Click on the generate button see different angles in different quadrants.

Use the slider [icon]/images/ggb/toolbar/mode_slider.png[/icon] to choose any [b]angle θ.[/b] As you move the point around the unit circle, observe how the right triangle changes.[br][br]The radius of the circle is [b]1[/b], so the hypotenuse of every triangle is always [b]1[/b].[br][br]This is why the unit circle is so useful in trigonometry![br][list][*]When you drop a [b]vertical line[/b] from the point to the x-axis, its length shows [b]sin(θ)[/b].[br]This segment measures how far the point is [i]up or down[/i] from the x-axis.[br][br][/*][*]When you draw a [b]horizontal line[/b] from the point to the y-axis, its length shows [b]cos(θ)[/b].[br]This segment measures how far the point is [i]left or right[/i] from the y-axis.[br][br][/*][*]The ratio of the sides gives [b]tan(θ) and [b]cot(θ)[/b][/b].[br]Notice how the green line extends this ratio beyond the triangle.[br][br][/*][*]In the same way, the extended purple and red segments represent [b]cosec(θ)[/b], and [b]sec(θ)[/b].[br][br][/*][/list][b]As you explore, focus on how each function corresponds to a specific length on the diagram.[br][br]The goal is to recall where the basic trig functions come from and how they are visualized on the unit circle.[/b]

As you move the slider from 0° all the way to 360°, watch what happens in the other quadrants as the angle keeps growing and not an acute angle anymore.[br][list][*]What do you notice about the values of sinθ and cosθ when θ is in the 1st quadrant?[/*][/list][list][*]As θ moves into the 2nd, 3rd and 4th quadrants, do you see some of the same [b]numerical values[/b] of sinθ and cosθ appearing again (possibly with different signs)?[br][br][/*][/list][b]And here is something to think about:[/b][br][br]From 0° to 90°, sinθ increases, but in other quadrants it cannot keep increasing because on the unit circle sine must stay between –1 and 1.[br][br]So if the angle keeps getting larger but the sine value cannot, [b]what should we expect when we see something like sin(150°)?[/b][br][br][b][size=150][i][color=#0000ff]In our main activities, we will explain how knowing the sine or cosine of one acute angle in the 1st quadrant can help you predict the sine or cosine of other angles around the circle![/color][/i][/size][/b][br]

Trigonometric Reduction Formulas in the Unit Circle

11th Grade

40-50 minutes

Constructivism and Cooperative Learning

Smart board/Projector, Student devices (phone/tablet/laptop), GeoGebra activity link, Whiteboard (in case of technical issues)

Generation of algebraic identities through the periodicity of trigonometric functions and symmetries on the unit circle (reflections about axes).

MAT.11.1.1.ç.[br]Trigonometrik referans fonksiyonlar ile bu fonksiyonlardan türetilen fonksiyonların grafik temsilleri (birim çember izdüşümleri) ile cebirsel temsilleri (indirgeme özdeşlikleri) arasındaki ilişkiyi ifade eder.

It is assumed that students can find the sine, cosine, tangent, and cotangent ratios of an acute angle in a right triangle; and knows that the function values represent the unit circle coordinates.

The student can explain why the function name does not change in reductions of [math]\text{$180^\circ\pm\alpha$ }[/math], and why the function name change in reductions of [math]90^{\circ}\pm\alpha[/math] and [math]270^{\circ}\pm\alpha[/math] .[br]The student can explain why the sign is adjusted according to the region, using the projections (geometric reflections) on the unit circle.

[i]Previous Lesson:[/i] Unit circle definition, trigonometric ratios, fundamental identities.[br][i][b]This Lesson:[/b][/i] The exploration of all reduction functions ([math]\text{$180^\circ\pm\alpha$ }[/math], [math]\text{$90^\circ\pm\alpha$ }[/math] and [math]\text{$270^\circ\pm\alpha$ }[/math])[br][i]Next Lesson:[/i] Graphs of periodic functions and phase shifts.

