**Narration**
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Welcome to this tutorial on **Trig Tour**, an **interactive PhET simulation**.
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In this tutorial, we will demonstrate **Trig Tour**, an **interactive PhET simulation**.
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Here I am using, **Ubuntu Linux OS** version 16.04
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**Java** version 1.8.0
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**Firefox Web Browser** version 60.0.2
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Learners should be familiar with trigonometry.
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Using this **simulation** we will learn how to,
Construct right triangles for a point moving around a unit circle

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Calculate trigonometric ratios, **cos**, **sin** and **tan**, of angle **theta**
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Graph **theta** versus **cos**, **sin** and **tan functions** of **theta** along **x** and **y axes**
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Let us begin.
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Use the given link to download the **simulation**.
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I have already downloaded the **Trig Tour simulation** to my **Downloads folder**.
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To open the **simulation**, right click on the **trig-tour html** file.
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Select the **Open With Firefox Web Browser** option.
The file opens in the **browser**.

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This is the **interface** for the **Trig Tour** simulation.
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The **interface** has four boxes:
**Values**

**Unit circle**

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**Functions**, **Special angles**, **labels** and **grid**
**Graph**

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The **reset button** takes you back to the starting point.
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In the **Functions** box, check **Special angles, Labels, Grid** and click **cos**.
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**Cosine function**
**Cosine** of an angle is the ratio of the lengths of the adjacent side to the hypotenuse.

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**Cosine** value is the **x co-ordinate** of a point moving around a unit circle.
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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.

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A unit circle is drawn in a **Cartesian coordinate system** with **x** and **y axes** in the **Unit Circle** box.
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A red point is seen at the circumference of the circle on the **x-axis**.
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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.

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

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You can see empty circles on the unit circle.
In the **Functions** box, uncheck **Special Angles**.

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

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Look at the graph.
The red point has moved to 30 degrees along the **cos function**.

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In the **Values** box, click **radians** radio button.
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This converts 30 degrees into **pi** divided by 6 radians for **theta** in the **Values** box.
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**Sine function**
**Sine** of an angle is the ratio of the lengths of the opposite side to the hypotenuse.

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**Sine** value is the **y-co-ordinate** of the point moving around the same unit circle.
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**Sine theta** is **y** divided by radius and hence, is **y** for the unit circle.
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Drag the red point back to the x axis.
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In the **Functions** box, click **sin**.
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Click the **degrees** radio button.
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As seen earlier, **x comma y** are **1 comma 0**.
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Note the definitions of **sine theta** given earlier.
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When **angle theta** is 0 **degrees**, the **y co-ordinate** of the red point is 0.
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The graph shows **angle theta** on the **x-axis** and the amplitude of the **sine theta function** on the **y-axis**.
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At **angle theta** of 0 **degrees**, as **sine theta** is 0, the red point has amplitude 0.
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In the **Unit Circle,** drag the red point counter clockwise to the next **special angle** 30 **degrees**.
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In the **Values** box, note that **x comma y** is squareroot of 3 divided by 2 comma half.
Remember how you can calculate these.

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In the graph, the red point has moved to 30 **degrees** along the **sine function**.
Its amplitude is 0.5 or half.

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

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**Tan theta** is the ratio of **sin theta** to **cos theta** and to **y** divided by **x**.
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Drag the red point back to the **x-axis** that is to **1 comma 0**.
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In the **Functions** box, click **tan**.
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When angle **theta** 0, **tan theta** is ratio of the **y co-ordinate** 0 to **x co-ordinate** 1 that is 0.
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The graph shows angle **theta** on the **x-axis** and the amplitude of the **tan theta function** on the **y-axis**.
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At **angle theta** 0, as **tan theta** is 0, the red point has amplitude of 0.
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In the **Unit Circle**, drag the red point counter clockwise to the **special angle** 90 **degrees** on the **y-axis**.
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In the **Values** box, **x comma y** has become **0 comma 1**.
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Note that **tan theta** is **plus or minus infinity** in the **Values** box.
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Now look at the graph.
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The red point has moved to 90 degrees where **tan theta** now falls on the vertical dotted line.
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This dotted line is the **vertical asymptote** of the **function**.
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It represents the value of **x** which the **function** approaches but never touches.
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Here, the **function** increases without bound towards **infinity** in both directions.
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Let us summarize.
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In this tutorial, we have demonstrated how to use the **Trig Tour Phet simulation**.
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Using this **simulation**, we have learnt to:
Construct right triangles for a point moving around a unit circle

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Calculate trigonometric ratios, **cos**, **sin** and **tan**, of angle **theta**
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Graph **theta** versus **cos**, **sin** and **tan functions** of **theta** along **x** and **y axes**.
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As an **assignment**, observe: **Cosine, sine **and** tangent** values for all **special angles**
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**Cosine, sin** and **tangent** graphs.
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Relationship between ratios for supplementary angles
The sum of supplementary angles is 180 degrees.

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The video at the following link summarizes the **Spoken Tutorial project**.
Please download and watch it

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The **Spoken Tutorial Project **team conducts workshops using spoken tutorials and gives certificates on passing online tests.
For more details, please write to us.

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Please post your timed queries in this forum.
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This project is partially funded by **Pandit Madan Mohan Malaviya National Mission on Teachers and Teaching**.
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**Spoken Tutorial Project** is funded by **NMEICT**, **MHRD**, Government of India.
More information on this mission is available at this link.

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This is **Vidhya Iyer** from **IIT Bombay**, signing off.
Thank you for joining.