Applications-of-GeoGebra/C2/Inverse-Trigonometric-Functions/English - Revision history

Madhurig at 10:24, 28 June 2018

2018-06-28T10:24:57Z

Madhurig at 08:03, 28 June 2018

2018-06-28T08:03:01Z

Madhurig at 12:30, 27 June 2018

2018-06-27T12:30:22Z

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Vidhya at 16:00, 11 June 2018

2018-06-11T16:00:04Z

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Vidhya: Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Inverse Trigonometric Functions'''. |- |..."

2018-04-15T12:19:54Z

Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Inverse Trigonometric Functions'''. |- |..."

New page

{|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 plot graphs of '''inverse trigonometric functions''' in '''GeoGebra'''.
|-
| | '''Slide Number 3'''
'''Pre-requisites'''

'''www.spoken-tutorial.org'''
| | To follow this '''tutorial''', you should be familiar with:
'''GeoGebra''' interface

'''Trigonometry '''and related graphs

If not, 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'''
|-
| | '''Slide Number 5'''
'''Inverse trigonometric functions'''
| | '''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'''

Double click on '''x axis''' in '''Graphics '''view >> '''Object Properties'''
| | Now let us change '''x Axis units''' to '''radians'''.

Double-click on '''x axis''' in '''Graphics '''view and then on '''Object Properties'''.
|-
| | Click on '''Preferences-Graphics''' >> '''x axis'''.
| | In '''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 '''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'''.
|-
| | Click on '''Slider''' tool >> click on '''Graphics''' view.
| | Click on '''Slider''' tool and then click on '''Graphics''' view.
|-
| | Point to the '''Slider dialog box'''.
| | '''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 '''Symbol menu'''.
|-
| | Point to''' Min, Max''' and '''Increment''' values.

Click '''OK''' button.
| | Type the '''Min''' value as minus 360 and '''Max''' plus 360 with '''Increment''' 1.

Click '''OK''' button.
|-
| | 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 '''input bar''', type '''alpha is equal to theta divided by 180 in parentheses pi'''.

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

Press '''Enter'''.
|-
| | Drag '''slider ϴ''' to -360 and then to 360.
| | Drag '''slider theta''' from minus 360 to plus 360.
|-
| | Point to values of '''α''' in '''Algebra''' view.
| | In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to plus 2 '''pi radians'''.
|-
| | 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 '''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'''.

Here, defines the '''sine function '''of''' x'''.
|-
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click on '''Graphics view''' to see 2 '''pi radians''' on either side of '''origin'''.
| | Under '''Move Graphics View''', click on '''Zoom Out''' tool.

Click on '''Graphics view''' to see 2 '''pi radians''' on either side of the '''origin'''.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' from minus 360 to plus 360.
|-
| | Point to''' fS sine function''' graph.
| | This graphs '''sine function''' of '''x'''.

'''Domain of x''' is between minus 2 '''pi''' and '''alpha'''.

That is,'''x''' lies between minus 2 '''pi''' and '''alpha'''.

Observe the graph of '''fS'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Inverse sine function'''

Type '''i_S: = Function[asin(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''i underscore S colon is equal to Function with capital F'''

Type the following words in square brackets.

'''a sin x in parentheses comma minus 1 comma 1”

Press '''Enter'''.
|-
| | Point to''' iS function''' graph.
| | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.

'''x''' lies between minus 1 and plus 1.

Observe the graph.
|-
| | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
| | In '''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'''.
|-
| | In '''Algebra''' view, right-click on''' PS, check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PS''', check '''Trace On''' option.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PS''', '''iS''' and '''FS'''.
| | Traces appear for '''inverse sine function''' graph for '''alpha'''.

'''fs''' also appears in '''Graphics''' view.

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 '''PS'''.
| | Click on and move '''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 function'''

Type '''f_C: = Function[cos(x), -2π, α]''' in '''input bar''' >> press '''Enter'''.
| | 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''' to 360.
|-
| | Point to''' fC function''' graph.
| | This graphs the '''cos x function'''.

