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 plot graphs of '''inverse trigonometric functions''' in '''GeoGebra'''.
+
||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'''
+
||'''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
  
−
'''Trigonometry '''and related graphs
+
'''Trigonometry'''
  
−
If not, 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'''
 
|-
 
|-
−
| | '''Slide Number 5'''
+
||Show the '''GeoGebra''' window.
−
'''Inverse trigonometric functions'''
+
||I have already opened the '''GeoGebra''' interface.
−
|  | '''Arcsine, arccosine, arctangent''' etc are '''inverse trigonometric functions'''.
+
−
 
+
−
These ratios of right triangle lengths help to calculate the angle
+
−
|-
+
−
|  | Show the '''GeoGebra''' window.
+
−
| | 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'''.
  
−
Double-click on '''x axis''' in '''Graphics '''view 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 '''Object Properties''' menu, click on '''Preferences-Graphics''' and then on '''x Axis'''.
+
||In the '''Object Properties''' menu, click on '''Preferences-Graphics''' and then on '''xAxis'''.
 
|-
 
|-
−
| | 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 '''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'''.
 
|-
 
|-
−
| | Click on '''Slider''' tool >> click on '''Graphics''' view.
+
||Point to the '''toolbar'''.
−
| | Click on '''Slider''' tool and then click on '''Graphics''' view.
+
||Note that the name appears when you place the mouse over any '''tool icon''' in the '''toolbar'''.
 
|-
 
|-
−
| | Point to the '''Slider dialog box'''.
+
||Click on '''Slider''' tool >> click on '''Graphics''' view.
−
| | '''Slider dialog-box''' appears.
+
||In the '''Graphics toolbar''', click on '''Slider''' and then in the top of '''Graphics''' view.
 
|-
 
|-
−
| | Point to Number radio button.
+
||Point to the '''Slider dialog box'''.
−
| | By default, '''Number''' radio-button is selected.
+
||A '''slider dialog-box''' appears.
 
|-
 
|-
−
| | Type '''Name''' as '''symbol theta ϴ'''.
+
||Point to Number radio button.
−
| | In the '''Name''' field, select '''theta''' from '''Symbol menu'''.
+
||By default, '''Number''' radio-button is selected.
 
|-
 
|-
−
| | Point to''' Min, Max''' and '''Increment''' values.
+
||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''' button.
+
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''' button.
+
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 '''input bar''', type '''alpha  is equal to theta divided by 180 in parentheses 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 95: Line 99:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to -360 and then to 360.
+
||Drag '''slider ϴ''' to -360 and then back to 360.
−
| | Drag '''slider theta''' from minus 360 to plus 360.
+
||Drag '''slider theta''' to 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 plus 2 '''pi radians'''.
+
||In '''Algebra''' view, observe how '''alpha''' changes from minus '''2pi''' to '''2pi 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 '''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 113: Line 117:
 
Type the following words in square brackets.
 
Type the following words in square brackets.
  
−
'''sin x in parentheses comma minus 2 pi comma alpha'''.
+
'''sin, x in parentheses, comma minus 2 pi comma alpha'''.
  
 
Press '''Enter'''.
 
Press '''Enter'''.
 +
|-
 +
||Drag 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'''.
  
−
Here, defines the '''sine function '''of''' x'''.
+
'''x''' is between  '''-2 pi''' and '''alpha''' which can take a maximum value of '''2pi'''.  
 +
 
 +
This is called the '''domain''' of the '''function'''.
 
|-
 
|-
−
|  | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
+
||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'''.
 
Click on '''Graphics view''' to see 2 '''pi radians''' on either side of '''origin'''.
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.
+
||In the menu that appears, click on '''Zoom Out'''.
  
