View source for Applications-of-GeoGebra/C2/Trigonometric-Ratios-and-Graphs/English

{|border=1
||'''Visual Cue'''
||'''Narration'''

|-
|  | '''Slide Number 1'''

'''Title Slide'''
|  | Welcome to this tutorial on '''Trigonometric Ratios and Graphs'''.
|-
|  | '''Slide Number 2'''

'''Learning Objectives'''
|  | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
Calculate '''trigonometric ratios'''

Plot corresponding graphs
|-
|  | '''Slide Number 3'''

'''Pre-requisites'''
|  | To follow this '''tutorial''', you should be familiar with
*'''GeoGebra''' interface

*Previous '''tutorials''' in this series

If not, for relevant '''tutorials''', please visit our website '''www.spoken-tutorial.org'''.
|-
| >'''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.

Point to unit circle and right triangle '''ACB''''.
|  | I have opened '''GeoGebra''' interface with a '''unit circle''' and a right triangle '''A C B prime.'''
|-
|  | '''Slide Number 5'''

'''Sine function'''

'''Sine''' of an angle is the ratio of the lengths of the opposite side to the '''hypotenuse'''.

Angle B'AC = αº = βº

In triangle AB'C,

'''sin(α) = B'C/AB' = y(B')/radius'''

Here, '''sin(α) = y co-ordinate''' of point '''B''''
|  | '''Sine''' of an angle is the ratio of the lengths of the opposite side to the '''hypotenuse'''.

'''Angle B prime A C''' is equal to '''alpha degrees''' and to '''beta degrees'''

In '''triangle A B prime C''', '''sine alpha''' equals ratio of the lengths '''B prime C''' to '''A B prime'''.

This is also equal to ratio of '''y co-ordinate''' of '''B prime''' to '''radius'''.

Here, '''sine alpha''' is '''y co-ordinate''' of point '''B prime'''.
|-
|  | Click on '''Options''' menu >> select '''Rounding''' >> '''3 Decimal Places'''.
|  | Click on '''Options''' menu.

Select '''Rounding''' and then '''3 Decimal Places'''.

All the ratios will now have 3 decimal places.
|-
|  | '''Setting up the sine function'''

In '''input bar''', type '''SINE= y(B')/radius'''>> press '''Enter'''
|  | Now let us show '''sine alpha''' values using the '''input bar'''.

In '''input bar''', type '''SINE''' is equal to '''y B prime''' in parentheses divided by '''radius'''.

Press '''Enter'''.
|-
|  | Point to '''sine''' values in '''Algebra''' view.
|  | '''Sine''' values are displayed in '''Algebra''' view.
|-
|  | Drag '''slider α''' to 0 and then to 360º.
|  | Drag '''alpha slider''' to 0 and then to 360 '''degrees'''.
|-
|  |
|  | Observe the change in '''sine''' values in '''Algebra''' view.

Observe that '''sine''' value remains positive as long as '''y axis''' values are positive.
|-
|  | '''Graphing the sine function'''

Click on '''Point''' >> click on '''Graphics''' view.
|  |
Click on '''Point''' tool.

Click on the screen outside the circle in '''Graphics view.'''
|-
|  | Point to point '''D'''.
|  | Point '''D''' appears outside the circle.
|-
|  | Drag '''slider α''' to 0.
|  | Set '''alpha''' to 0 '''degrees''' on the '''slider'''.
|-
|  | Right click on '''D''' >> Select '''Object Properties''' >> '''Color''' tab >> red.
|  | Right-click on '''D''' and click on '''Object Properties'''.

Select '''Color''' tab and choose red.
|-
|  | Close the '''Preferences''' window.
|  | Close the '''Preferences''' window.
|-
|  | Again right-click on '''D''' and check '''Trace On''' option.
|  | Again, right-click on '''D''' and check '''Trace On''' option.
|-
|  | In '''Algebra''' view, double click on '''D'''.
|  | In '''Algebra''' view, double click on '''D'''.
|-
|  | Delete '''co-ordinates''' of '''D'''.
|  | Delete '''co-ordinates''' of '''D'''.
|-
|  | Select '''symbol α''' >> click on the letter '''α''' >> Insert '''α''' as '''x co-ordinate''' of '''D'''.
|  | Select '''symbol alpha''', click on the letter '''alpha'''.

Insert '''alpha''' as '''x co-ordinate''' of '''D'''.
|-
|  | Type '''SINE''' as '''y co-ordinate''' of '''D''' >> press '''Enter'''
|  | Type '''SINE''' as '''y co-ordinate''' of '''D''', and press '''Enter'''.
|-
|  | Point to '''D (α, SINE)''' in the '''Algebra''' view.
|  | '''D''' has been changed to '''alpha comma SINE'''.
|-
|  |
|  | '''GeoGebra''' will convert '''alpha''' into '''radians'''.