'''x''' lies between minus 2 '''pi''' and '''alpha'''.

Observe the graph.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | '''Inverse cosine function'''

Type '''i_C: = Function[acos(x), -1, 1]''' in '''input bar''' >> press '''Enter'''.
| | In '''input bar''', type the following command:

'''i underscore C colon is equal to Function with capital F'''

Type the following words in square brackets.

'''acos x in parentheses comma minus 1 comma 1'''.

Press '''Enter'''.

Here, '''IC''' defines the '''inverse cosine''' (or '''arccosine''') '''function''' of '''x'''.
|-
| | 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.

Observe the graph.
|-
| | '''Point on cosine function'''.

Type '''P_C = (cos(α), α)''' in '''input bar''' >> press '''Enter'''
| | 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 '''Pc'''.
| | This creates a point '''PC'''.
|-
| | In '''Algebra''' view, right-click on '''PC''' check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PC''', check '''Trace On''' option.
|-
| | Drag '''slider ϴ''' from 0 to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PC''', '''iC''' and '''FC'''.
| | Traces appear for '''inverse cosine function''' graph for '''alpha'''.

'''FC''' also appears in '''Graphics''' view.

Compare '''iC''' and traces of '''PC'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Click on and move '''Graphics''' view to erase traces of '''PC'''.
| | Click on and move '''Graphics''' view to erase traces of '''PC'''.
|-
| | In '''Algebra''' view, uncheck '''fC''', '''iC''' and '''PC''' 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'''.
| | In '''input bar''', type the following command:

'''f underscore T colon is equal to Function with capital F'''

Type the following words in square brackets.

'''Tan x in parentheses comma minus 2 pi comma alpha'''.

Press '''Enter'''.

Here, '''fT''' defines the '''tangent function''' of '''x'''.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fT tangent function''' graph.
| | 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'''.
| | In '''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'''.

Here, '''IT''' defines the '''inverse tangent''' (or '''arctangent''') '''function''' of '''x'''.
|-
| | Point to '''iT function''' graph.
| | This graphs the '''inverse tangent''' function of '''x'''.

'''x''' lies between minus '''infinity''' and plus '''infinity'''.

Observe the graph.
|-
| | '''Point on tangent function'''

'''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 '''PT'''.
| | This creates point '''PT'''.
|-
| | In '''Algebra''' view, right-click on '''PT''', check '''Trace On''' option.
| | In '''Algebra''' view, right-click on '''PT''', check '''Trace On''' option.
|-
| | Drag '''slider ϴ''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to traces of '''PT''', '''iT''' and '''FT'''.
| | Traces appear for '''inverse tangent function''' graph for '''alpha'''.

'''FT''' also appears in '''Graphics''' view.

Compare '''iT''' and traces of '''PT'''.
|-
| | Drag '''slider theta''' back to -360.
| | Drag '''slider theta''' back to minus 360.
|-
| | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''PT'''.
| | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''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''' 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 '''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 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.
| | 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 '''COSIN''' as '''caption'''.
| | In the '''Caption''' field, type '''COSIN'''.
|-
| | Click on '''Objects''' >> select '''fS, iS''' and '''PS''' >> '''apply'''.
| | Click on '''Objects''' drop-down menu to select '''fc, ic''' and '''Pc''', one by one, click '''Apply'''.
|-
| | Point to '''check box''' “'''COSIN'''”.
| | A '''check-box''' “'''COSIN'''” is created in '''Graphics''' view.

It gives us option to display or hide '''cosine, arccosine''' graphs and point '''Pc'''.
|-
| | Click on '''check box'''.

Click on the top of the grid in '''Graphics''' view.
| | 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 '''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 on '''Graphics''' view.