−
Click on '''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'''.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to 360.
+
||Again, click on '''Move Graphics View''' and drag the background to see the graph properly.
−
| | Drag '''slider theta''' from minus 360 to plus 360.
+
||Again, click on '''Move Graphics View''' and drag the background to see the graph properly.  
 
|-
 
|-
−
| | Point to''' f<sub>S</sub> sine function''' graph.
+
||Drag '''slider ϴ''' back to -360.
−
| | This graphs '''sine function''' of '''x'''.
+
||Drag '''slider theta''' back to minus 360.
 +
|-
 +
||'''Slide Number 5'''
 +
 
 +
'''Inverse Trigonometric Functions'''
 +
 
 +
e.g., If '''sin<sup>-1</sup>z''' (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. 
  
−
'''Domain of x''' is between minus 2 '''pi''' and '''alpha'''.
+
So a '''principal value''' has to be defined and the '''domain''' has to be restricted.
  
−
That is,'''x''' lies between minus 2 '''pi''' and '''alpha'''.
+
To get the '''inverse function''' graph, interchange '''x''' and '''y''' axes.
  
−
Observe the graph of '''fS'''.
+
Next, change curvature of '''trigonometric function graph'''.
 +
 
 +
You can pause and refer to the example in the '''additional material''' provided for this '''tutorial'''.
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||
−
| | Drag '''slider theta''' back to minus 360.
+
||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 '''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 150: Line 199:
 
Type the following words in square brackets.
 
Type the following words in square brackets.
  
−
'''a sin x in parentheses comma minus 1 comma 1”
+
'''asin, x in parentheses, comma minus 1 comma 1'''
  
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
|  | Point to''' i<sub>S</sub> function''' graph.
+
||Drag the boundary to see '''Algebra''' view properly.  
−
| | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.
+
||Drag the boundary to see '''Algebra''' view properly.
 +
|-
 +
||Point to''' i<sub>S</sub> function''' graph.
 +
||This graphs the '''inverse sine''' (or '''arc sine''') function of '''x'''.
  
−
'''x''' lies between minus 1 and plus 1.
+
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.
 
Observe the graph.
 
|-
 
|-
−
|  | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
+
||Drag the boundary to see '''Graphics''' view properly.  
−
| | In '''input bar''', type the following command:
+
||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'''
 
'''P underscore S colon is equal to'''
Line 172: Line 229:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to''' P<sub>S</sub>'''.
+
||Point to''' P<sub>S</sub>'''.
−
| | This creates point '''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''' 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 '''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 '''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.
  
 
Compare '''iS''' and traces of '''PS'''.
 
Compare '''iS''' and traces of '''PS'''.
−
|-
 
−
|  | Drag '''slider theta''' back to -360.
 
−
|  | Drag '''slider theta''' back to minus 360.
 
−
|-
 
−
|  | Click on and move '''Graphics''' view to erase traces of '''P<sub>S</sub>'''.
 
−
|  | Click on and move '''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 '''fS,''' '''iS''', and '''PS''' to hide them.
 
−
|-
 
−
|  | '''Cosine function'''
 
  
−
Type '''f_C: = Function[cos(x), -2π, α]''' in '''input bar''' >> press '''Enter'''.
+
Note that the '''domain''' for the graph that '''PS''' traces is not restricted from -1 to 1.
−
|  | In '''input bar''', type the following command:
+
−
 
+
−
'''f underscore C colon is equal to Function with capital F'''
+
−
 
+
−
Type the following words in square brackets.
+
−
'''cos x in parentheses comma minus 2 pi comma alpha'''.
+
−
 
+
−
Press '''Enter'''.
+
−
 
+
−
Here, '''fC''' defines the '''cosine function'''.
+
 
|-
 
|-
−
| | Drag '''slider ϴ''' to 360.
+
||Drag '''slider theta''' back to -360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Point to''' f<sub>C</sub> function''' graph.
+
||Click and drag the background in '''Graphics''' view to erase traces of '''P<sub>S</sub>'''.
−
| | This graphs the '''cos x function'''.
+
||Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.
−
 
