View source for Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English

{|border=1
||'''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<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.  

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''' i<sub>S</sub> 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''' P<sub>S</sub>'''.
|  | 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 '''PS''', check the '''Trace On''' option.
|-
|  | Drag '''slider ϴ''' to 360.
|  | Drag '''slider theta''' to 360.
|-
|  | 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'''.

'''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 '''P<sub>S</sub>'''.
|  | 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 '''fS,''' '''iS''', and '''PS''' to hide them.
|-
|  | '''Cosine and Inverse Cosine Functions'''

'''Cosine function f<sub>C</sub>''' in '''domain [-2π, α]'''

'''Inverse cosine function i<sub>C</sub>''' in '''domain [-1,1]

'''P<sub>C</sub> (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 '''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.  
|-
|  | 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.
|-
|  | Drag '''slider theta''' back to -360.
|  | Drag '''slider theta''' back to minus 360.
|-
|  | '''Tangent and Inverse Tangent Functions'''
'''Tangent function f<sub>T</sub>''' in '''domain [-2π, α]'''

'''Inverse tangent function i<sub>T</sub>''' in '''domain [-∞, ∞]'''

'''P<sub>T</sub> (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 '''i<sub>T</sub> 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 '''P<sub>T</sub>''' and '''i<sub>T</sub>'''.
|  | Compare traces of '''PT''' and '''iT'''.
|-
|  | Drag '''slider theta''' back to -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 '''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 '''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 '''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 the 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.
|  | 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 '''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'''.
|-
|  | 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 '''P<sub>T</sub>'''.
|-
|  | 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 '''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.
|-
|  | 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 '''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 '''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.
|-
|  | 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'''

Create'''check-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.
|-
|}