[b]Before Phase (5-10 min):[/b] [br]Pre Activity: Trigonometric Functions in Regions is introduced. The purpose of this activity is to recall where the basic trigonometric functions come from and how they are visualized on the unit[br]circle. That is, students observe how the functions and given lengths change on the unit circle by changing the slider. Then, students answer the activity questions (2 questions). The teacher asks the following question to introduce the main activity: Can different angles have the same sine or cosine value?[br][br][b][i]Note: [/i][/b]Pre-Activity instruction and exploration sections within the application have not been included in this lesson for detailed discussion. Detailed explanations are provided for students who use this book by themselves so they can learn from the application with only guidance by the instructions. [br][br][b]During Phase (30 min):[/b] [br][b][i]1[sup]st[/sup] Part:[/i][/b] The teacher first opens Main Activity-1: Cofunction Angle 180° in this phase. The purpose of this[br]activity is to demonstrate why trigonometric function names remain the same when using the horizontal X-axis (180°) as a reference. The teacher demonstrates the first three steps of the activity's instructions on the smart board with the students. [br]•       Set the slider to an angle between 90° and 270° (e.g., 120°).[br]•       Notice that the applet shows the x-axis as your reference line. It is calculating how far your angle is from 180°. The angle is rewritten as (180°±x) (e.g.: 180 - 60°). [br]•       You will see the same x angle drawn in the 1st quadrant as a simple right triangle.[br][br]Ask the students the questions in steps 4 and 5 and ask the students to share their ideas with the whole class.[br]•       Compare the two triangles. Look at the triangle in the 1st quadrant and look at the triangle of the (180° ± x) angle. Which sides correspond to sin(x)? Which sides correspond to cos(x)?[br]•       Determine how the signs of sine and cosine change when the angle is in Quadrant II and Quadrant III. [br][br]Students examine the equations in the applet and solve the questions in the activity (5 questions). Students provide the answers to the questions, and the teacher checks the answers with the whole class. Once the teacher is sure that the questions have been understood and completed, they ask how the[br]variable angle equality used in the activity can be expressed in general terms and asks students to write down their answers. Possible student answer: sin(180° ± x) = ± sin(x). The answers from the students are discussed with the class. After observing that the topic has been understood and the correct[br]answer has been obtained, the teacher has the students note the correct generalization and moves on to the second main activity.[br][br][b][i]2[sup]nd[/sup] Part: [/i][/b]The teacher, then, opens Main Activity-2: Cofunction Angle 90° in this phase. The purpose of this[br]activity is to demonstrate why trigonometric function names change when using the vertical Y-axis (90°) as a reference. The teacher demonstrates the first step of the activity's instructions on the smart board with the students. [br]•       Set the slider to an angle between 0° and 180° (e.g., 50°). The gray angle shows how far your angle is from the 90° line. Notice how the applet rewrites the angle as (90° - + x).[br][br]Ask the students the questions in steps 2 and 3 and ask the students to share their ideas with the whole class.[br]•       Look at the right triangle formed by this gray angle near the 90° reference axis. Examine the sides of this small triangle. Which side is opposite to the gray angle? Which side is adjacent to the gray angle? How[br]do these compare to the opposite and adjacent sides of the original black triangle?[br]•       Use your observations to decide which function value (sin or cos) the small triangle’s sides now represent.[br][br]Students examine the equations in the applet and solve the questions in the activity (7 questions). Again, students provide the answers to the questions, and the teacher checks the answers with the whole class. Once the teacher is sure that the questions have been understood and completed, they ask how the variable angle equality used in the activity can be expressed in general terms and asks students to write down their answers. Possible student answer: sin(90° ± x) = ± cos(x). The answers from the students are discussed with the class. After observing that the topic has been understood and the correct answer has been obtained, the teacher has the students note the correct generalization and moves on to the last main activity. [br][br][b][i]3[sup]rd[/sup] Part: [/i][/b]Similarly, the teacher opens Main Activity-3: Cofunction Angle 270° in this phase. The purpose of this activity is to demonstrate why trigonometric function names change and signs reverse when using the vertical Y-axis (270°) as a reference. The teacher demonstrates the first two steps of the activity's instructions on the smart board with the students. [br]•  Set the slider to an angle between 180° and 360° (e.g. 240°).[br]•       Think of the negative Y-axis (270°) as your new reference line. The gray angle shows how[br]far your angle is from 270°. So, the angle is rewritten as (270+-x) (e.g. 270 -30⁰)[br][br]Ask the students the questions in steps 3, 4, 5 and ask the students to share their ideas with the whole class.[br]•        Focus on the right triangle formed by this gray angle near the 270° reference axis. Examine the sides of this small triangle? What is the length of the vertical side, sin(x)? What is the length of the[br]horizontal side, cos(x)? How do these compare to the sides of the original black triangle?[br]•       Use your observations to guess which function value (sin or cos) the small triangle’s sides now represent. [br]•        Determine how the signs of sine and cosine change when the angle is in 3rd or 4th Quadrant.[br][br]Students examine the equations in the applet and solve the questions in the activity (2 questions). The students provide the answers to the questions, and the teacher checks the answers with the whole class. Again, once the teacher is sure that the questions have been understood and completed, they ask how the variable angle equality used in the activity can be expressed in general terms and asks students to write down their answers. Possible student answer: sin(270° ± x) = ± cos(x). The answers from the students are[br]discussed with the class. After observing that the topic has been understood and the correct answer has been obtained, the teacher has the students note the correct generalization and moves on the after phase.[br][br][b]After (5 min): [/b][br]The teacher asks students to verbally explain the generalizations they made during phase and share them with the entire class. The expected answers from students are as follows:[br]•        When we measure an angle from the X-axis as 180°, the triangle acts like a mirror across the Y-axis, so the "opposite" side (vertical) stays vertical, and the "adjacent" side (horizontal) stays horizontal. The roles of the sides do not swap, so the function name stays the same[i]. sin(180° ± x)= ± sin(x), cos(180° ± x) = ± cos(x), tan(180° ± x) = ± tan(x) and cot(180° ± x) = ± cot(x).[/i] [br]•        When we measure an angle from the vertical Y-axis as 90° or 270°, the triangle "tips over." The side that was "Opposite" to the angle becomes "Adjacent" to the complementary angle. This geometric swap is why Sine (Opposite) becomes Cosine (Adjacent). [i]sin(90° ± x)= ± cos(x), cos(90° ± x) = ± sin(x), tan(90° ± x) = ± cot(x) and cot(90° ± x) = ± tan(x). [/i][br][br][br]