The '''alpha''' value in '''radians''' is the '''x co-ordinate''' of '''D'''.

Its '''y co-ordinate''' is the '''SINE''' value of '''alpha'''.

This will make '''D''' trace the '''sine function''' as you change '''angle alpha'''.
|-
|  |
|  | We want to see 2 '''pi radians''' along the positive side of the '''x axis'''.
|-
|  | Under '''Move Graphics View''', click once on '''Zoom Out''' and then twice in '''Graphics''' view.
|  | Under '''Move Graphics View''', click once on '''Zoom Out''' and then twice in '''Graphics''' view.
|-
|  | Click on '''Move Graphics View''' tool.

Click on '''Graphics''' background and when hand '''symbol''' appears, move '''Graphics''' view.
|  | Click on '''Move Graphics View''' tool.

Click on '''Graphics''' background and when hand '''symbol''' appears, move '''Graphics''' view.
|-
|  | Point to circle and 2 '''pi radians''' on right side of origin on '''x axis'''.
| | You should see the circle and 2 '''pi radians''' along positive side of '''x axis'''.
|-
|  | Drag '''slider α''' from 0º to 360º.

|  | Increase '''alpha''' on the '''slider''' from 0 to 360 '''degrees''' 2 '''pi radians'''.
|-
|  | Point to traces of '''D'''.
|  | Point '''D''' will trace the '''sine function''' graph.
|-
|  | Point to '''SINE''' values in '''Algebra''' view.
|  | '''Sine''' values remain positive as long as '''y''' values are positive.
|-
|  | In '''input bar''', type '''d(x) = sin(x)''' >> press '''Enter'''.
|  | In '''input bar''', type '''d x''' in parentheses is equal to '''sin x''' in parentheses and press '''Enter'''.
|-
|  | Point to the '''sine function''' graph beyond '''−2π''' and '''+2π''' '''radians'''.
|  | '''Sine function''' will be graphed beyond '''minus 2 pi''' and '''plus 2 pi radians'''.
|-
|  | Click on and move '''Graphics''' view to see '''d of x''' beyond '''minus''' and '''plus''' 2 '''pi radians'''.
|  | Click on and move '''Graphics''' view to see '''d of x''' beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
|  | Point to '''D'''.
|  | Note that this will erase traces of '''D'''.
|-
|  | Click on and move '''Graphics''' view to see circle and '''plus''' 2 '''pi radian''' along '''x axis'''.
|  | Click on and move '''Graphics''' view to see circle and '''plus''' 2 '''pi radians''' along '''x axis'''.
|-
|  | Drag '''slider α''' to 0 '''degrees''' to see traces of '''D'''.
|  | Again drag '''slider alpha''' to 0 '''degrees''' to see traces of '''D'''.
|-
|  | Point to '''d(x)''' and traces of '''D'''.
|  | Compare '''d of x''' with traces of '''D'''.
|-
|  | '''Slide Number 6'''

'''Cosine function'''

'''Cosine''' of an angle is the ratio of the lengths of the adjacent side to the hypotenuse.

cos(α) = AC/AB' = x(B')/radius

In this unit circle, cos(α) = x co-ordinate of point B'
|  |
'''Cosine''' of an angle is the ratio of the lengths of the adjacent side to the '''hypotenuse'''.

'''Cos alpha''' is equal to the following ratios.

Length of '''AC to''' length of '''AB prime''' and '''x co-ordinate''' of '''B prime''' to '''radius'''.

In this '''unit circle, cos alpha''' corresponds to '''x co-ordinate''' of point '''B prime.'''
|-
|  | Right-click on point '''D''' and uncheck '''Trace On''' option.
|  | Right-click on point '''D''' and uncheck '''Trace On''' option.
|-
|  | Click on and move '''Graphics''' view slightly to erase traces of '''D'''.
|  | Click on and move '''Graphics''' view slightly to erase traces of '''D'''.
|-
|  | In '''input bar''', type '''COSINE = x(B')/radius''' >> press '''Enter'''.

|  | In '''input bar''', type the following line.

'''COSINE''' is equal to '''x B prime''' in parentheses divided by '''radius'''.

Press '''Enter'''.
|-
|  | Point to '''cosine''' value in '''Algebra''' view.
|  | '''Cosine''' value is displayed in '''Algebra''' view.
|-
|  | Drag '''slider α''' from 0º to 360º.
|  | Drag '''slider alpha''' from 0 to 360 '''degrees'''.
|-
|  | Point to the '''cosine''' value in the '''Algebra''' view.
|  | Observe how '''cosine''' values change in '''Algebra''' view.
|-
|  | Point to positive side of '''x axis'''.
|  | Note how '''cosine''' remains positive as long as '''x axis''' values are positive.
|-
|  | '''Graphing the cosine function'''

Click on '''Point''' tool and click outside the circle.
|  |
Click on '''Point''' tool and click outside the circle.

