Difference between revisions of "Applications-of-GeoGebra/C3/3D-Geometry/English"

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||In this '''tutorial''', we will learn how to use '''GeoGebra''' to view:
 
||In this '''tutorial''', we will learn how to use '''GeoGebra''' to view:
  
−
And construct different structures in '''3D space'''
+
And construct different structures in 3D space
  
 
Solids of rotation of polynomial functions
 
Solids of rotation of polynomial functions
Line 53: Line 53:
 
The axes are '''x''' (in red), '''y''' (in green) and '''z''' (in blue).
 
The axes are '''x''' (in red), '''y''' (in green) and '''z''' (in blue).
  
−
All points in '''3D''' space are denoted by their '''x y z coordinates'''.
+
All points in 3D space are denoted by their x y z coordinates.
  
 
The point of intersection of the three axes is the '''origin O 0 comma 0 comma 0'''.
 
The point of intersection of the three axes is the '''origin O 0 comma 0 comma 0'''.
Line 64: Line 64:
 
|-
 
|-
 
||Show the '''GeoGebra''' window.
 
||Show the '''GeoGebra''' window.
−
||Let us draw a '''3D''' pyramid in '''GeoGebra'''.
+
||Let us draw a 3D pyramid in '''GeoGebra'''.
  
 
I have already opened a new window in '''GeoGebra'''.
 
I have already opened a new window in '''GeoGebra'''.
Line 89: Line 89:
 
This creates point '''A''' at the origin.
 
This creates point '''A''' at the origin.
 
|-
 
|-
−
||Then click on '''(2,0)''' to create point '''B'''.
+
||Click on '''(2,0)''' to create point '''B'''.
 
||Then click on 2 comma 0 to create point '''B'''.
 
||Then click on 2 comma 0 to create point '''B'''.
 
|-
 
|-
−
||Click on '''(2,2)''' for point '''C''', and on '''(0,2)''' to draw point '''D'''.
+
||Click on '''(2,2)''' for point '''C''' >>
 +
on '''(0,2)''' to draw point '''D'''.
 
||Click on 2 comma 2 for '''C''' and on 0 comma 2 to draw '''D'''.
 
||Click on 2 comma 2 for '''C''' and on 0 comma 2 to draw '''D'''.
 
|-
 
|-
Line 98: Line 99:
 
||Finally, click again on '''A'''.
 
||Finally, click again on '''A'''.
 
|-
 
|-
−
||Point to quadrilateral '''q1''' in '''2D''' and '''3D Graphics''' views.
+
||Point to quadrilateral '''q1''' in '''2D''' >> '''3D Graphics''' views.
 
||Note that a quadrilateral '''q1''' is seen in '''2D''' and '''3D Graphics''' views.
 
||Note that a quadrilateral '''q1''' is seen in '''2D''' and '''3D Graphics''' views.
  
Line 107: Line 108:
 
||Click on the '''Move''' tool.
 
||Click on the '''Move''' tool.
 
|-
 
|-
−
||Click in '''2D Graphics''' and drag the background.
+
||Click in '''2D Graphics''' >> drag the background.
 
||Click in '''2D Graphics''' and drag the background.
 
||Click in '''2D Graphics''' and drag the background.
 
|-
 
|-
Line 113: Line 114:
 
||Drag the boundary to see '''3D Graphics''' properly.
 
||Drag the boundary to see '''3D Graphics''' properly.
 
|-
 
|-
−
||Click in '''3D Graphics''' and under '''Pyramid''', on the '''Extrude to Pyramid or Cone''' tool.
+
||Click in '''3D Graphics''' and under '''Pyramid''',  
 +
on the '''Extrude to Pyramid or Cone''' tool.
  
 
Click on the square in '''3D Graphics''' view.
 
Click on the square in '''3D Graphics''' view.
−
| | Click in '''3D Graphics''' and under '''Pyramid''', on the '''Extrude to Pyramid or Cone''' tool.
+
|| Click in '''3D Graphics''' and under '''Pyramid''', on the '''Extrude to Pyramid or Cone''' tool.
  
 
In '''3D Graphics''' view, click on the square.
 