[i]In the 1[sup]st[/sup] part of the during phase:[br][/i][br][b]Misconceived:[/b] The student thinks the name should change: [math]\text{$\sin(180^\circ- x)}[/math] = [math]\text{\cos(x)$}[/math] or [math]\text{$\cos(180^\circ- x) = \sin(x)$}[/math][br][b]Teacher Support: [/b]Compare the region I and region II triangles in the applet. Does the length of the vertical side representing [math]\sin\alpha[/math] become a horizontal side when we take symmetry with respect to [math]180^{\circ}[/math], or does it remain vertical? Is the vertical length (sin) compared to the vertical length (sin) or the horizontal length (cos)?[br][br][b]Partially:[/b] The student remembers that the names remain constant but the signs in Region II are incorrect: [math]\cos(180^{\circ}-x)=+\cos(x)[/math].[br][b][b]Teacher Support:[/b][/b] "Is the [math]x[/math]-coordinate (i.e., the [math]\cos[/math] value) of a point in region II positive or negative? By looking at the unit circle, you see that we are to the left of the point [math](-1,1)[/math]. So, what should the [math]\cos[/math] value be?"[br][i]In the 2[sup]nd[/sup] part of the during phase: [/i][b][br]Misconceived: [/b]The student does not apply the name substitution rule: [math]\cos(90^{\circ}-x)=\cos(x)[/math]. [br][b]Teacher Support: [/b]Focus on the small gray angle in representative triangle created by [math]90^{\circ}[/math] shown in the applet. Doesn't the opposite side (sin) in this triangle now represent the adjacent side (cos) of the original angle [math]x[/math]? What does it mean for the triangle to be 'lying'?[br][br][b]Partially:[/b] The student makes the name change but misremembers the sign in Quadrant I: [math]\sin(90^{\circ}-x)=-\cos(x)[/math].[br][b]Teacher Support:[/b] "Which region are we in now (between [math]0^{\circ}[/math] and [math]90^{\circ}[/math])? What should the values ​​of [math]\sin[/math] and [math]\cos[/math] be in region I? Why?"[br][br][i]In the 3[sup]rd[/sup] part of the during phase:[br][br][/i][b]Misconceived: [/b]The student makes the name change correctly, but for region III, the sign for [math]\sin[/math] is positive: [math]\text{$\sin(270^\circ- x) = +\cos(x)$[br]}[/math].[br][b]Teacher Support:[/b] "In which region does the angle [math]270^{\circ}-x[/math] fall? Are the y-coordinates (i.e., the value of sin) positive or negative in that region? Note that the sign depends on the sign of the initial [math]\sin(270^{\circ}-x)[/math] in the region, not on the sign of [math]\text{$\cos(x)$ }[/math] found after the reduction."[br][br][b]Satisfactory:[/b] The student finds the name change and the sign correct: [math]\text{$\cos(270^\circ+ x) =}\text{+\sin(x)$}[/math] is positive in Quadrant IV).[br][b]Extension Question:[/b] "So, what would [math]sec(270^{\circ}+x)[/math] be equal to?