+
−
'''x''' lies between minus 2 '''pi''' and '''alpha'''.
+
−
 
+
−
Observe the graph.
+
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||In '''Algebra''' view, uncheck '''f<sub>S</sub>''', '''i<sub>S</sub>''' and '''P<sub>S</sub>'''.
−
| | Drag '''slider theta''' back to minus 360.
+
||In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
 
|-
 
|-
−
| | '''Inverse cosine function'''
+
||'''Slide Number 6'''
  
−
Type '''i_C: = Function[acos(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
+
'''Cosine and Inverse Cosine Functions'''
−
|  | In '''input bar''', type the following command:
+
  
−
'''i underscore C colon is equal to Function with capital F'''
+
'''Cosine function f<sub>C</sub>''' in '''domain [-2π, α]'''
  
−
Type the following words in square brackets.
+
'''Inverse cosine function i<sub>C</sub>''' in '''domain [-1,1]
  
−
'''acos x in parentheses comma minus 1 comma 1'''.
+
'''P<sub>C</sub> (cos(α),α)'''
 +
||'''Cosine and Inverse Cosine Functions'''
  
−
Press '''Enter'''.
+
Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.
  
−
Here, '''IC''' defines the '''inverse cosine''' (or '''arccosine''') '''function''' of '''x'''.
+
Its '''domain''' should be from  '''-2 pi''' to '''alpha'''.   
−
|-
+
−
|  | Point to '''iC function''' graph.
+
−
| | This graphs the '''inverse cosine''' (or '''arccosine''') '''function''' of '''x'''.
+
  
−
The '''domain''' of '''x''' is from minus 1 to plus 1.
+
Graph the '''inverse cosine function iC" in the '''domain''' from -1 to 1.
  
−
Observe the graph.
+
Create a point '''PC''' whose '''co-ordinates''' are '''cos alpha comma alpha'''.
−
|-
+
−
|  | '''Point on cosine function'''.
+
  
−
Type '''P_C = (cos(α), α)''' in '''input bar''' >> press '''Enter'''
+
The '''domain''' of the inverse cosine''' graph that '''PC''' traces will go beyond -1 and 1.   
−
|  | In '''input bar''', type the following command:
+
−
 
+
−
'''P underscore C colon is equal to'''
+
−
 
+
−
Type the following words in parentheses.
+
−
 
+
−
'''cos alpha in parentheses comma alpha'''
+
−
 
+
−
 
+
−
 
+
−
Press '''Enter'''.
+
 
|-
 
|-
−
| | Point to '''P<sub>c</sub>'''.
+
||Point to '''f<sub>C</sub>, i<sub>C</sub> and traces of P<sub>C</sub> in '''Graphics''' view.
−
| | This creates a point '''PC'''.
+
||The '''cosine''' and '''inverse cosine functions''' should look like this.
 
|-
 
|-
−
| | In '''Algebra''' view, right-click on '''P<sub>C</sub>''' check '''Trace On''' option.
+
||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, right-click on '''PC''', check '''Trace On''' option.
+
||In '''Algebra view, uncheck '''fC, iC and PC and move the background to erase traces of PC.
 
|-
 
|-
−
| | Drag '''slider ϴ''' from 0 to 360.
+
||Drag '''slider theta''' back to -360.
−
| | Drag '''slider theta''' to 360.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Point to traces of '''P<sub>C</sub>''', '''i<sub>C</sub>''' and '''F<sub>C</sub>'''.
+
||'''Slide Number 7'''
−
|  | Traces appear for '''inverse cosine function''' graph for '''alpha'''.
+
  
−
'''FC''' also appears in '''Graphics''' view.
+
'''Tangent and Inverse Tangent Functions'''
 +
'''Tangent function f<sub>T</sub>''' in '''domain [-2π, α]'''
  
−
Compare '''iC''' and traces of '''PC'''.
+
'''Inverse tangent function i<sub>T</sub>''' in '''domain [-∞, ∞]'''
−
|-
+
−
|  | Drag '''slider theta''' back to -360.
+
−
|  | Drag '''slider theta''' back to minus 360.
+
−
|-
+
−
|  | Click on and move '''Graphics''' view to erase traces of '''P<sub>C</sub>'''.
+
−
|  | Click on and move '''Graphics''' view to erase traces of '''PC'''.
+
−
|-
+
−
|  | In '''Algebra''' view, uncheck '''f<sub>C</sub>''', '''i<sub>C</sub>''' and '''P<sub>C</sub>''' to hide them.
+
−
|  | In '''Algebra''' view, uncheck '''fC, iC''' and '''PC''' to hide them.
+
−
|-
+
−
|  | '''Tangent function'''
+
  
−
Type '''f_T: = Function[tan(x), -2π, α]''' in '''input bar''' >> press '''Enter'''.
+
'''P<sub>T</sub> (tan(α),α)'''
−
| | In '''input bar''', type the following command:
+
||'''Tangent and Inverse Tangent Functions'''
  
−
'''f underscore T colon is equal to Function with capital F'''
+
Now graph the '''tangent function fT'''.
  
−
Type the following words in square brackets.
+
Its domain should also be from  '''-2 pi''' to '''alpha'''.  
  
−
'''Tan x in parentheses comma minus 2 pi comma alpha'''.
+
We will look at the graph for the '''inverse tangent function iT'''.
  
−
Press '''Enter'''.
+
Its domain will be from minus infinity to infinity. 
 +
 
 +
Create a point '''PT''' whose '''co-ordinates''' are '''tan alpha comma alpha'''.
  
−
Here, '''fT''' defines the '''tangent function''' of '''x'''.
+
The '''domain''' of the '''inverse tangent''' graph that '''PT''' traces will go beyond -1 and 1.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to 360.
+
||
−
| | Drag '''slider theta''' to 360.
+
||Let us look at the '''inverse tangent function''' graph in the domain from -1 to 1.
 
|-
 
|-
−
| | Point to '''f<sub>T</sub> tangent function''' graph.
+
||'''Inverse tangent function'''
−
|  | This graphs '''tangent function''' of '''x''' in the '''domain''' from minus 2 '''pi''' to '''alpha'''.
+
−
|-
+
−
|  | Drag '''slider theta''' back to -360. 
+
−
|  | Drag '''slider theta''' back to minus 360.
+
−
|-
+
−
|  | '''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'''.
−
| | In '''input bar''', type the following command:
+
||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'''
 
'''i underscore T colon is equal to Function with capital F'''
Line 319: Line 326:
 
Type the following words in square brackets.
 
Type the following words in square brackets.
  
−
'''atan x in parentheses comma minus infinity comma infinity'''
+
'''atan, x in parentheses, comma minus infinity comma infinity'''
  
 
Press '''Enter'''.
 
Press '''Enter'''.
−
 
−
Here, '''IT''' defines the '''inverse tangent''' (or '''arctangent''') '''function''' of '''x'''.
 
 
|-
 
|-
−
| | 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 plus '''infinity'''.
+
'''x''' lies between minus '''infinity''' and '''infinity'''.
  
 
Observe the graph.
 
Observe the graph.
 
|-
 
|-
−
| | '''Point on tangent function'''
+
||Drag '''slider ϴ''' to 360.
−
 
+
||Drag '''slider theta''' to 360.
−
'''Type P_T = (tan(α), α)''' in '''input bar''' >> press '''Enter'''
+
−
 
+
−
| | In '''input bar''', type the following command:
+
−
 
+
−
'''P underscore T colon is equal to'''
+
−
 
+
−
Type the following words in parentheses.
+
−
 
+
−
'''Tan alpha in parentheses comma alpha'''
+
−
 
+
−
Press '''Enter'''.
+
 
|-
 
|-
−
| | Point to '''P<sub>T</sub>'''.
+
||Point to traces of '''P<sub>T</sub>''' and '''i<sub>T</sub>'''.
−
| | This creates point '''PT'''.
+
||Compare traces of '''PT''' and '''iT'''.
 