In '''3D Graphics''' view, click on the square.
Line 142: Line 144:
 
'''Rotation of a Polynomial'''
 
'''Rotation of a Polynomial'''
  
−
Let us rotate '''f(x)= ¬2x<sup>4</sup>-x<sup>3</sup>+3x<sup>2</sup>'''
+
Let us rotate '''f(x)= -2x<sup>4</sup>-x<sup>3</sup>+3x<sup>2</sup>'''
  
 
Part in second '''quadrant''' ('''XY''' plane) about '''x-axis'''
 
Part in second '''quadrant''' ('''XY''' plane) about '''x-axis'''
Line 149: Line 151:
 
Let us rotate '''f of x''' equals minus '''2 x raised to 4''' minus '''x cubed''' plus '''3 x squared'''.
 
Let us rotate '''f of x''' equals minus '''2 x raised to 4''' minus '''x cubed''' plus '''3 x squared'''.
  
−
We will rotate the part that lies in the second '''quadrant''', in '''XY''' plane, about the '''x-axis'''.
+
We will rotate the part that lies in the second quadrant, in '''XY''' plane, about the '''x-axis'''.
 
|-
 
|-
 
||Show the '''GeoGebra''' window.
 
||Show the '''GeoGebra''' window.
−
 
−
 
−
 
 
||I have already opened a new window in '''GeoGebra'''.
 
||I have already opened a new window in '''GeoGebra'''.
  
 
We will initially work with '''Algebra''' and '''2D Graphics''' views and open '''3D Graphics''' view later.
 
We will initially work with '''Algebra''' and '''2D Graphics''' views and open '''3D Graphics''' view later.
 
|-
 
|-
−
||In '''input bar''', type the following line and press '''Enter'''.
+
||In '''input bar''', type the following line  
 +
 
 +
>> press '''Enter'''.
  
 
'''f(x) = -2 x^4 -x^3+3 x^2'''
 
'''f(x) = -2 x^4 -x^3+3 x^2'''
Line 170: Line 171:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
||Under '''Perpendicular Line''', click on '''Parallel line''' and on the '''y-axis'''.
+
||Under '''Perpendicular Line''', click on '''Parallel line'''  
 +
 
 +
>> on the '''y-axis'''.
 
||Under '''Perpendicular Line''', click on '''Parallel line''' and on the '''y-axis'''.
 
||Under '''Perpendicular Line''', click on '''Parallel line''' and on the '''y-axis'''.
 
|-
 
|-
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||Point '''A''' appears.
 
||Point '''A''' appears.
 
|-
 
|-
−
||Click on '''Slider''' and in '''Graphics''' view.
+
||Click on '''Slider''' >> in '''Graphics''' view.
 
||Click on '''Slider''' and in '''Graphics''' view.
 
||Click on '''Slider''' and in '''Graphics''' view.
 
|-
 
|-
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||Leave '''a''' as the '''Name'''.
 
||Leave '''a''' as the '''Name'''.
 
|-
 
|-
−
||Change '''Min''' value to -1.5, '''Max''' value to 0 and '''Increment''' to 0.05.
+
||Change '''Min''' value to -1.5, '''Max''' value to 0
 +
 
 +
'''Increment''' to 0.05.
 
|| Change '''Min''' value to '''minus''' 1.5, '''Max''' value to 0 and '''Increment''' to 0.05.
 
|| Change '''Min''' value to '''minus''' 1.5, '''Max''' value to 0 and '''Increment''' to 0.05.
 
|-
 
|-
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||Point to '''slider a'''.
 
||Point to '''slider a'''.
  
−
Point to the part of '''function''' and '''x-axis''' in the second '''quadrant'''.
+
Point to the part of '''function''',
 +
 
 +
'''x-axis''' in the second '''quadrant'''.
 
||This creates '''slider a''', which changes the value of '''a''' from -1.5 to 0.
 
||This creates '''slider a''', which changes the value of '''a''' from -1.5 to 0.
  
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As its '''x co-ordinate a''' changes, '''C''' moves below point '''B''' along the '''x-axis'''.
 