1. Using your page's general settings, press the “Show Axis” and “Show Grid” buttons. For the grid, select the major grid lines section.[br]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”.[br]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 90° and 270° respectively. [br]4.  Create a point on the circle and in the second 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 second and third quadrants of the coordinate plane.[br]5.  Similarly, place a point at (1, 0), and name it as A.[br]6. Use the “Segment” tool to create a new PO segment. [br]7. Measure the AOP angle (α) with the “Angle” tool and set the “Angle Between” setting to 0° and 360° in the settings section.[br]8. Using the “Perpendicular Line” tool, create a line passing through point P and the x-axis.[br]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.[br]10.  Hide the line you created in the 8[sup]th[/sup] step from the settings by deselecting the show object. [br]11. Using the segment tool again, create the BO segment. You now have a right triangle that can only move in the second and third quadrants of the coordinate plane.[br]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 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”.[br]13.  Now we will change the appearance of the AOP angle. To do this, first 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.[br]14.  Create a new point on the circle with coordinates (-1, 0), and name it as C.[br]15.  Use the “Angle” tool to create the COP angle, and name it as β.[br]16.  Now we will create a new angle that moves dynamically with angle β and is located in the first quadrant. To do this, first use the “Rotate Around Point” tool to create a point by clicking on point A and then point O, and name it as P'. [br]17.  Go to the Settings section and enter “Rotate(A, β)” in the “Definition” section. Ensure that when you change the slider, the angle rotates by the same degree from point A with angle β.[br]18.  Then hide the β angle, create a new segment between the OP’ points and measure the AOP’ angle using the “Angle” tool.[br]19.  Now, we are trying to create a representative triangle in the first quadrant. To do this, create another perpendicular line from point P’ to the x-axis using the “Perpendicular Line” tool.[br]20.  Place a point where the line intersects the x-axis, and name it as D. Use the segment tool to create a new P’D segment.[br]21.  Create a new segment as OD and make the line invisible in the settings section.[br]22.  Change the style of the sides of the triangle P’OD 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. [br]23.  Let's label the P’D edge and OD edge as “a” and “b” respectively and P’O as “1” in the “Caption” section of the settings. [br]24.  Change the AOP’ 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 AOP perspective.[br]25.  Now let's dynamically write the AOP angle we created. To do this, let's create a new text using the “Text” tool as “text1”and go to the geometry section in the “Advanced” tab, open a new (empty box) and enter “[b]If(α < 180°, "180°- " β, α [/b][b]≟ 180[/b][b]°, "180[/b][b]°-0[/b][b]°", 180[/b][b]° < [/b][b]α, "180[/b][b]°+" [/b][b]β)[/b]”. [br]26.  Go to the settings section of the AOP angle, select text1 from the “Use Text as Caption” section. [br]27.  Go to the settings of text1 and make it invisible by deselecting “Show Object”. With this step, your main triangle is complete. [br]28.  Now change the color of the AOP angle and slider to blue by using the “Color” tab in the settings section. This will make the relationship between the slider and the angle visually apparent.