Lesson Plan

Auteur :Esma Nur Karaca - Hafsa Metin, Esma Nur Karaca

Lesson Title

Trigonometric Reduction Formulas in the Unit Circle

Grade Level

11th Grade

Duration

40-50 minutes

Instructional Strategies

Constructivism and Cooperative Learning

Materials

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

Mathematical Big Ideas

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

Learning Objectives

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

Prior Knowledge

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.

Articulation of Understanding

The student can explain why the function name does not change in reductions of

, and why the function name change in reductions of

and

. The student can explain why the sign is adjusted according to the region, using the projections (geometric reflections) on the unit circle.

Continuity

Previous Lesson: Unit circle definition, trigonometric ratios, fundamental identities. This Lesson: The exploration of all reduction functions (

and

) Next Lesson: Graphs of periodic functions and phase shifts.

Flow of The Lesson

Before Phase (5-10 min): 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 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? Note: 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. During Phase (30 min): 1^st Part: The teacher first opens Main Activity-1: Cofunction Angle 180° in this phase. The purpose of this 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. • Set the slider to an angle between 90° and 270° (e.g., 120°). • 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°). • You will see the same x angle drawn in the 1st quadrant as a simple right triangle. Ask the students the questions in steps 4 and 5 and ask the students to share their ideas with the whole class. • 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)? • Determine how the signs of sine and cosine change when the angle is in Quadrant II and Quadrant III. 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 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 answer has been obtained, the teacher has the students note the correct generalization and moves on to the second main activity. 2^nd Part: The teacher, then, opens Main Activity-2: Cofunction Angle 90° in this phase. The purpose of this 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. • 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). Ask the students the questions in steps 2 and 3 and ask the students to share their ideas with the whole class. • 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 do these compare to the opposite and adjacent sides of the original black triangle? • Use your observations to decide which function value (sin or cos) the small triangle’s sides now represent. 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. 3^rd Part: 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. • Set the slider to an angle between 180° and 360° (e.g. 240°). • Think of the negative Y-axis (270°) as your new reference line. The gray angle shows how far your angle is from 270°. So, the angle is rewritten as (270+-x) (e.g. 270 -30⁰) Ask the students the questions in steps 3, 4, 5 and ask the students to share their ideas with the whole class. • 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 horizontal side, cos(x)? How do these compare to the sides of the original black triangle? • Use your observations to guess which function value (sin or cos) the small triangle’s sides now represent. • Determine how the signs of sine and cosine change when the angle is in 3rd or 4th Quadrant. 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 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. After (5 min): 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: • 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. sin(180° ± x)= ± sin(x), cos(180° ± x) = ± cos(x), tan(180° ± x) = ± tan(x) and cot(180° ± x) = ± cot(x). • 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). sin(90° ± x)= ± cos(x), cos(90° ± x) = ± sin(x), tan(90° ± x) = ± cot(x) and cot(90° ± x) = ± tan(x).

Possible Students’ Responses Examples (Expected or Observed):

In the 1^st part of the during phase: Misconceived: The student thinks the name should change:

Teacher Support: Compare the region I and region II triangles in the applet. Does the length of the vertical side representing

become a horizontal side when we take symmetry with respect to

, or does it remain vertical? Is the vertical length (sin) compared to the vertical length (sin) or the horizontal length (cos)? Partially: The student remembers that the names remain constant but the signs in Region II are incorrect:

. Teacher Support: "Is the

-coordinate (i.e., the

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

. So, what should the

value be?" In the 2^nd part of the during phase: Misconceived: The student does not apply the name substitution rule:

. Teacher Support: Focus on the small gray angle in representative triangle created by

shown in the applet. Doesn't the opposite side (sin) in this triangle now represent the adjacent side (cos) of the original angle

? What does it mean for the triangle to be 'lying'? Partially: The student makes the name change but misremembers the sign in Quadrant I:

. Teacher Support: "Which region are we in now (between

and

)? What should the values of

and

be in region I? Why?" In the 3^rd part of the during phase: Misconceived: The student makes the name change correctly, but for region III, the sign for

is positive:

. Teacher Support: "In which region does the angle

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

in the region, not on the sign of

found after the reduction." Satisfactory: The student finds the name change and the sign correct: