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

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(Created page with "{|border=1 ||'''Visual Cue''' ||'''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this tutorial on '''Inverse Trigonometric Functions'''. |- |...")
 
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'''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'''
Line 21: Line 27:
 
'''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'''
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'''GeoGebra 5.0.388.0-d'''
 
'''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.
 
|  | Show the '''GeoGebra''' window.
Line 46: Line 46:
 
|  | 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 '''x Axis'''.
 
|-
 
|-
 
|  | Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
 
|  | Check the '''Distance''' option, select '''π/2''' >> select '''Ticks first option'''
−
|  | Check the '''Distance''' option, select '''pi''' divided by 2 and then '''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.
Line 61: Line 61:
  
 
'''GeoGebra''' will convert '''degrees''' of angle '''alpha''' to '''radians'''.
 
'''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.
 
|  | Click on '''Slider''' tool >> click on '''Graphics''' view.
−
|  | Click on '''Slider''' tool and then 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'''.
 
|  | Point to the '''Slider dialog box'''.
−
|  | '''Slider dialog-box''' appears.
+
|  | A '''slider dialog-box''' appears.
 
|-
 
|-
 
|  | Point to Number radio button.
 
|  | Point to Number radio button.
Line 72: Line 75:
 
|-
 
|-
 
|  | Type '''Name''' as '''symbol theta ϴ'''.
 
|  | Type '''Name''' as '''symbol theta ϴ'''.
−
|  | In the '''Name''' field, select '''theta''' from '''Symbol menu'''.
+
|  | In the '''Name''' field, select '''theta''' from the '''Symbol menu'''.
 
|-
 
|-
 
|  | Point to''' Min, Max''' and '''Increment''' values.
 
|  | Point to''' Min, Max''' and '''Increment''' values.
  
−
Click '''OK''' 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'''.
Line 89: Line 92:
  
 
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 98:
 
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''' from minus 360 and then back to 360.
 
|-
 
|-
 
|  | Point to values of '''α''' in '''Algebra''' view.
 
|  | Point to values of '''α''' in '''Algebra''' view.
−
|  | In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to plus 2 '''pi radians'''.
+
|  | In '''Algebra''' view, observe how '''alpha''' changes from minus 2 '''pi''' to 2 '''pi radians''' as you change '''theta'''.
 
|-
 
|-
 
|  | Drag '''slider ϴ''' to minus 360.
 
|  | Drag '''slider ϴ''' to minus 360.
Line 107: Line 110:
  
 
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 116:
 
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 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'''. 
  
−
Here, defines the '''sine function '''of''' x'''.
+
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.
 
|  | 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.
 +
|-
 +
|  | '''Inverse Trigonometric Functions'''
  
−
'''Domain of x''' is between minus 2 '''pi''' and '''alpha'''.
+
e.g., If '''sin<sup>-1</sup>z''' (or '''arcsin z''') '''= w''', then '''z = sin w'''
  
−
That is,'''x''' lies between minus 2 '''pi''' and '''alpha'''.
+
Restrict '''domain''' of trigonometric function, define '''principal value'''
  
−
Observe the graph of '''fS'''.
+
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'''.
 
|-
 
|-
−
|  | 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 196:
 
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'''.
 +
|-
 +
|  | Drag the boundary to see '''Algebra''' view properly. 
 +
|  | Drag the boundary to see '''Algebra''' view properly.
 
|-
 
|-
 
|  | Point to''' i<sub>S</sub> function''' graph.
 
|  | Point to''' i<sub>S</sub> function''' graph.
 
|  | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.
 
|  | This graphs '''inverse sine''' (or '''arc sine''') function of '''x'''.
  