As its '''x co-ordinate a''' changes, '''C''' moves below point '''B''' along the '''x-axis'''.
 
|-
 
|-
−
||Under '''Line''', click on '''Segment''' >> click on '''B''' and '''C''' to join them.
+
||Under '''Line''', click on '''Segment'''  
 +
 
 +
>> click on '''B''' and '''C''' to join them.
 
||Under '''Line''', click on '''Segment''' and click on '''B''' and '''C''' to join them.
 
||Under '''Line''', click on '''Segment''' and click on '''B''' and '''C''' to join them.
 
|-
 
|-
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||Click on '''Move Graphics View''' and drag the background to the left.
 
||Click on '''Move Graphics View''' and drag the background to the left.
 
|-
 
|-
−
||Click on '''View''' >> check '''3D Graphics''' to see the '''3D Graphics''' view.
+
||Click on '''View'''  
 +
 
 +
>> check '''3D Graphics''' to see the '''3D Graphics''' view.
 
||Click on '''View''' and check '''3D Graphics''' to see the '''3D Graphics''' view.
 
||Click on '''View''' and check '''3D Graphics''' to see the '''3D Graphics''' view.
 
|-
 
|-
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Click to see an arrow aligned with the '''y-axis'''.
 
Click to see an arrow aligned with the '''y-axis'''.
 
|-
 
|-
−
||Then, drag along the '''y axis''' to see the curve properly.
+
||Drag along the '''y axis''' to see the curve properly.
 
||Drag to pull the '''y-axis''' in or outwards to see the curve.
 
||Drag to pull the '''y-axis''' in or outwards to see the curve.
 
|-
 
|-
−
||Type '''Circle[C,f(a),xAxis]''' in the '''input bar''' >> press '''Enter'''.
+
||Type '''Circle[C,f(a),xAxis]''' in the '''input bar'''
 +
 
 +
>> press '''Enter'''.
 
||In the '''input bar''', type the following line.
 
||In the '''input bar''', type the following line.
  
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Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
||Right-click on circle '''c''' in '''Algebra''' view and check '''Trace On''' option.
+
||Right-click on circle '''c''' in '''Algebra''' view  
 +
 
 +
>> check '''Trace On''' option.
 
||In '''Algebra''' view, right-click on circle '''c''' and check '''Trace On''' option.
 
||In '''Algebra''' view, right-click on circle '''c''' and check '''Trace On''' option.
 
|-
 
|-
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||Observe the solid traced as '''a''' changes.
 
||Observe the solid traced as '''a''' changes.
 
|-
 
|-
−
||Point to '''2D''' and '''3D Graphics''' views.
+
||Point to '''2D''' >> '''3D Graphics''' views.
 
||Watch both '''2D''' and '''3D Graphics''' views.
 
||Watch both '''2D''' and '''3D Graphics''' views.
 
|-
 
|-
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|-
 
|-
 
||Point to the part of '''f''' in the second '''quadrant'''.
 
||Point to the part of '''f''' in the second '''quadrant'''.
−
||The part of '''function f''' that is in the second '''quadrant''' in '''2D,''' rotates around the '''x-axis'''.
+
||The part of '''function f''' that is in the second '''quadrant''' in 2D, rotates around the '''x-axis'''.
 
|-
 
|-
 
||Drag '''3D Graphics''' to see it from another angle.
 
||Drag '''3D Graphics''' to see it from another angle.
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|-
 
|-
 
||Show the '''GeoGebra''' window.
 
||Show the '''GeoGebra''' window.
−
||Finally, let us look at '''trigonometric functions''' in '''3D.'''
+
||Finally, let us look at '''trigonometric functions''' in 3D.
  
 
I have already opened a new window in '''GeoGebra'''.
 
I have already opened a new window in '''GeoGebra'''.
Line 314: Line 329:
 
||Drag the boundary to see '''2D Graphics''' properly.
 
||Drag the boundary to see '''2D Graphics''' properly.
 
|-
 
|-
−
||Click in '''2D Graphics'''>> '''Slider''' tool and in '''Graphics''' view.
+
||Click in '''2D Graphics'''
 +
 
 +
>> '''Slider''' tool and in '''Graphics''' view.
 
||Click in '''2D Graphics''', then on the '''Slider''' tool and in '''Graphics''' view.
 
||Click in '''2D Graphics''', then on the '''Slider''' tool and in '''Graphics''' view.
 
|-
 
|-
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||This creates a '''slider t''' which will change '''t''' from minus 6 to 16.   
 
||This creates a '''slider t''' which will change '''t''' from minus 6 to 16.   
 
|-
 
|-
−
||In the '''input bar''', type '''f(t)=cos(t)''' >> press '''Enter'''.
+
||In the '''input bar''', type '''f(t)=cos(t)'''  
 +
 
 +
>> press '''Enter'''.
 
