
| | Type '''SIN''' as '''caption'''.
+
||Type '''SIN''' as '''caption'''.
−
| | In the '''Caption''' field, type '''SIN.'''
+
||In the '''Caption''' field, type '''SIN.'''
 
|-
 
|-
−
| | Click on '''Objects''' >> select '''f<sub>S</sub>, i<sub>S</sub>''' and '''P<sub>S</sub>''' >> '''apply'''
+
||Click on '''Objects''' >> select '''f<sub>S</sub>, i<sub>S</sub>''' and '''P<sub>S</sub>''' >> '''apply'''
−
| | 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'''.
+
||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'''.
 
|-
 
|-
−
| | Point to '''check box''' “'''SIN'''”.
+
||Point to '''check box''' '''SIN'''.
−
| | A '''check-box''' “'''SIN'''” is created in '''Graphics''' view.
+
||A '''check-box''' '''SIN''' is created in '''Graphics''' view.
  
 
It gives us the option to display or hide '''sine, arcsine''' graphs and point '''P<sub>S</sub>'''.
 
It gives us the option to display or hide '''sine, arcsine''' graphs and point '''P<sub>S</sub>'''.
 
|-
 
|-
−
| | Click on '''Check Box'''.
+
||Click on '''Check Box'''.
  
 
Click on the top of the grid in '''Graphics''' view.
 
Click on the top of the grid in '''Graphics''' view.
−
| | Again, click on '''Check Box'''.
+
||Again, click on '''Check Box'''.
  
 
Click on the top of the grid in '''Graphics''' view.
 
Click on the top of the grid in '''Graphics''' view.
 
|-
 
|-
−
| | Point to the '''dialog box'''.
+
||Point to the '''dialog box'''.
−
| | A '''Check-Box to Show/Hide Objects dialog-box''' appears.
+
||A '''Check-Box to Show/Hide Objects dialog-box''' appears.
 
|-
 
|-
−
| | Type '''TAN''' as '''caption'''.
+
||Type '''TAN''' as '''caption'''.
−
| | In the '''Caption''' field, type '''TAN'''.
+
||In the '''Caption''' field, type '''TAN'''.
 
|-
 
|-
−
| | Click on '''Objects''' >> select '''f<sub>T</sub>, i<sub>T</sub>''' and '''P<sub>T</sub>''' >> '''apply'''.
+
||Click on '''Objects''' >> select '''f<sub>T</sub>, i<sub>T</sub>''' and '''P<sub>T</sub>''' >> '''apply'''.
−
| | 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'''.
+
||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'''.
 
|-
 
|-
−
| | Point to '''check box''' “'''TAN'''”.
+
||Point to '''check box''' '''TAN'''.
−
| | A '''check-box''' “'''TAN'''” is created in '''Graphics''' view.
+
||A '''check-box''' '''TAN''' is created in '''Graphics''' view.
  
 
It gives us the option to display or hide '''tangent, arctangent''' graphs and point '''P<sub>T</sub>'''.
 
It gives us the option to display or hide '''tangent, arctangent''' graphs and point '''P<sub>T</sub>'''.
 
|-
 
|-
−
| | Click on '''Move''' tool to uncheck all boxes.
+
||Click on '''Move''' tool to uncheck all boxes.
−
| | In the '''toolbar''', click on the first '''Move''' button and uncheck all boxes.
+
||In the '''toolbar''', click on the first '''Move''' button and uncheck all boxes.
 
|-
 
|-
−
| | Check “'''SIN'''” box.
+
||Check '''SIN''' box.
−
| | Check the “'''SIN'''” box.
+
||Check the '''SIN''' box.
 
|-
 
|-
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
 
|-
 
|-
−
| | Point to '''f<sub>S</sub>, i<sub>S</sub>''' and traces of '''P<sub>S</sub>''' in '''Graphics''' view.
+
||Point to '''f<sub>S</sub>, i<sub>S</sub>''' and traces of '''P<sub>S</sub>''' in '''Graphics''' view.
−
| | Observe '''fS, iS''' and traces of '''PS''' appear in '''Graphics''' view.
+
||Observe '''fS, iS''' and traces of '''PS''' appear in '''Graphics''' view.
 
