Creating The Main Activity-3: Cofunction Angle 270°
1. Using your page's general settings, press the “Show Axis” and “Show Grid” buttons. For the grid, select the major grid lines section.
2. From the “Circle: Center & Radius” section, create a circle with the center point (0, 0) and a radius of 1 cm. Name the center point “O”.
3. Using the “Slider” tool, create a slider and select the angle type, then from the settings of it, name the slider as “angle” and from the section “Slider” set min and max points to 180° and 360° respectively.
4. Create a point on the circle and in the fourth quadrant of the coordinate plane, and name it as P. In the “Definition” section of this point, write “(cos(angle), sin(angle))”. This will ensure that your point P can only move on the circle within the third and fourth quadrants of the coordinate plane.
5. Similarly, place a point at (1, 0), and name it as A.
6. Use the “Segment” tool to create a new PO segment.
7. Measure the POA angle (α) with the “Angle” tool.
8. Using the “Perpendicular Line” tool, create a line passing through point P and the x-axis.
9. Place a point where the line you created intersects the x-axis, and named it as B. Again, using the segment tool, create a new PB segment.
10. Hide the line you created in the 8th step from the settings by deselecting the show object.
11. Using the segment tool again, create the BO segment. You now have a right triangle that can only move in the third and fourth quadrants of the coordinate plane.
12. Let's change the style of this right triangle a little. To make its edges more distinct, go to the “Color” tab in the settings section of each segment of the triangle and make them black. Change the name of the PO segment to “1” in the “Caption” section. Similarly, let's call the PB segment “a” and the OB segment “b”.
13. Now we will change the appearance of the POA angle. To do this, first adjust the angle between 180° and 360° in the settings section, then open the “Style” tab in the settings section and change the appearance of the angle to a counterclockwise rotating arrow shape in the “Decoration” section.
14. Create a new point on the circle with coordinates (0, -1), and name it as C.
15. Use the “Angle” tool to create the COP angle, and name it as β.
16. Use the “Text” tool to create a text, and name it as text1, and go to the geometry section in the “Advanced” tab, open a new (empty box) and enter “If(α ≟ 180°, "180°", α ≟
360°, "360°", α ≟ 270°, "270°", 180° < α < 270°, "270° - " + β, 270° < α < 360°, "270° + " β)”.
17. Go to the settings section of the POA angle, select text1 from the “Use Text as Caption” section.
18. Go to the settings of text1 and make it invisible by deselecting “Show Object”. With this step, your main triangle is complete.
19. Now change the color of the POA angle and your slider to blue using the “Color” segment in the settings section.
20. Now we will create another similar right triangle. This time, create another perpendicular line from point P to the y-axis using the “Perpendicular Line” tool.
21. Place a point where the line intersects the y-axis, and name it as D. Use the segment tool to create a new PD segment.
22. Create a new segment as OD and make the line from the 20th step invisible in the settings section.
23. Change the style of the PD and OD sides of this new right triangle you created to dashed lines using the “Line Style” section in the “Style” tab in the settings section. You can also make their colors red in the “Color” tab.
24. Let's label the OD edge and PD edge as “a” and “b” respectively in the “Caption” section of the settings.
25. Change the COP angle in the “Style” section of the settings by increasing its size and set its color to gray in the “Color” segment. This will make it easier to distinguish from the POA perspective.
26. Next, we need to add the dynamic texts we want to appear on the sides of the page when we change the slider. To do this, create a new text, name it as text2.
27. Write the following inside text2: “cos(text1)= sin(β)= b If(β ≟ 90°, "=1", β ≟ 0°,
"=0")”. The text highlighted in bold is dynamic text, meaning it will appear inside the dynamic boxes we opened in the “Advanced” section. The remaining parts are static, meaning they will be written as they are.
28. Create a new text as text3. Write “sin(text1)= -cos(β)= -a If(β ≟ 0°, "= -1", β ≟ 90°,
"=0")” inside it. In this text, the bold text is also dynamic text, while the rest is static text.
29. Create a new text as text4. Inside it, write “tan(text1)= -cot(β)= -\frac{a}{b} If(β ≟ 0°, "=\frac{1}{0}=undefined", β ≟ 90°, "=\frac{0}{1}=0")”.
30. Create another new text as text5. Inside it, write “cot(text1)= -tan(β)= -\frac{b}{a} If(β ≟ 90°, "= \frac{1}{0}=undefined", β ≟ 0°, "= \frac{0}{1}=0")”.
31. For all text2, text3, text4 and text5 select the “Serif” and “La TeX formula” boxes.
32. Again, for all text2, text3, text4 and text5 open their settings and write “360° ≥ α ≥ 270°” in the “Condition to Show Object” field in the “Advanced” tab.
33. These texts we created are for the fourth quadrant of our triangle on the coordinate plane. Therefore, we should place these texts one below the other on the side of the coordinate plane corresponding to region 4.
34. Now, in a similar way, we will write 4 more texts, and these will be for region 3, and we will also arrange them on the side of region 3. This way, the similarities and differences between the two regions will be easier to see.
35. We will create a new text, named “text6”, and its content will be similar to text2, as follows: “cos(text1)= sin(β)= -b If(β ≟ 90°, "= -1")”.
36. We will create a new text, named “text7”, and its content will be similar to text3, as follows: “sin(text1)= cos(β)= -a If(β ≟ 90°, "=0")”.
37. We will create a new text, named “text8”, and its content will be similar to text4, as follows: “tan(text1)=cot(β)= \frac{a}{b} If(β ≟ 90°, "=\frac{0}{-1}=0")”.
38. We will create a new text, named “text9”, and its content will be similar to text5
as follows: “cot(text1)=tan(β)= \frac{b}{a} If(β ≟ 90°, "=\frac{-1}{0}=undefined")”.
39. text6, text7, text8, and text9 will all be in “Serif” and “La TeX formula” format, and the “Condition to Show Object” field will contain “270° > α ≥180°”.
40. After placing the texts in a regular order on the screen, your activity is ready, change the “angle” and see what is happening!