||In the '''input bar''', type '''f t''' in parentheses equals '''cos t''' in parentheses and press '''Enter'''.
 
||In the '''input bar''', type '''f t''' in parentheses equals '''cos t''' in parentheses and press '''Enter'''.
 
|-
 
|-
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||Click in '''2D Graphics'''.
 
||Click in '''2D Graphics'''.
 
|-
 
|-
−
||Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''2D Graphics'''.
+
||Under '''Move Graphics View''', click on '''Zoom Out'''  
 +
 
 +
>> click in '''2D Graphics'''.
 
||Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''2D Graphics'''.
 
||Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''2D Graphics'''.
 
|-
 
|-
Line 399: Line 420:
 
Click on '''Slider''' tab.
 
Click on '''Slider''' tab.
  
−
Under '''Animation''', for '''Repeat''', choose option “'''Increasing'''” from dropdown menu.
+
Under '''Animation''', for '''Repeat''',  
 +
 
 +
choose option '''Increasing''' from dropdown menu.
 
|| A '''Preferences''' dialog-box opens.
 
|| A '''Preferences''' dialog-box opens.
  
Line 409: Line 432:
 
||Close the '''Preferences''' dialog box.
 
||Close the '''Preferences''' dialog box.
 
|-
 
|-
−
||Right-click on point '''A''' in '''Algebra''' view >> select '''Trace On'''.
+
||Right-click on point '''A''' in '''Algebra''' view  
 +
 
 +
>> select '''Trace On'''.
 
||In '''Algebra''' view, right-click on '''A''' and select '''Trace On'''.
 
||In '''Algebra''' view, right-click on '''A''' and select '''Trace On'''.
 
|-
 
|-
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||Let us summarize.
 
||Let us summarize.
 
|-
 
|-
−
||'''Slide Number 6'''
+
||'''Slide Number 7'''
  
 
'''Summary'''
 
'''Summary'''
 
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to view:
 
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to view:
  
−
And construct different structures in '''3D''' space
+
And construct different structures in 3D space
  
−
Solids of rotation of '''polynomial functions'''
+
Solids of rotation of polynomial functions
  
−
'''Trigonometric functions''' in '''3D''' space
+
Trigonometric functions in 3D space
 
|-
 
|-
−
||'''Slide Number 7'''
+
||'''Slide Number 8'''
  
 
'''Assignment'''
 
'''Assignment'''
Line 461: Line 486:
 
||As an assignment:
 
||As an assignment:
  
−
Construct a prism and a cylinder anywhere in '''3D''' space.
+
Construct a prism and a cylinder anywhere in 3D space.
  
 
Draw lines to pierce the structures and find their intersection points.
 
Draw lines to pierce the structures and find their intersection points.
Line 469: Line 494:
 
Show the solid formed due to rotation of the peak, in the first '''quadrant''', in the '''XY''' plane.
 
Show the solid formed due to rotation of the peak, in the first '''quadrant''', in the '''XY''' plane.
 
|-
 
|-
−
||'''Slide Number 8'''
+
||'''Slide Number 9'''
  
 
'''Assignment'''
 
'''Assignment'''
  
−
You tried to fly a kite off a cliff. The kite got dumped into the lake below
+
You tried to fly a kite off a cliff.  
 +
 
 +
The kite got dumped into the lake below
  
 
You gave out 325 feet of string
 
You gave out 325 feet of string
  
−
The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees
+
The angle of declination from where you stand at the cliff’s edge  
 +
 
 +
to the kite is 15 degrees
  
 
How high is the cliff?
 
How high is the cliff?
Line 490: Line 519:
 
How high is the cliff?
 
How high is the cliff?
 
|-
 
|-
−
||'''Slide Number 8'''
+
||'''Slide Number 10'''
  
 
'''About Spoken Tutorial Project'''
 
'''About Spoken Tutorial Project'''
Line 497: Line 526:
 
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:
 
||The '''Spoken Tutorial '''project''' '''team:
  
−
<nowiki* conducts workshops using spoken tutorials and</nowiki>
+
conducts workshops using spoken tutorials and</nowiki>
  
−
<nowiki>* gives certificates on passing online tests.</nowiki>
+
gives certificates on passing online tests.</nowiki>
  
 
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 523: Line 552:
 
||Please post your timed queries on this forum.
 
||Please post your timed queries on this forum.
 