|-
 
|-
−
| | Uncheck '''SIN''' box.
+
||Uncheck '''SIN''' box.
−
| | Uncheck '''SIN''' box.
+
||Uncheck '''SIN''' box.
 
|-
 
|-
−
| | Click on and move '''Graphics''' view slightly to erase traces of '''P<sub>S</sub>'''.
+
||Click on and move '''Graphics''' view slightly to erase traces of '''P<sub>S</sub>'''.
−
| | Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.
+
||Click on and move '''Graphics''' view slightly to erase traces of '''PS'''.
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||Drag '''slider theta''' back to -360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Check “'''TAN'''” box.
+
||Check '''TAN''' box.
−
| | Check “'''TAN'''” box.
+
||Check '''TAN''' box.
 
|-
 
|-
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
 
|-
 
|-
−
| | Point to '''f<sub>T</sub>, i<sub>T</sub>''' and traces of '''P<sub>T</sub>''' in '''Graphics''' view.
+
||Point to '''f<sub>T</sub>, i<sub>T</sub>''' and traces of '''P<sub>T</sub>''' in '''Graphics''' view.
−
| | Observe '''fT, iT''' and traces of '''PT''' appear in '''Graphics''' view.
+
||Observe '''fT, iT''' and traces of '''PT''' appear in '''Graphics''' view.
 
|-
 
|-
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Check the '''SIN''' box.
+
||Check the '''SIN''' box.
−
| | Check the '''SIN''' box.
+
||Check the '''SIN''' box.
 
|-
 
|-
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' to 360.
 
|-
 
|-
−
| | Point to all the '''functions''' in '''Graphics''' view.
+
||Point to all the '''functions''' in '''Graphics''' view.
−
| | Observe all the '''functions''' appearing in '''Graphics''' view.
+
||Observe all the '''functions''' appearing in '''Graphics''' view.
 
|-
 
|-
−
| |
+
||
−
| | Let us summarize.
+
||Let us summarize.
 
|-
 
|-
−
| | '''Slide Number 6'''
+
||'''Slide Number 6'''
  
 
'''Summary'''
 
'''Summary'''
−
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
+
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
  
 
Graph '''trigonometric functions'''
 
Graph '''trigonometric functions'''
Line 445: Line 447:
 
Create'''check-boxes''' to group and show/hide '''functions'''
 
Create'''check-boxes''' to group and show/hide '''functions'''
 
|-
 
|-
−
| | '''Slide Number 7'''
+
||'''Slide Number 7'''
  
 
'''Assignment'''
 
'''Assignment'''
−
| | As an assignment:
+
||As an assignment,
  
−
Plot graphs of  
+
Plot graphs of,
  
 
'''Secant''' and '''arcsecant'''
 
'''Secant''' and '''arcsecant'''
Line 460: Line 462:
 
For hints, you can refer to the '''additional material''' provided.   
 
For hints, you can refer to the '''additional material''' provided.   
 
|-
 
|-
−
| | '''Slide Number 8'''
+
||'''Slide Number 8'''
  
 
'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
−
| | The video at the following link summarizes the '''Spoken Tutorial project'''.
+
||The video at the following link summarizes the '''Spoken Tutorial project'''.
  
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
−
| | '''Slide Number 9'''
+
||'''Slide Number 9'''
  
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
−
| | The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.
+
||The '''Spoken Tutorial Project '''team conducts workshops and gives certificates.
  
 
For more details, please write to us.
 
For more details, please write to us.
 
|-
 
|-
−
| | '''Slide Number 10'''
+
||'''Slide Number 10'''
  
 
'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
Line 488: Line 490:
 
Someone from our team will answer them.
 
Someone from our team will answer them.
  