−
'''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.
 +
|-
 +
|  | Drag the boundary to see '''Graphics''' view properly. 
 +
|  | Drag the boundary to see '''Graphics''' view properly.
 
|-
 
|-
 
|  | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
 
|  | Type '''P_S = (sin(α), α)''' in '''input bar''' >> press '''Enter'''
−
|  | In '''input bar''', type the following command:
+
|  | In the '''input bar''', type the following command:
  
 
'''P underscore S colon is equal to'''
 
'''P underscore S colon is equal to'''
Line 173: Line 227:
 
|-
 
|-
 
|  | 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.
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|-
 
|-
 
|  | 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'''.
 +
 +
Note that the '''domain''' for the graph that '''PS''' traces is not restricted from -1 to 1. 
 
|-
 
|-
 
|  | Drag '''slider theta''' back to -360.
 
|  | Drag '''slider theta''' back to -360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
|  | Click on and move '''Graphics''' view to erase traces of '''P<sub>S</sub>'''.
+
|  | Click and drag the background in '''Graphics''' view to erase traces of '''P<sub>S</sub>'''.
−
|  | Click on and move '''Graphics''' view to erase traces of '''PS'''.
+
|  | Click and drag the background in '''Graphics''' view to erase traces of '''PS'''.
 
|-
 
|-
 
|  | In '''Algebra''' view, uncheck '''f<sub>S</sub>''', '''i<sub>S</sub>''' and '''P<sub>S</sub>'''.
 
|  | In '''Algebra''' view, uncheck '''f<sub>S</sub>''', '''i<sub>S</sub>''' and '''P<sub>S</sub>'''.
 
|  | In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
 
|  | In '''Algebra''' view, uncheck '''fS,''' '''iS''', and '''PS''' to hide them.
 
|-
 
|-
−
|  | '''Cosine function'''
+
|  | '''Cosine and Inverse Cosine Functions'''
  
−
Type '''f_C: = Function[cos(x), -2π, α]''' in '''input bar''' >> press '''Enter'''.
+
'''Cosine function f<sub>C</sub>''' in '''domain [-2π, α]'''
−
|  | In '''input bar''', type the following command:
+
  
−
'''f underscore C colon is equal to Function with capital F'''
+
'''Inverse cosine function i<sub>C</sub>''' in '''domain [-1,1]
  
−
Type the following words in square brackets.
+
'''P<sub>C</sub> (cos(α),α)'''
−
'''cos x in parentheses comma minus 2 pi comma alpha'''.
+
|  | '''Cosine and Inverse Cosine Functions'''
  
−
Press '''Enter'''.
+
Follow the steps shown for '''SINE''' to graph the '''cosine function fC'''.
  
−
Here, '''fC''' defines the '''cosine function'''.
+
Its '''domain''' should be from -2 '''pi''' to '''alpha'''.   
−
|-
+
−
|  | Drag '''slider ϴ''' to 360.
+
−
|  | Drag '''slider theta''' to 360.
+
−
|-
+
−
| | Point to''' f<sub>C</sub> function''' graph.
+
−
|  | This graphs the '''cos x function'''.
+
  
−
'''x''' lies between minus 2 '''pi''' and '''alpha'''.
+
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'''. 
−
|-
+
−
|  | 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'''.
+
The '''domain''' of the inverse cosine''' graph that '''PC''' traces will go beyond -1 and 1.
−
|  | 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.
+
|  | Point to '''f<sub>C</sub>, i<sub>C</sub> and traces of P<sub>C</sub> in '''Graphics''' view.
−
|  | This graphs the '''inverse cosine''' (or '''arccosine''') '''function''' of '''x'''.
+
|  | The '''cosine''' and '''inverse cosine functions''' should look like this.
−
 
+
−
The '''domain''' of '''x''' is from minus 1 to plus 1.
+
−
 
+
−
Observe the graph.
+
 
|-
 
|-
−
|  | '''Point on cosine function'''.
+
|  | 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.
−
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 '''P<sub>c</sub>'''.
+
−
|  | This creates a point '''PC'''.
+
−
|-
+
−
|  | In '''Algebra''' view, right-click on '''P<sub>C</sub>''' 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 '''P<sub>C</sub>''', '''i<sub>C</sub>''' and '''F<sub>C</sub>'''.
+
−
|  | 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 -360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
|  | Click on and move '''Graphics''' view to erase traces of '''P<sub>C</sub>'''.
+
|  | '''Tangent and Inverse Tangent Functions'''
−
|  | Click on and move '''Graphics''' view to erase traces of '''PC'''.
+
'''Tangent function f<sub>T</sub>''' in '''domain [-2π, α]'''
−
|-
+
−
|  | 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'''.
+
'''Inverse tangent function i<sub>T</sub>''' in '''domain [-∞, ∞]'''
−
|  | In '''input bar''', type the following command:
+
  
−
'''f underscore T colon is equal to Function with capital F'''
+
'''P<sub>T</sub> (tan(α),α)'''
 +
|  | '''Tangent and Inverse Tangent Functions'''
 +
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 -infinity to infinity.
  