|-
 
|-
−
||'''Slide Number 11'''
+
||'''Slide Number 13'''
  
 
'''Acknowledgement'''
 
'''Acknowledgement'''

Latest revision as of 17:19, 25 October 2018

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on 3D Geometry.
Slide Number 2

Learning Objectives

In this tutorial, we will learn how to use GeoGebra to view:

And construct different structures in 3D space

Solids of rotation of polynomial functions

Trigonometric functions in 3D space

Slide Number 3

System Requirement

Here I am using:

Ubuntu Linux OS version 16.04

GeoGebra 5.0.481.0-d

Slide Number 4

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with:

GeoGebra interface

Geometry

For relevant tutorials, please visit our website.

Slide Number 5

Rectangular Co-ordinate System

[[Image:]]

This image shows the rectangular coordinate system.

It is made up of mutually perpendicular axes and planes formed by them.

The axes are x (in red), y (in green) and z (in blue).

All points in 3D space are denoted by their x y z coordinates.

The point of intersection of the three axes is the origin O 0 comma 0 comma 0.

The gray rectangle in the image depicts the XY plane.

The planes divide space into 8 octants.

Point A is in the XOYZ octant and has the coordinates 4 comma 4 comma 2.

Show the GeoGebra window. Let us draw a 3D pyramid in GeoGebra.

I have already opened a new window in GeoGebra.

This time, we work with Algebra, 2D Graphics and 3D Graphics views.

Under View, select 3D Graphics. Under View, select 3D Graphics.
Click in 2D Graphics View to draw in 2D. Click in 2D Graphics View to draw in 2D.
Drag the boundary to see 2D Graphics properly. Drag the boundary to see 2D Graphics properly.
Click in 2D Graphics. Click in 2D Graphics.
In 2D Graphics view, click on Polygon tool and click on origin (0,0).

Point to A.

In 2D Graphics view, click on the Polygon tool and click on origin 0 comma 0.

This creates point A at the origin.

Click on (2,0) to create point B. Then click on 2 comma 0 to create point B.
Click on (2,2) for point C >>

on (0,2) to draw point D.

Click on 2 comma 2 for C and on 0 comma 2 to draw D.
Click again on point A. Finally, click again on A.
Point to quadrilateral q1 in 2D >> 3D Graphics views. Note that a quadrilateral q1 is seen in 2D and 3D Graphics views.


The length of each side is 2 units.

Click on Move tool. Click on the Move tool.
Click in 2D Graphics >> drag the background. Click in 2D Graphics and drag the background.
Drag the boundary to see 3D Graphics properly. Drag the boundary to see 3D Graphics properly.
Click in 3D Graphics and under Pyramid,

on the Extrude to Pyramid or Cone tool.

Click on the square in 3D Graphics view.

Click in 3D Graphics and under Pyramid, on the Extrude to Pyramid or Cone tool.

In 3D Graphics view, click on the square.

An Altitude text box opens up, type 3 and click OK. In the Altitude text-box that opens, type 3 and click OK.
Point to pyramid e in 3D Graphics view.

Point to base.

Point to E (1,1,3).

Show altitude.

A pyramid e appears in 3D Graphics view.

Its base is the quadrilateral q1.

Its apex is E 1 comma 1 comma 3.

Its altitude or height is 3 units.

Slide Number 6

Rotation of a Polynomial

Let us rotate f(x)= -2x4-x3+3x2

Part in second quadrant (XY plane) about x-axis

Rotation of a Polynomial

Let us rotate f of x equals minus 2 x raised to 4 minus x cubed plus 3 x squared.

We will rotate the part that lies in the second quadrant, in XY plane, about the x-axis.

Show the GeoGebra window. I have already opened a new window in GeoGebra.

We will initially work with Algebra and 2D Graphics views and open 3D Graphics view later.

In input bar, type the following line

>> press Enter.

f(x) = -2 x^4 -x^3+3 x^2

In the input bar, type the following line.

To type the caret symbol, hold Shift key down and press 6.

Spaces here denote multiplication.

Press Enter.

Under Perpendicular Line, click on Parallel line

>> on the y-axis.

Under Perpendicular Line, click on Parallel line and on the y-axis.
Keeping the cursor on the x-axis. Keep the cursor on the x-axis.
Drag it along until you reach the intersection of f and x-axis.

Point to the label function f, x-axis that appears.