It gives us option to display or hide '''tangent, arctangent''' graphs and point '''PT'''.
|-
| | Click on '''Move''' tool to uncheck all boxes.
| | Click on '''Move''' tool to uncheck all boxes.
|-
| | Check “'''SIN'''” box.
| | Check “'''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 “'''COSIN'''” box.
| | Check “'''COSIN'''” box.
|-
| | Drag '''slider theta''' to 360.
| | Drag '''slider theta''' to 360.
|-
| | Point to '''fC, iC''' and traces of '''PC''' in '''Graphics''' view.
| | Observe '''fC, iC''' and traces of '''PC''' appear in '''Graphics''' view.
|-
| | Uncheck '''COSIN''' box.
| | Uncheck '''COSIN''' box.
|-
| | Click on and move '''Graphics''' view slightly to erase traces of '''PC'''.
| | Click on and move '''Graphics''' view slightly to erase traces of '''PC'''.
|-
| | Drag '''slider theta''' back to minus 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 '''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'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to graph:

'''Sine, cosine, tangent functions''' of '''alpha'''

Inverse '''sine, cosine, tangent functions''' of '''alpha'''

View or hide them using '''check-boxes'''
|-
| | '''Slide Number 7'''

'''Assignment'''
| | As an assignment:

Plot graphs of '''inverse functions''' of '''secant''', '''cosecant''' and '''cotangent'''.
|-
| | '''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.
|-
|}

@@ Line 35: / Line 35: @@
 '''System Requirement'''
 ||Here I am using:
-'''Ubuntu Linux OS version. 14.04'''
+'''Ubuntu Linux OS version 14.04'''
 '''GeoGebra 5.0.388.0-d'''
@@ Line 49: / Line 49: @@
 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 '''xAxis'''.
 |-
 ||Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
@@ Line 78: / Line 78: @@
 ||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'''.
@@ Line 152: / Line 152: @@
 ||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.
@@ Line 181: / Line 181: @@
 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
+To get the '''inverse function''' graph, interchange '''x''' and '''y''' axes.
 Next, change curvature of '''trigonometric function graph'''.
@@ Line 199: / Line 199: @@
 Type the following words in square brackets.
-'''asin, x in parentheses, comma minus 1 comma 1”
+'''asin, x in parentheses, comma minus 1 comma 1'''
 Press '''Enter'''.
@@ Line 207: / Line 207: @@
 |-
 ||Point to''' i<sub>S</sub> function''' graph.
-||This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.
+||This graphs the '''inverse sine''' (or '''arc sine''') function of '''x'''.
 Note that '''x''' and '''y axes''' are interchanged for this '''inverse sine function'''.
@@ Line 271: / Line 271: @@
 Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.
-Its '''domain''' should be from -2 '''pi''' to '''alpha'''.
+Its '''domain''' should be from  '''-2 pi''' to '''alpha'''.
 Graph the '''inverse cosine function iC" in the '''domain''' from -1 to 1.
@@ Line 300: / Line 300: @@
 Now graph the '''tangent function fT'''.
-Its domain should also be from -2 '''pi''' to '''alpha'''.
+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.
+Its domain will be from minus infinity to infinity.
 Create a point '''PT''' whose '''co-ordinates''' are '''tan alpha comma alpha'''.
@@ Line 316: / Line 316: @@
 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.
 In the '''symbol menu''', click on the '''infinity symbol''' in the third row and third column from the right.
@@ Line 357: / Line 356: @@
 |-
 ||cursor on the window.
-||Let us create check boxes to make it easier to group and  see different functions at a time.
+||Let us create check boxes to make it easier to group and see different functions at a time.
 |-
@@ Line 405: / Line 404: @@
 |-
 ||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 un-check all boxes.
 |-
 ||Check '''SIN''' box.