|-
 
|-
−
| | In '''Algebra''' view, right-click on '''P<sub>T</sub>''', check '''Trace On''' option.
+
||Drag '''slider theta''' back to -360.
−
| | In '''Algebra''' view, right-click on '''PT''', check '''Trace On''' option.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Drag '''slider ϴ''' to 360.
+
||Drag the background.
−
| | Drag '''slider theta''' to 360.
+
||Drag the background slightly to the erase the traces of '''PT'''
 
|-
 
|-
−
| | Point to traces of '''P<sub>T</sub>''', '''i<sub>T</sub>''' and '''F<sub>T</sub>'''.
+
||In '''Algebra''' view, uncheck '''f<sub>T</sub>''' and '''P<sub>T</sub>'''.
−
| | Traces appear for '''inverse tangent function''' graph for '''alpha'''.
+
||In '''Algebra''' view, uncheck '''fT''' and '''PT'''.
−
 
+
−
'''FT''' also appears in '''Graphics''' view.
+
−
 
+
−
Compare '''iT''' and traces of '''PT'''.
+
 
|-
 
|-
−
| | Drag '''slider theta''' back to -360.
+
||In '''Algebra''' view, check '''f<sub>S</sub>,  i<sub>S</sub>, i<sub>C</sub>, P<sub>S</sub>''' and '''P<sub>C</sub>''' to show them again.
−
| | Drag '''slider theta''' back to minus 360.
+
||In '''Algebra''' view, check '''fS''',  '''iS''', and '''PS''' to show them again.
 
|-
 
|-
−
| | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''P<sub>T</sub>'''.
+
||cursor on the window.
−
| | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''PT'''.
+
||Let us create check boxes to make it easier to group and see different functions at a time.
 
|-
 
|-
−
|  | 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.
+
||'''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'''.
 
−
|  | Under '''Slider''', click on '''Check-box''' tool.
 
−
 
−
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 '''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'''.
 
−
|-
 
−
|  | Point to '''check box''' “'''SIN'''”.
 
−
|  | A '''check-box''' “'''SIN'''” is created in '''Graphics''' view.
 
−
 
−
It gives us option to display or hide '''sine, arcsine''' graphs and point '''P<sub>S</sub>'''.
 
−
|-
 
−
|  | Click on '''check box'''.
 
  
 
Click on the top of the grid in '''Graphics''' view.
 
Click on the top of the grid in '''Graphics''' view.
−
| | Click on '''check box''' tool.
+
||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.
+
||A'''Check-Box to Show/Hide Objects dialog-box''' appears.
 
|-
 
|-
−
| | Type '''COSIN''' as '''caption'''.
+
||Type '''SIN''' as '''caption'''.
−
| | In the '''Caption''' field, type '''COSIN'''.
+
||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>c</sub>, i<sub>c</sub>''' and '''P<sub>c</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''' “'''COSIN'''”.
+
||Point to '''check box''' '''SIN'''.
−
| | A '''check-box''' “'''COSIN'''” is created in '''Graphics''' view.
+
||A '''check-box''' '''SIN''' is created in '''Graphics''' view.
  
−
It gives us option to display or hide '''cosine, arccosine''' graphs and point '''P<sub>c</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.
−
| | Click on '''check box''' tool.
+
||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'''.
−
| | '''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 on '''Graphics''' view.
+
||A '''check-box''' '''TAN''' is created in '''Graphics''' view.
  
−
It gives us 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.
−
| | Click on '''Move''' tool to uncheck all boxes.
+
||In the '''toolbar''', click on the first '''Move''' button and un-check all boxes.
 
|-
 
|-
−
| | Check “'''SIN'''” box.
+
||Check '''SIN''' box.
−
| | Check “'''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 the '''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 “'''COSIN'''” box.
+
||Check '''TAN''' box.
−
| | Check “'''COSIN'''” box.
+
||Check the '''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>C</sub>, i<sub>C</sub>''' and traces of '''P<sub>C</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 '''fC, iC''' and traces of '''PC''' appear in '''Graphics''' view.
+
||Observe '''fT, iT''' and traces of '''PT''' appear in '''Graphics''' view.
 