−
| | Please post your timed queries on this forum.
+
||Please post your timed queries on this forum.
 
|-
 
|-
−
| | '''Slide Number 11'''
+
||'''Slide Number 11'''
  
 
'''Acknowledgement'''
 
'''Acknowledgement'''
−
| | The '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.
+
||The '''Spoken Tutorial Project''' is funded by '''NMEICT, MHRD''', Government of India.
  
 
More information on this mission is available at this link.
 
More information on this mission is available at this link.
 
|-
 
|-
−
| |
+
||
−
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.
+
||This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.
  
 
Thank you for joining.
 
Thank you for joining.
 
|-
 
|-
 
|}
 
|}

Revision as of 18:00, 27 June 2018

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Inverse Trigonometric Functions.
Slide Number 2

Learning Objectives

In this tutorial, we will learn to use GeoGebra to

Plot graphs of inverse trigonometric functions

Compare them to graphs of trigonometric functions

Create check-boxes to group and show or hide functions

Slide Number 3

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with:

GeoGebra interface

Trigonometry

For relevant tutorials, please visit our website.

Slide Number 4

System Requirement

Here I am using:

Ubuntu Linux OS version. 14.04

GeoGebra 5.0.388.0-d

Show the GeoGebra window. I have already opened the GeoGebra interface.
Switching x axis to radians

Double click on x axis in Graphics view >> Object Properties

Now let us change x Axis units to radians.

In Graphics view, double-click on the x axis and then on Object Properties.

Click on Preferences-Graphics >> x axis. In the Object Properties menu, click on Preferences-Graphics and then on x Axis.
Check the Distance option, select π/2 >> select Ticks first option Check the Distance option, select pi divided by 2 and then the Ticks first option.
Close the Preferences box. Close the Preferences box.
Point to x-axis. Units of x-axis are in radians with interval of pi divided by 2 as shown.

GeoGebra will convert degrees of angle alpha to radians.

Point to the toolbar. Note that the name appears when you place the mouse over any tool icon in the toolbar.
Click on Slider tool >> click on Graphics view. In the Graphics toolbar, click on Slider and then in the top of Graphics view.
Point to the Slider dialog box. A slider dialog-box appears.
Point to Number radio button. By default, Number radio-button is selected.
Type Name as symbol theta ϴ. In the Name field, select theta from the Symbol menu.
Point to Min, Max and Increment values.

Click OK.

Type the Min value as minus 360 and Max plus 360 with Increment 1.

Click OK.

Point to slider ϴ. This creates a number slider theta from minus 360 to plus 360.
In input bar, type α = (ϴ /180) π.

Point to space between the right parenthesis and pi for multiplication.

Press Enter

In the input bar, type alpha is equal to theta divided by 180 in parentheses, and then pi.

Note how GeoGebra inserts a space between the right parenthesis and pi for multiplication.

Press Enter.

Drag slider ϴ to -360 and then back to 360. Drag slider theta from minus 360 and then back to 360.
Point to values of α in Algebra view. In Algebra view, observe how alpha changes from minus 2 pi to 2 pi radians as you change theta.
Drag slider ϴ to minus 360. Drag slider theta back to minus 360.
Sine function

In input bar, type f_S: = Function[sin(x), -2π, α] >> press Enter.

In the input bar, type the following command:

f underscore S colon is equal to Function with capital F

Type the following words in square brackets.

sin, x in parentheses, comma minus 2 pi comma alpha.

Press Enter.

Drag the boundary to see Algebra view properly. Drag the boundary to see Algebra view properly.
Point to fS in Algebra view. Here, fS defines the sine function of x.

x is between -2 pi and alpha which can take a maximum value of 2 pi.

This is called the domain of the function.

Drag the boundary to see Graphics View properly. Drag the boundary to see Graphics View properly.
Drag slider theta from minus 360 to 360. Drag slider theta from minus 360 to 360.
Point to fS sine function graph. This graphs the sine function of x.
In the toolbar, click on the bottom right triangle of the last button.

Point to Move Graphics View button.

In the toolbar, click on the bottom right triangle of the last button.

Note that this button is called Move Graphics View.