−
Here, '''fT''' defines the '''tangent function''' of '''x'''.
+
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.
 
|-
 
|-
−
|  | 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.
+
−
|  | 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'''
 
|  | '''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 319:
 
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'''
 
−
 
−
'''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>'''.
 
−
|  | This creates point '''PT'''.
 
−
|-
 
−
|  | In '''Algebra''' view, right-click on '''P<sub>T</sub>''', check '''Trace On''' option.
 
−
|  | In '''Algebra''' view, right-click on '''PT''', check '''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>T</sub>''', '''i<sub>T</sub>''' and '''F<sub>T</sub>'''.
+
|  | Point to traces of '''P<sub>T</sub>''' and '''i<sub>T</sub>'''.
−
|  | Traces appear for '''inverse tangent function''' graph for '''alpha'''.
+
|  | Compare traces of '''PT''' and '''iT'''.
−
 
+
−
'''FT''' also appears in '''Graphics''' view.
+
−
 
+
−
Compare '''iT''' and traces of '''PT'''.
+
 
|-
 
|-
 
|  | Drag '''slider theta''' back to -360.
 
|  | Drag '''slider theta''' back to -360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
|  | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''P<sub>T</sub>'''.
+
|  | In '''Algebra''' view, uncheck '''f<sub>T</sub>''' and '''P<sub>T</sub>'''.
−
|  | Click on '''Move Graphics View''' tool and move '''Graphics''' view to erase traces of '''PT'''.
+
|  | In '''Algebra''' view, uncheck '''fT''' and '''PT'''.
 
|-
 
|-
 
|  | In '''Algebra''' view, check '''f<sub>S</sub>, f<sub>C</sub>, i<sub>S</sub>, i<sub>C</sub>, P<sub>S</sub>''' and '''P<sub>C</sub>''' to show them again.
 
|  | In '''Algebra''' view, check '''f<sub>S</sub>, f<sub>C</sub>, i<sub>S</sub>, i<sub>C</sub>, P<sub>S</sub>''' and '''P<sub>C</sub>''' to show them again.
Line 375: Line 349:
 
Under '''Slider''', click on check box tool.
 
Under '''Slider''', click on check box tool.
  
−
Click on the top of the grid in '''Graphics view'''.
+
Click on the top of the grid in '''Graphics''' view.
−
|  | Under '''Slider''', click on '''Check-box''' 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'''.
Line 392: Line 366:
 
|  | A '''check-box''' “'''SIN'''” is created in '''Graphics''' view.
 
|  | 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>'''.
+
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 '''COSIN''' as '''caption'''.
+
−
|  | In the '''Caption''' field, type '''COSIN'''.
+
−
|-
+
−
|  | 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'''.
+
−
|-
+
−
|  | 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 '''P<sub>c</sub>'''.
+
−
|-
+
−
|  | 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'''.
 
|  | Type '''TAN''' as '''caption'''.
Line 432: Line 385:
 
|-
 
|-
 
|  | 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 uncheck 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.
Line 455: Line 408:
 
|-
 
|-
 
|  | Drag '''slider theta''' back to -360.
 
|  | 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 '''f<sub>C</sub>, i<sub>C</sub>''' and traces of '''P<sub>C</sub>''' 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 '''P<sub>C</sub>'''.
 