|-
 
|-
−
| | Uncheck '''COSIN''' box.
+
||Drag '''slider theta''' back to minus 360.
−
| | Uncheck '''COSIN''' box.
+
||Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
| | Click on and move '''Graphics''' view slightly to erase traces of '''P<sub>C</sub>'''.
+
||Check the '''SIN''' box.
−
| | Click on and move '''Graphics''' view slightly to erase traces of '''PC'''.
+
||Check the '''SIN''' box.
 
|-
 
|-
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' to 360.
−
| | Drag '''slider theta''' back to minus 360.
+
||Drag '''slider theta''' to 360.
 
|-
 
|-
−
| | Check “'''TAN'''” box.
+
||Point to all the '''functions''' in '''Graphics''' view.
−
| | Check “'''TAN'''” box.
+
||Observe the '''functions''' appearing in '''Graphics''' view.
 
|-
 
|-
−
| | Drag '''slider theta''' to 360.
+
||
−
| | Drag '''slider theta''' to 360.
+
||Let us summarize.
 
|-
 
|-
−
| | Point to '''f<sub>T</sub>, i<sub>T</sub>''' and traces of '''P<sub>T</sub>''' in '''Graphics''' view.
+
||'''Slide Number 8'''
−
|  | 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 '''SIN''' and '''COSIN''' boxes.
+
−
|  | Check '''SIN''' and '''COSIN''' boxes.
+
−
|-
+
−
|  | 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'''
 
'''Summary'''
−
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to graph:
+
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
  
−
'''Sine, cosine, tangent functions''' of '''alpha'''
+
Graph '''trigonometric functions'''
  
−
Inverse '''sine, cosine, tangent functions''' of '''alpha'''
+
Graph '''inverse trigonometric functions'''
  
−
View or hide them using '''check-boxes'''
+
Create '''check-boxes''' to group and show/hide '''functions'''
 
|-
 
|-
−
| | '''Slide Number 7'''
+
||'''Slide Number 9'''
  
 
'''Assignment'''
 
'''Assignment'''
−
| | As an assignment:
+
||As an assignment,
 +
 
 +
Plot graphs of,
 +
 
 +
'''Secant''' and '''arcsecant'''
 +
 
 +
'''Cosecant''' and '''arccosecant'''
 +
 
 +
'''Cotangent''' and '''arccotangent'''
  
−
Plot graphs of '''inverse functions''' of '''secant''', '''cosecant''' and '''cotangent'''.
+
For hints, you can refer to the '''additional material''' provided.
 
|-
 
|-
−
| | '''Slide Number 8'''
+
||'''Slide Number 10'''
  
 
'''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 11'''
  
 
'''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 12'''
  
 
'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
Line 545: Line 502:
 
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 13'''
  
 
'''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.
 
|-
 
|-
 
|}
 
|}

Latest revision as of 15:54, 28 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 xAxis.
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 to minus 360 and then back to 360.
Point to values of α in Algebra view. In Algebra view, observe how alpha changes from minus 2pi to 2pi 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 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 2pi.

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.
Slide Number 5

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 the 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.
Slide Number 6

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.
Slide Number 7

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 minus 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.
Drag the background. Drag the background slightly to the erase the traces of PT
In Algebra view, uncheck fT and PT. In Algebra view, uncheck fT and PT.
In Algebra view, check fS, iS, iC, PS and PC to show them again. In Algebra view, check fS, iS, and PS to show them again.
cursor on the window. Let us create check boxes to make it easier to group and see different functions at a time.
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. ACheck-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 un-check 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 the 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 the 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 the functions appearing in Graphics view.
Let us summarize.
Slide Number 8

Summary

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

Graph trigonometric functions

Graph inverse trigonometric functions

Create check-boxes to group and show/hide functions

Slide Number 9

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 10

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Slide Number 11

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Slide Number 12

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Slide Number 13

Acknowledgement

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This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Vidhya

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