Under Move Graphics View, click on Zoom Out tool.

Click on Graphics view to see 2 pi radians on either side of origin.

In the menu that appears, click on Zoom Out.

Click in Graphics view to see 2 pi radians on either side of the origin.

Again, click on Move Graphics View and drag the background to see the graph properly. Again, click on Move Graphics View and drag the background to see the graph properly.
Drag slider ϴ back to -360. Drag slider theta back to minus 360.
Inverse Trigonometric Functions

e.g., If sin-1z (or arcsin z) = w, then z = sin w

Restrict domain of trigonometric function, define principal value

Interchange x and y axes

Change curvature of trigonometric function graph

Inverse Trigonometric Functions

For example, If inverse sine of z (also known as arcsin of z) is w.

Then, z is sin w.

w can have multiple values.

So a principal value has to be defined and the domain has to be restricted.

To get the inverse function graph, interchange x and y axes

Next, change curvature of trigonometric function graph.

You can pause and refer to the example in the additional material provided for this tutorial.

Let us go back to the GeoGebra window.
Inverse sine function

Type i_S: = Function[asin(x), -1, 1] in input bar >> press Enter.

In the input bar, type the following command:

i underscore S colon is equal to Function with capital F

Type the following words in square brackets.

asin, x in parentheses, comma minus 1 comma 1”

Press Enter.

Drag the boundary to see Algebra view properly. Drag the boundary to see Algebra view properly.
Point to iS function graph. This graphs inverse sine (or arc sine) function of x.

Note that x and y axes are interchanged for this inverse sine function.

Its domain (set of x values) lies between minus 1 and 1.

Observe the graph.

Drag the boundary to see Graphics view properly. Drag the boundary to see Graphics view properly.
Type P_S = (sin(α), α) in input bar >> press Enter 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.

Point to PS. This creates point PS on the inverse sine graph.

On the sine function graph, PS would be alpha comma sine alpha.

In Algebra view, right-click on PS, check Trace On option. In Algebra view, right-click on PS, check the Trace On option.
Drag slider ϴ to 360. Drag slider theta to 360.
Point to traces of PS, iS and FS. Traces appear for the inverse sine function graph for alpha.

fs also appears in Graphics view.

Compare iS and traces of PS.

Note that the domain for the graph that PS traces is not restricted from -1 to 1.

Drag slider theta back to -360. Drag slider theta back to minus 360.
Click and drag the background in Graphics view to erase traces of PS. Click and drag the background in Graphics view to erase traces of PS.
In Algebra view, uncheck fS, iS and PS. In Algebra view, uncheck fS, iS, and PS to hide them.
Cosine and Inverse Cosine Functions

Cosine function fC in domain [-2π, α]

Inverse cosine function iC in domain [-1,1]

PC (cos(α),α)

Cosine and Inverse Cosine Functions

Follow the steps shown for SINE to graph the cosine function fC.

Its domain should be from -2 pi to alpha.

Graph the inverse cosine function iC" in the domain from -1 to 1.

Create a point PC whose co-ordinates are cos alpha comma alpha.

The domain of the inverse cosine graph that PC traces will go beyond -1 and 1.

Point to fC, iC and traces of PC in Graphics view. The cosine and inverse cosine functions should look like this.
In Algebra view, uncheck fC, iC and PC and move the background to erase traces of PC. In Algebra view, uncheck fC, iC and PC and move the background to erase traces of PC.
Drag slider theta back to -360. Drag slider theta back to minus 360.
Tangent and Inverse Tangent Functions

Tangent function fT in domain [-2π, α]

Inverse tangent function iT in domain [-∞, ∞]

PT (tan(α),α)

Tangent and Inverse Tangent Functions

Now graph the tangent function fT.

Its domain should also be from -2 pi to alpha.

We will look at the graph for the inverse tangent function iT.

Its domain will be from -infinity to infinity.

Create a point PT whose co-ordinates are tan alpha comma alpha.

The domain of the inverse tangent graph that PT traces will go beyond -1 and 1.

