PhET-Simulations-for-Mathematics/C2/Trig-Tour/English-timed

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Time Narration
00:01 Welcome to this tutorial on Trig Tour, an interactive PhET simulation.
00:07 In this tutorial, we will demonstrate Trig Tour, an interactive PhET simulation.
00:15 Here I am using, Ubuntu Linux OS version 16.04
00:22 Java version 1.8.0
00:26 Firefox Web Browser version 60.0.2
00:32 Learners should be familiar with trigonometry.
00:36 Using this simulation we will learn how to,

Construct right triangles for a point moving around a unit circle

00:47 Calculate trigonometric ratios, cos, sin and tan, of angle theta
00:54 Graph theta versus cos, sin and tan functions of theta along x and y axes
01:01 Let us begin.
01:03 Use the given link to download the simulation.
01:08 I have already downloaded the Trig Tour simulation to my Downloads folder.
01:14 To open the simulation, right click on the trig-tour html file.
01:20 Select the Open With Firefox Web Browser option.

The file opens in the browser.

01:29 This is the interface for the Trig Tour simulation.
01:34 The interface has four boxes:

Values

Unit circle

01:41 Functions, Special angles, labels and grid

Graph

01:48 The reset button takes you back to the starting point.
01:53 In the Functions box, check Special angles, Labels, Grid and click cos.
02:05 Cosine function

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

02:15 Cosine value is the x co-ordinate of a point moving around a unit circle.
02:23 The center of this unit circle is the origin 0 comma 0.

cosine theta is x divided by radius and hence, is x for the unit circle.

02:38 A unit circle is drawn in a Cartesian coordinate system with x and y axes in the Unit Circle box.
02:49 A red point is seen at the circumference of the circle on the x-axis.
02:55 A blue arrow is seen along the x-axis pointing to the red point.

This corresponds to a radius of 1 for the unit circle.

03:07 The Values box contains important values.
03:12 The angle ϴ (theta) can be given in degrees or radians.
03:17 Click the degrees radio button.
03:20 x comma y are co-ordinates 1 comma 0 of the red point at angle theta equals 0 degrees.
03:30 When angle theta equals 0 degrees, x co-ordinate of the red point is 1.
03:38 x-axis of the graph shows angle theta.
03:43 y-axis of the graph shows the amplitude of the cos theta function.
03:49 At an angle theta of 0 degrees, cos theta is 1.
03:54 The red point is at the highest amplitude of 1.
03:59 In the Values box, click the radians radio button.
04:04 x axis of the theta vs cos theta graph is converted into radians.
04:11 Remember that pi radians are equal to 180 degrees.
04:17 One full rotation of 360 degrees is equal to 2 pi radians.

Again, click the degrees radio button.

04:29 You can see empty circles on the unit circle.

In the Functions box, uncheck Special Angles.

04:39 Observe how the empty circles disappear.
04:43 Again, check Special Angles.
04:47 These circles are angles made by the red point with the x-axis as it moves along the circle.
04:56 Important angles have been chosen as Special angles.
05:01 In the Unit Circle, drag the red point counter-clockwise (CCW) to the next special angle.
05:09 The red point has moved 30 degrees in the counter-clockwise direction along the circle.
05:16 In the Values box, x comma y is the squareroot of 3 divided by 2 comma half.
05:25 In the unit circle, according to Pythagoras’ theorem, x squared plus y squared is 1.
05:34 Two square lengths in the Cartesian plane is equal to 1 as radius of unit circle is 1.
05:44 y covers only 1 square length and hence, is half.
05:50 x covers 1 full and almost three-fourths of a second square.
05:57 The squareroot of 3 divided by 2 is 0.866.

This is the value of x.

06:07 Look at the graph.

The red point has moved to 30 degrees along the cos function.

06:15 In the Values box, click radians radio button.
06:20 This converts 30 degrees into pi divided by 6 radians for theta in the Values box.
06:29 Sine function

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

06:39 Sine value is the y-co-ordinate of the point moving around the same unit circle.
06:47 Sine theta is y divided by radius and hence, is y for the unit circle.
06:56 Drag the red point back to the x axis.
07:00 In the Functions box, click sin.
07:04 Click the degrees radio button.
07:07 As seen earlier, x comma y are 1 comma 0.
07:13 Note the definitions of sine theta given earlier.
07:18 When angle theta is 0 degrees, the y co-ordinate of the red point is 0.
07:25 The graph shows angle theta on the x-axis and the amplitude of the sine theta function on the y-axis.
07:34 At angle theta of 0 degrees, as sine theta is 0, the red point has amplitude 0.
07:43 In the Unit Circle, drag the red point counter clockwise to the next special angle 30 degrees.
07:51 In the Values box, note that x comma y is squareroot of 3 divided by 2 comma half.

Remember how you can calculate these.

08:04 In the graph, the red point has moved to 30 degrees along the sine function.

Its amplitude is 0.5 or half.

08:17 Tangent function

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

08:27 Tan theta is the ratio of sin theta to cos theta and to y divided by x.
08:35 Drag the red point back to the x-axis that is to 1 comma 0.
08:44 In the Functions box, click tan.
08:48 When angle theta 0, tan theta is ratio of the y co-ordinate 0 to x co-ordinate 1 that is 0.
09:00 The graph shows angle theta on the x-axis and the amplitude of the tan theta function on the y-axis.
09:09 At angle theta 0, as tan theta is 0, the red point has amplitude of 0.
09:17 In the Unit Circle, drag the red point counter clockwise to the special angle 90 degrees on the y-axis.
09:27 In the Values box, x comma y has become 0 comma 1.
09:33 Note that tan theta is plus or minus infinity in the Values box.
09:40 Now look at the graph.
09:43 The red point has moved to 90 degrees where tan theta now falls on the vertical dotted line.
09:53 This dotted line is the vertical asymptote of the function.
09:58 It represents the value of x which the function approaches but never touches.
10:05 Here, the function increases without bound towards infinity in both directions.
10:13 Let us summarize.
10:16 In this tutorial, we have demonstrated how to use the Trig Tour Phet simulation.
10:23 Using this simulation, we have learnt to:

Construct right triangles for a point moving around a unit circle

10:33 Calculate trigonometric ratios, cos, sin and tan, of angle theta
10:39 Graph theta versus cos, sin and tan functions of theta along x and y axes.
10:46 As an assignment, observe: Cosine, sine and tangent values for all special angles
10:53 Cosine, sin and tangent graphs.
10:57 Relationship between ratios for supplementary angles

The sum of supplementary angles is 180 degrees.

11:08 The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it

11:17 The Spoken Tutorial Project team conducts workshops using spoken tutorials and gives certificates on passing online tests.

For more details, please write to us.

11:29 Please post your timed queries in this forum.
11:33 This project is partially funded by Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching.
11:42 Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

11:55 This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Nancyvarkey