Applications-of-GeoGebra/C2/Trigonometric-Ratios-and-Graphs/English
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 = α^{0} =β^{0} In triangle AB'C, sin(α) = B'C/AB' = y(B')/radius Here, sin(α) = y co-ordinate of point B' |
Sine function
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 360^{0}. | Drag alpha slider to 0 and then to 360 degrees. |
Point to sine values in Algebra view. | 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. |
Point to the values in Algebra and Graphics view. | 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. |
Point to positive side of x axis. | 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 0^{0} to 360^{0}. | 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 function
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 0^{0} to 360^{0}. | 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 0^{0}. | 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 0^{0} to 360^{0}. | 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 Number 7
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 function
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 0^{0} to 360^{0}. |
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 0^{0} to 360^{0}. | 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 360^{0} to 0^{0}. | 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
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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. |