'''Point E''' appears outside the circle.
|-
|  | Drag '''slider α''' to 0º.
|  | Drag '''slider alpha''' to 0 '''degrees'''.
|-
|  | Right click on '''E''' >> Select '''Object Properties'''>> '''Color''' tab >> Brown.
|  | Right-click on '''E''', click on '''Object Properties'''.

Select '''Color''' tab and choose brown.
|-
|  | Close the '''Preferences''' window.
|  | Close the '''Preferences''' window.
|-
|  | Right-click on '''E,''' check '''Trace On''' option.
|  | Right-click on '''E''', check '''Trace On''' option.
|-
|  | In '''Algebra''' view, double click on '''E'''.
|  | In '''Algebra''' view, double click on '''E'''.
|-
|  | Delete '''co-ordinates''' of '''E'''.
|  | Delete '''co-ordinates''' of '''E'''.
|-
|  | Select '''symbol α''' >> click on the letter '''α''' >> insert '''α''' as '''x co-ordinate''' of '''E'''
|  | Select '''symbol alpha''', click on the letter '''alpha'''.

Insert '''alpha''' as '''x co-ordinate''' of '''E'''.
|-
|  | Type '''COSINE''' instead of the '''y co-ordinate''' of '''E''' >> press '''Enter'''
|  | Type '''COSINE''' instead of '''y co-ordinate''' of '''E''', and press '''Enter'''.
|-
|  | Point to '''E''' ('''α, COSINE''') in '''Algebra''' view.
|  | '''E''' has been changed to '''alpha comma COSINE'''.
|-
|  | Drag '''slider α''' from 0º to 360º.
|  | Drag '''slider alpha''' from 0 to 360 '''degrees'''.
|-
|  | Point to traces of '''E'''.
|  | Point '''E''' will trace the '''cosine function''' graph.
|-
|  | In '''input bar,''' type '''e(x) = cos(x)''' >> press '''Enter'''.
|  | In input bar, type '''e x in parentheses is equal to cos x in parentheses'''.

Press '''Enter'''.
|-
|  | Point to '''cosine function e(x)'''.
|  | '''Cosine function e of x''' will be graphed beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
|  | Click on and move '''Graphics''' view to see '''e(x''') beyond '''−2π''' and '''+2π radians'''.
|  | Click and move '''Graphics''' view to see '''e of x''' beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
|  | Point to '''E'''.
|  | This will erase traces of '''E'''.
|-
|  | Click on and move '''Graphics''' view to see +2 '''pi radians''' along '''x axis'''.
|  | Click on and move '''Graphics '''view to see '''plus''' 2 '''pi radians''' along '''x axis'''.
|-
|  | Drag '''slider α''' to 0 '''degrees''' to see traces of '''E'''.
|  | Again drag '''slider alpha''' to 0 '''degrees''' to see traces of '''E'''.
|-
|  | Point to '''e(x)''' and traces of '''E'''.
|  | Compare the graph of '''e of x''' with traces of '''E'''.
|-
|  | Right-click on point '''E''' >> Uncheck '''Trace on'''
|  | Right-click on '''E''' and uncheck '''Trace On''' option.
|-
|  | Click in and move '''Graphics''' view slightly to erase traces of '''E'''.
|  | Click on and move '''Graphics''' view slightly to erase traces of '''E'''.
|-
|  | '''Slide Number7'''

'''Tangent function'''

'''Tangent''' of an angle is the ratio of lengths of the opposite side to the adjacent side

tan(α) = sin(α)/cos(α) = B'C/AC

tan(α) = y(B')/x(B')
|  |
'''Tangent''' of an angle is the ratio of lengths of the opposite side to the adjacent side.

'''Tan alpha''' is the ratio of '''sine alpha''' to '''cos alpha''' and the ratio of lengths of '''B prime C''' to '''AC'''.

'''Tan alpha''' is also the ratio of the '''y co-ordinate''' to '''x co-ordinate''' of '''B prime'''.
|-
|  | In '''input bar''', type '''TANGENT = y(B')/x(B')''' >> press '''Enter'''.
|  | In '''input bar''', type the following line.

'''TANGENT''' is equal to '''y B prime''' in parentheses divided by '''x B prime''' in parentheses.