−
|  | 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.
 
|  | Drag '''slider theta''' back to minus 360.
 
|-
 
|-
Line 487: Line 422:
 
|  | Drag '''slider theta''' back to minus 360.
 
|  | Drag '''slider theta''' back to minus 360.
 
|-
 
|-
−
|  | Check '''SIN''' and '''COSIN''' boxes.
+
|  | Check the '''SIN''' box.
−
|  | Check '''SIN''' and '''COSIN''' boxes.
+
|  | Check the '''SIN''' box.
 
|-
 
|-
 
|  | Drag '''slider theta''' to 360.
 
|  | Drag '''slider theta''' to 360.
Line 502: Line 437:
  
 
'''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 7'''
Line 515: Line 450:
 
|  | As an assignment:
 
|  | As an assignment:
  
−
Plot graphs of '''inverse functions''' of '''secant''', '''cosecant''' and '''cotangent'''.
+
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'''
 
|  | '''Slide Number 8'''

Revision as of 21:30, 11 June 2018

Visual Cue Narration
Slide Number 1

Title Slide

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

Learning Objectives

In this tutorial, we will learn to use GeoGebra to

Plot graphs of inverse trigonometric functions

Compare them to graphs of trigonometric functions

Create check-boxes to group and show or hide functions

Slide Number 3

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with:

GeoGebra interface

Trigonometry

For relevant tutorials, please visit our website.

Slide Number 4

System Requirement

Here I am using:

Ubuntu Linux OS version. 14.04

GeoGebra 5.0.388.0-d

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

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

Now let us change x Axis units to radians.

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

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

GeoGebra will convert degrees of angle alpha to radians.

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

Click OK.

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

Click OK.

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

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

Press Enter

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

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

Press Enter.

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

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

In the input bar, type the following command:

f underscore S colon is equal to Function with capital F

Type the following words in square brackets.

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

Press Enter.

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

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

This is called the domain of the function.

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

Point to Move Graphics View button.

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

Note that this button is called Move Graphics View.

Under Move Graphics View, click on Zoom Out tool.

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

In the menu that appears, click on Zoom Out.

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

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

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

Restrict domain of trigonometric function, define principal value

Interchange x and y axes

Change curvature of trigonometric function graph

Inverse Trigonometric Functions

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

Then, z is sin w.

w can have multiple values.

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

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

Next, change curvature of trigonometric function graph.

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

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

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

In the input bar, type the following command:

i underscore S colon is equal to Function with capital F

Type the following words in square brackets.

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

Press Enter.

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

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

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

Observe the graph.

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

P underscore S colon is equal to

Type the following words in parentheses.

sin alpha in parentheses comma alpha

Press Enter.

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

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

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

fs also appears in Graphics view.

Compare iS and traces of PS.

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

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

Cosine function fC in domain [-2π, α]

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

PC (cos(α),α)

Cosine and Inverse Cosine Functions

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

Its domain should be from -2 pi to alpha.

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

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

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

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

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

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

PT (tan(α),α)

Tangent and Inverse Tangent Functions

Now graph the tangent function fT.

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

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

Its domain will be from -infinity to infinity.

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

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

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

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

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

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

In the input bar, type the following command:

i underscore T colon is equal to Function with capital F

Type the following words in square brackets.

atan, x in parentheses, comma minus infinity comma infinity

Press Enter.

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

x lies between minus infinity and infinity.

Observe the graph.

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

Under Slider, click on check box tool.

Click on the top of the grid in Graphics view.

Under Slider, click on Check-box.

Click on the top of the grid in Graphics view.

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

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

Click on Check Box.

Click on the top of the grid in Graphics view.

Again, click on Check Box.

Click on the top of the grid in Graphics view.

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

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

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

Summary

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

Graph trigonometric functions

Graph inverse trigonometric functions

Createcheck-boxes to group and show/hide functions

Slide Number 7

Assignment

As an assignment:

Plot graphs of

Secant and arcsecant

Cosecant and arccosecant

Cotangent and arccotangent

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

Slide Number 8

About Spoken Tutorial project

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

Please download and watch it.

Slide Number 9

Spoken Tutorial workshops

The Spoken Tutorial Project team conducts workshops and gives certificates.

For more details, please write to us.

Slide Number 10

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site.

Choose the minute and second where you have the question.

Explain your question briefly.

Someone from our team will answer them.

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

Acknowledgement

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

More information on this mission is available at this link.

This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Vidhya

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