@@ Line 100: / Line 100: @@
 |-
 ||Drag '''slider ϴ''' to -360 and then back to 360.
-||Drag '''slider theta''' from minus 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 2 '''pi''' to 2 '''pi radians''' as you change '''theta'''.
+||In '''Algebra''' view, observe how '''alpha''' changes from minus  '''2pi''' to  '''2pi radians''' as you change '''theta'''.
 |-
 ||Drag '''slider ϴ''' to minus 360.
@@ Line 121: / Line 121: @@
 Press '''Enter'''.
 |-
-||Drag the boundary to see '''Algebra''' view properly.
+||Drag boundary to see '''Algebra''' view properly.
 ||Drag the boundary to see '''Algebra''' view properly.
 |-
@@ Line 127: / Line 127: @@
 ||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 '''2pi'''.
 This is called the '''domain''' of the '''function'''.
@@ Line 160: / Line 160: @@
 ||Drag '''slider theta''' back to minus 360.
 |-
-||'''Inverse Trigonometric Functions'''
+||'''Slide Number 5'''
 e.g., If '''sin<sup>-1</sup>z''' (or '''arcsin z''') '''= w''', then '''z = sin w'''
@@ Line 256: / Line 258: @@
 ||In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
 |-
-||'''Cosine and Inverse Cosine Functions'''
+||'''Slide Number 6'''
 '''Cosine function f<sub>C</sub>''' in '''domain [-2π, α]'''
@@ Line 284: / Line 288: @@
 ||Drag '''slider theta''' back to minus 360.
 |-
-||'''Tangent and Inverse Tangent Functions'''
+||'''Slide Number 7'''
 '''Tangent function f<sub>T</sub>''' in '''domain [-2π, α]'''
@@ Line 340: / Line 346: @@
 ||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 '''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.
-||In '''Algebra''' view, check '''fS, fC, iS, iC, PS''' and '''PC''' to show them again.
+||In '''Algebra''' view, check '''fS''',  '''iS''', and '''PS''' to show them again.
 |-
 ||'''Check boxes'''
@@ Line 357: / Line 370: @@
 |-
 ||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 '''SIN''' as '''caption'''.
@@ Line 378: / Line 391: @@
 |-
 ||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'''.
@@ Line 404: / Line 417: @@
 |-
 ||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>'''.
@@ Line 413: / Line 426: @@
 |-
 ||Check '''TAN''' box.
-||Check '''TAN''' box.
+||Check the '''TAN''' box.
 |-
 ||Drag '''slider theta''' to 360.
@@ Line 431: / Line 444: @@
 |-
 ||Point to all the '''functions''' in '''Graphics''' view.
-||Observe all the '''functions''' appearing in '''Graphics''' view.
+||Observe the '''functions''' appearing in '''Graphics''' view.
 |-
 ||
 ||Let us summarize.
 |-
-||'''Slide Number 6'''
+||'''Slide Number 8'''
 '''Summary'''
@@ Line 445: / Line 458: @@
 Graph '''inverse trigonometric functions'''
-Create'''check-boxes''' to group and show/hide '''functions'''
+Create '''check-boxes''' to group and show/hide '''functions'''
 |-
-||'''Slide Number 7'''
+||'''Slide Number 9'''
 '''Assignment'''
@@ Line 456: / Line 469: @@
 '''Secant''' and '''arcsecant'''
-'''Cosecant''' and arccosecant'''
+'''Cosecant''' and '''arccosecant'''
 '''Cotangent''' and '''arccotangent'''
@@ Line 462: / Line 475: @@
 For hints, you can refer to the '''additional material''' provided.
 |-
-||'''Slide Number 8'''
+||'''Slide Number 10'''
 '''About Spoken Tutorial project'''
@@ Line 469: / Line 482: @@
 Please download and watch it.
 |-
-||'''Slide Number 9'''
+||'''Slide Number 11'''
 '''Spoken Tutorial workshops'''
@@ Line 476: / Line 489: @@
 For more details, please write to us.
 |-
-||'''Slide Number 10'''
+||'''Slide Number 12'''
 '''Forum for specific questions:'''
@@ Line 492: / Line 505: @@
 ||Please post your timed queries on this forum.
 |-
-||'''Slide Number 11'''
+||'''Slide Number 13'''
 '''Acknowledgement'''