Press '''Enter'''.
|-
|  | Point to the '''tangent''' value in '''Algebra''' view.
|  | '''Tangent''' value is displayed in '''Algebra''' view.
|-
|  | '''Setting up the tangent function'''

Drag '''alpha slider''' from 0º to 360º.
|  | Drag '''alpha slider''' from 0 to 360 '''degrees'''.
|-
|  | Point to the '''Tangent''' values in''' Algebra''' view.
|  | Observe how '''tangent''' values change in '''Algebra''' view.
|-
|  | Click on''' Point''' tool and click outside the circle.
|  | Click on '''Point''' tool and click outside the circle.
|-
|  | Point to point '''F'''.
|  | Point '''F''' appears outside the circle.
|-
|  | Drag '''α slider''' to 0.
|  | Set '''alpha''' to 0 '''degrees''' on the '''slider'''.
|-
|  | Right-click on '''F''' >> Select '''Object Properties''' >> '''Color''' tab >> green.
|  | Right-click on '''F''' and select '''Object Properties'''.

Select '''Color''' tab and choose green.
|-
|  | Close the '''Preferences''' window.
|  | Close the '''Preferences''' window.
|-
|  | Again right-click on '''F''', check '''Trace On''' option.
|  | Again right-click on '''F''', check '''Trace On''' option.
|-
|  | In '''Algebra''' view, scroll down and double click on '''F'''.
|  | In '''Algebra''' view, scroll down and double click on '''F'''.
|-
|  | Delete '''co-ordinates''' of '''F'''.
|  | Delete '''co-ordinates''' of '''F'''.
|-
|  | Select '''symbol α''' >> click on the letter '''α''' >> insert '''α''' as '''x co-ordinate''' of '''F'''
|  | Select '''symbol alpha''', click on the letter '''alpha'''.

Insert '''alpha''' as '''x co-ordinate''' of '''F'''.
|-
|  | Type '''TANGENT''' as '''y co-ordinate''' of '''F''' >>press '''Enter'''
|  | Type '''TANGENT''' as '''y co-ordinate''' of '''F''', and press '''Enter'''.
|-
|  | Point to '''F''' ('''α, TANGENT''') in the '''Algebra''' view.
|  | '''F''' has been changed to '''alpha comma TANGENT'''.
|-
|  | Point to '''F'''.
|  | Point '''F''' will trace the '''tangent function''' graph as '''alpha''' value changes.
|-
|  | Drag '''α slider''' value from 0º to 360º.
|  | Increase '''alpha''' on the '''slider''' from 0 to 360 '''degrees''' 2 '''pi radians'''.
|-
|  | Point to traces of '''F''' from 0 to π/2 '''radians'''.
|  | '''F''' increases from '''origin''' to '''infinity'''.

Note '''vertical asymptote''' at '''pi''' divided by 2 '''radians'''.
|-
|  | Point to the graphs.
|  | '''Tangent''' value is plus '''infinity''' at '''pi''' divided by 2 '''radians'''.

It is minus '''infinity''' at 3 '''pi''' divided by 2 '''radians'''.
|-
|  | Type '''f(x) = tan(x)''' in '''input bar''' >> press '''Enter'''.
|  | In '''input bar''', type '''f x in parentheses is equal to tan x in parentheses''' and press '''Enter'''.
|-
|  | Point to '''f(x)'''.
|  | The '''tangent function''' is graphed beyond minus 2 '''pi''' and plus 2 '''pi radians'''.
|-
|  | Click on and move '''Graphics''' view beyond '''−2π''' and '''+2π radians'''.
|  | Click on and move '''Graphics''' view to see graph of '''f of x''' beyond '''minus''' 2 '''pi''' and '''plus''' 2 '''pi radians'''.
|-
|  | Click on and move '''Graphics''' background to see plus 2 '''pi radians''' along '''x axis'''.  
|  | Click on and move '''Graphics''' view to see '''plus 2''' '''pi radians''' along '''x axis'''.
|-
|  | Drag '''α slider''' value from 360º to 0º.
|  | Drag '''slider alpha''' back to 0 '''degrees''' to see traces of '''F'''.
|-
|  | Point to '''f(x)''' and traces of '''F'''.
|  | Also compare the '''tangent function f of x''' with traces of '''F'''.
|-
|  |
|  | Let us summarize.
|-
|  | '''Slide Number 8'''

'''Summary'''
|  | In this tutorial, we have learnt

how to use '''GeoGebra''' to calculate and graph '''sin alpha''', '''cos alpha''' and '''tan alpha'''
|-
|  | '''Slide Number 9'''

'''Assignment'''
|  | Assignment

Try these steps to graph '''secant, cosecant''' and '''cotangent functions'''.

Analyze the link between '''sine''' values for '''supplementary angles''' (angles whose sum is 180 '''degrees''').

Analyze the link between '''sine''' and '''cosine''' values for '''supplementary angles'''.
|-
| '''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 and gives certificates.

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.
|-
|}