Drag it along until you see function f, x-axis at the intersection of f and x-axis.
Click on this intersection point. Click on this intersection point.
Point to A. Point A appears.
Click on Slider >> in Graphics view. Click on Slider and in Graphics view.
A Slider dialog box opens. A Slider dialog-box opens.
Leave a as the Name. Leave a as the Name.
Change Min value to -1.5, Max value to 0

Increment to 0.05.

Change Min value to minus 1.5, Max value to 0 and Increment to 0.05.
Click OK. Click OK.
Point to slider a.

Point to the part of function,

x-axis in the second quadrant.

This creates slider a, which changes the value of a from -1.5 to 0.

It will focus on the part of the graph in the second quadrant.

In the input bar, type (a,f(a)).

Press Enter.

In the input bar, type the following in parentheses.

a comma f a in parentheses.

Press Enter.

Point to B. This creates point B whose x coordinate is the value of a.

Its y-coordinate lies along the curve described by the function f between x equals 1.5 and 0.

Right-click on slider a >> check Animation On. Right-click on slider a and check Animation On.
Point to B and slider a. Point B travels along function f as a changes.
Right-click on slider a >> uncheck Animation On. Right-click on slider a and uncheck Animation On.
In the input bar, type (a,0) >> press Enter. In the input bar, type a comma 0 in parentheses and press Enter.
Point to C and slider a. This creates point C.

As its x co-ordinate a changes, C moves below point B along the x-axis.

Under Line, click on Segment

>> click on B and C to join them.

Under Line, click on Segment and click on B and C to join them.
Click on Move Graphics View >> drag background to the left. Click on Move Graphics View and drag the background to the left.
Click on View

>> check 3D Graphics to see the 3D Graphics view.

Click on View and check 3D Graphics to see the 3D Graphics view.
Point to 2D Graphics and 3D Graphics views. Note that what is drawn in 2D Graphics appears in the XY plane, in 3D Graphics.
Click in 3D Graphics view >> on Rotate 3D Graphics View.

Rotate the 3D Graphics view to see the curve properly.

Click in 3D Graphics view and on Rotate 3D Graphics View.

Rotate 3D Graphics to see the curve properly.

Place cursor on y-axis (in green).

Click to see an arrow aligned with the y-axis.

Place the cursor on the y-axis (in green).

Click to see an arrow aligned with the y-axis.

Drag along the y axis to see the curve properly. Drag to pull the y-axis in or outwards to see the curve.
Type Circle[C,f(a),xAxis] in the input bar

>> press Enter.

In the input bar, type the following line.
Point to circle c with center at point C. This creates circle c with center at point C.

Its radius is equal to f of a corresponding to the value of a on slider a.

Its rotation is around the x-axis.

Press Enter.

Right-click on circle c in Algebra view

>> check Trace On option.

In Algebra view, right-click on circle c and check Trace On option.
Right-click on slider a >> select Animation On option. Right-click on slider a and select Animation On option.
Point to the solid traced as a changes. Observe the solid traced as a changes.
Point to 2D >> 3D Graphics views. Watch both 2D and 3D Graphics views.
Point to Segment BC, x-axis and function f. Segment BC moves between the x-axis and function f.
Point to the part of f in the second quadrant. The part of function f that is in the second quadrant in 2D, rotates around the x-axis.
Drag 3D Graphics to see it from another angle. Drag 3D Graphics to see it from another angle.
Show the GeoGebra window. Finally, let us look at trigonometric functions in 3D.

I have already opened a new window in GeoGebra.

Click on 3D Graphics tool under View. Under View, click on 3D Graphics.
Drag the boundary to see 2D Graphics properly. Drag the boundary to see 2D Graphics properly.
Click in 2D Graphics

>> Slider tool and in Graphics view.

Click in 2D Graphics, then on the Slider tool and in Graphics view.
Point to the slider dialog box and Number radio button selection.

Type t in the Name field.

A slider dialog-box opens.

By default, the Number radio-button is selected.

In the Name field, type t.

Set Min to -6, Max to 16 and increment of 0.1.

Click OK.

Set Min to minus 6, Max to 16 and increment of 0.1.

Click OK.

Point to slider t. This creates a slider t which will change t from minus 6 to 16.
In the input bar, type f(t)=cos(t)

>> press Enter.