[br]29.  Next, we need to add the dynamic texts we want to appear on the above page when we change the slider. To do this, create a new text, name it as text2. [br]30.  Write the following inside text2: “cos([b]text1[/b])= -cos([b]β[/b])= -b [b]If([/b][b]β [/b][b]≟ 90[/b][b]°, "=0", [/b][b]β [/b][b]≟ 0[/b][b]°, "= -1")[/b]”. The text highlighted in [b]bold[/b] is dynamic text, meaning it will appear inside the[br]dynamic boxes we opened in the “Advanced” section. The remaining parts are static, meaning they will be written as they are.[br]31.  Create a new text as text3. Write “sin([b]text1[/b])= sin([b]β[/b])= a [b]If([/b][b]β [/b][b]≟ 90[/b][b]°, "=1", [/b][b]β [/b][b]≟ 0[/b][b]°, "= 0")[/b]” inside it. In this text, the [b]bold[/b] text is also dynamic text, while the rest is static text. [br]32.  Create a new text as text4. Inside it, write “tan([b]text1[/b])= -tan(β)= -\frac{a}{b} [b]If(β [/b][b]≟ 90[/b][b]°, "=\frac{1}{0}=undefined", [/b][b]β [/b][b]≟ 0[/b][b]°, "=\frac{0}{-1}=0")[/b]”.[br]33. Create another new text as text5. Inside it, write “cot([b]text1[/b])= -cot(β)= -\frac{b}{a} [b]If(β [/b][b]≟ 0[/b][b]°, "=\frac{-1 {0}=undefined", [/b][b]β [/b][b]≟ 90[/b][b]°, "=\frac{0}{1}=0")[/b]”.[br]34.  For all text2, text3, text4 and text5 select the “Serif” and “La TeX formula” boxes.[br]35.  Again, for all text2, text3, text4 and text5 open their settings and write “α ≤ 180°” in the “Condition to Show Object” field in the “Advanced” tab.[br]36.  These texts we created are for the second 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 2.[br]37.  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.[br]38.  We will create a new text, named “text6”, and its content will be similar to text2, as follows: “cos([b]text1[/b])= -cos([b]β[/b])= -b [b]If([/b][b]β [/b][b]≟ 90[/b][b]°, "=0")[/b]”.[br]39.  We will create a new text, named “text7”, and its content will be similar to text3, as follows: “sin([b]text1[/b])= -sin(β)= -a [b]If(β[/b] [b]≟ 90[/b][b]°, "=-1")[/b]”.[br]40.  We will create a new text, named “text8”, and its content will be similar to text4, as follows: “tan([b]text1[/b])= tan(β)= \frac{a}{b} [b]If(β [/b][b]≟ 90[/b][b]°, "=\frac{-1}{0}=undefined")[/b]”.[br]41.  We will create a new text, named “text9”, and its content will be similar to text5 as follows: “cot([b]text1[/b])=[br]cot(β)= \frac{b}{a} [b]If(β [/b][b]≟ 90[/b][b]°, "=\frac{0}{-1}=0")[/b]”.[br]42.  text6, text7, text8, and text9 will all be in “Serif” and “La TeX formula” format, and the “Condition to Show Object” field will contain “α >180°”.  [br]43.  After placing the texts in a regular order on the screen, your activity is ready, change the “angle” and see what is happening![br][br][br]

The core author of the book's first activity, [b]Pre Activity: Trigonometric Functions in Region[/b], is [b]Jesus Perez[/b]. Additionally, we completed the applet by adding [b]dynamic check boxes [/b]and writing [b]instructions[/b] for the GeoGebra activity.

File:Sin x = sin (180- x) and cos (180- x) = -cos x.png[br][url=https://commons.wikimedia.org/wiki/File:Sin_x_%3D_sin_%28180-_x%29_and_cos_%28180-_x%29_%3D_-cos_x.png]https://commons.wikimedia.org/wiki/File:Sin_x_%3D_sin_%28180-_x%29_and_cos_%28180-_x%29_%3D_-cos_x.png[/url][br]"This file is licensed under the [url=https://en.wikipedia.org/wiki/en:Creative_Commons]Creative Commons[/url] [url=https://creativecommons.org/licenses/by-sa/4.0/deed.en]Attribution-Share Alike 4.0 International[/url] license."

The remaining parts of the book were made by authors Hafsa Metin and Esma Nur Karaca.