Let us look at the inverse tangent function graph in the domain from -1 to 1.
Inverse tangent function

Type i_T: = Function[atan(x), -∞, ∞] in input bar >> press Enter.

To type infinity, click in the input bar and on symbol alpha appearing at the right end of the bar.

In the symbol menu, click on the infinity symbol in the third row and third column from the right.

In the input bar, type the following command:

i underscore T colon is equal to Function with capital F

Type the following words in square brackets.

atan, x in parentheses, comma minus infinity comma infinity

Press Enter.

Point to iT function graph. This graphs the inverse tangent function of x.

x lies between minus infinity and infinity.

Observe the graph.

Drag slider ϴ to 360. Drag slider theta to 360.
Point to traces of PT and iT. Compare traces of PT and iT.
Drag slider theta back to -360. Drag slider theta back to minus 360.
In Algebra view, uncheck fT and PT. In Algebra view, uncheck fT and PT.
In Algebra view, check fS, fC, iS, iC, PS and PC to show them again. In Algebra view, check fS, fC, iS, iC, PS and PC to show them again.
Check boxes

Under Slider, click on check box tool.

Click on the top of the grid in Graphics view.

Under Slider, click on Check-box.

Click on the top of the grid in Graphics view.

Point to the dialog box. Check-Box to Show/Hide Objects dialog-box appears.
Type SIN as caption. In the Caption field, type SIN.
Click on Objects >> select fS, iS and PS >> apply Click on Objects drop-down menu to select fS, iS and PS, one by one, click Apply.
Point to check box SIN. A check-box SIN is created in Graphics view.

It gives us the option to display or hide sine, arcsine graphs and point PS.

Click on Check Box.

Click on the top of the grid in Graphics view.

Again, click on Check Box.

Click on the top of the grid in Graphics view.

Point to the dialog box. A Check-Box to Show/Hide Objects dialog-box appears.
Type TAN as caption. In the Caption field, type TAN.
Click on Objects >> select fT, iT and PT >> apply. Click on Objects drop-down menu to select fT, iT and PT, one by one, click Apply.
Point to check box TAN. A check-box TAN is created in Graphics view.

It gives us the option to display or hide tangent, arctangent graphs and point PT.

Click on Move tool to uncheck all boxes. In the toolbar, click on the first Move button and uncheck all boxes.
Check SIN box. Check the SIN box.
Drag slider theta to 360. Drag slider theta to 360.
Point to fS, iS and traces of PS in Graphics view. Observe fS, iS and traces of PS appear in Graphics view.
Uncheck SIN box. Uncheck SIN box.
Click on and move Graphics view slightly to erase traces of PS. Click on and move Graphics view slightly to erase traces of PS.
Drag slider theta back to -360. Drag slider theta back to minus 360.
Check TAN box. Check TAN box.
Drag slider theta to 360. Drag slider theta to 360.
Point to fT, iT and traces of PT in Graphics view. Observe fT, iT and traces of PT appear in Graphics view.
Drag slider theta back to minus 360. Drag slider theta back to minus 360.
Check the SIN box. Check the SIN box.
Drag slider theta to 360. Drag slider theta to 360.
Point to all the functions in Graphics view. Observe all the functions appearing in Graphics view.
Let us summarize.
Slide Number 6

Summary

In this tutorial, we have learnt how to use GeoGebra to:

Graph trigonometric functions

Graph inverse trigonometric functions

Createcheck-boxes to group and show/hide functions

Slide Number 7

Assignment

As an assignment,

Plot graphs of,

Secant and arcsecant

Cosecant and arccosecant

Cotangent and arccotangent

For hints, you can refer to the additional material provided.

Slide Number 8

About Spoken Tutorial project

The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

Slide Number 9

Spoken Tutorial workshops

The Spoken Tutorial Project team conducts workshops and gives certificates.

For more details, please write to us.

Slide Number 10

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

Please post your timed queries on this forum.
Slide Number 11

Acknowledgement

The Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Vidhya

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