In the input bar, type f t in parentheses equals cos t in parentheses and press Enter.
Click in 2D Graphics. Click in 2D Graphics.
Under Move Graphics View, click on Zoom Out

>> click in 2D Graphics.

Under Move Graphics View, click on Zoom Out and click in 2D Graphics.
Click on Move Graphics View >> drag the background. Click on Move Graphics View and drag the background.
Point to the cosine function of f(t) in 2D and 3D Graphics views. You can see the graph of the cosine function of f of t, in 2D and 3D Graphics views.
Type g(t)=sin(t) in the input bar >> press Enter. Similarly, in the input bar, type g t in parentheses equals sin t in parentheses.

Press Enter.

Point to sine function graph (g(t)). Sine function graph of g of t appears.
Type h(t)=t/4 in the input bar >> press Enter. In the input bar, type h t in parentheses equals t divided by 4 and press Enter.
Point to line (h(t)). Line h of t is of the form y equals mx where slope m is 1 divided by 4.
Click in 3D Graphics view. Click in 3D Graphics view.
Click on Point tool >> click in the gray area in 3D Graphics view.

Point to point A.

Click on the Point tool and click in the gray area in 3D Graphics view.

This creates point A.

Drag the boundary to see its co-ordinates properly. Drag the boundary to see its co-ordinates properly.
Double click on point A in Algebra view.

Change the co-ordinates to (f(t),g(t),h(t)).

Press Enter.

In Algebra view, double-click on A.

Change the coordinates to the following. Press Enter.

Point to A. The x- coordinate of A is cos t.

The y-coordinate is sin t and t divided by 4 is its z coordinate.

Right-click on slider t >> click on Object Properties. Right-click on slider t and click on Object Properties.
A Preferences dialog-box opens.

Click on Slider tab.

Under Animation, for Repeat,

choose option Increasing from dropdown menu.

A Preferences dialog-box opens.

Click on Slider tab.

Under Animation, for Repeat, choose option “Increasing” from the dropdown menu.

Close the Preferences dialog box. Close the Preferences dialog box.
Right-click on point A in Algebra view

>> select Trace On.

In Algebra view, right-click on A and select Trace On.
Right-click on slider t >> check Animation On. Right-click on slider t and check Animation On.
Point to point A and the helix in 3D Graphics view.

Point to point A’s co-ordinates in Algebra view.

Point A traces a helix in 3D space with coordinates mentioned earlier.
Click in Rotate 3D Graphic View >> rotate the background.

Rotate 3D Graphics view.

Click in Rotate 3D Graphic View and rotate the background.

Rotate 3D Graphics view so you are looking down the z-axis at the XY plane.

Point to the traces of point A (cos(t), sin(t)). Note that the traces of A are the circumference of a unit circle.

Point A moves along the circle as angle t changes.

In 2D, its coordinates are cos t comma sin t.

Let us summarize.
Slide Number 7

Summary

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

And construct different structures in 3D space

Solids of rotation of polynomial functions

Trigonometric functions in 3D space

Slide Number 8

Assignment

Construct a prism and a cylinder.

Draw lines to pierce the structures and find their intersection points.

Graph the polynomial, f(x)=x5-7x4+9x3+23x2-50x+24.

Show the solid formed due to rotation of peak in first quadrant in XY plane.

As an assignment:

Construct a prism and a cylinder anywhere in 3D space.

Draw lines to pierce the structures and find their intersection points.

Graph the given polynomial.

Show the solid formed due to rotation of the peak, in the first quadrant, in the XY plane.

Slide Number 9

Assignment

You tried to fly a kite off a cliff.

The kite got dumped into the lake below

You gave out 325 feet of string

The angle of declination from where you stand at the cliff’s edge

to the kite is 15 degrees

How high is the cliff?

As another assignment,

You tried to fly a kite off a cliff. The kite got dumped into the lake below.

You gave out 325 feet of string.

The angle of declination from where you stand at the cliff’s edge to the kite is 15 degrees.

How high is the cliff?

Slide Number 10

About Spoken Tutorial Project

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

Please download and watch it.

Slide Number 11

Spoken Tutorial workshops

The Spoken Tutorial project team:

conducts workshops using spoken tutorials and</nowiki>

gives certificates on passing online tests.</nowiki>

For more details, please write to us.

Slide Number 12

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 13

Acknowledgement

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, Snehalathak, Vidhya

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