Download Exact Trigonometric Ratios: Triangles to the Unit Circle

Primary Type: Lesson Plan
Status: Published
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Resource ID#: 34876
Exact Trigonometric Ratios: Triangles to the Unit Circle
This is a lesson in which the student learns and practices exact trig values. They are explained using triangles, then the unit circle, and then guided
practice to be sure these concepts are mastered since they will be used in future problems in PreCalculus and Calculus.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Internet Connection,
Interactive Whiteboard, LCD Projector, Adobe Flash
Player, Microsoft Office
Instructional Time: 2 Hour(s)
Freely Available: Yes
Keywords: Sine, Cosine, Tangent, Cosecant, Secant, Cotangent, Special Right Triangles, Unit Circle, Quadrantals,
Reference Triangle
Instructional Design Framework(s): Direct Instruction, Demonstration, Structured Inquiry (Level 2), Guided
Inquiry (Level 3), Cooperative Learning
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Linamen 4 Homework doc.docx
Linamen 4 Homework Solutions doc.docx
Linamen 4Mat Activities Directions.doc
Linamen4 Practice sheet.doc
Linamen 4_Exact Trig Ratios.pptx
Linamen4 trig degree mat key2.pdf
Linamen4 trig degree mat.pdf
Linamen4 trig radian mat.pdf
Linamen4 trig solution squares.pdf
Linamen 4 Homework Solutions.jpg
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to use their knowledge of special right triangles to determine geometrically the values of the trigonometric ratios for sine, cosine, and tangent the
common angles of 30, 60 90 and their multiples in radian form in terms of pi. They will use the unit circle to find the trigonometric ratio for any special angles
represented using the unit circle.
When asked to determine a trigonometric ratio of an angle, the student will be able to determine the exact value by drawing an appropriate reference triangle, label the
sides of the triangle appropriately, and then determine the ratio.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be able to:
page 1 of 3 determine the missing side of a right triangle using the Pythagorean Theorem.
determine Sine, Cosine, and Tangent of an angle in a triangle when the three sides are given.
name the sides of the special right triangles given one side.
determine the reference angle and triangle when given an angle in degrees or radians.
Guiding Questions: What are the guiding questions for this lesson?
How would you find cosecant, secant, and cotangent if you have determined sine, cosine and tangent for an angle?
How would you explain how to determine the values of the other trigonometric functions if you are given the value of one trigonometric ratio? Could there be more
than one solution?
What special memory techniques would you use to remember the sine, cosine, and tangent of the special right triangles?
In which quadrants would you find each trigonometric ratio positive?
How are the sine, cosine, and tangent determined for the quadrantals? (A quadrantal is an angle in standard position whose terminal side lies on one of the
coordinate axes.)
If the sine of an angle is zero, what is the cosecant?
Teaching Phase: How will the teacher present the concept or skill to students?
This lesson will be presented by PowerPoint to an interactive whiteboard. A practice sheet is provided for students to use. When placed in a sheet protector to be used
with a dry erase pen, students can practice and erase.
Slide 1: Learning Goal and lesson agenda
Slides 2 - 4: Prior Knowledge check with answers
Slides 5 - 6: Trigonometric Ratios are defined
Slide 9: Special Right Triangle Review and Definition of the trig ratios in terms of x, y, and r.
Slide 11: Practice the signs of the trig ratios in each quadrant and then use "All Students Take Calculus" as a memory technique.
Slide 12: Determine the trig ratios for the quadrantals. Review division by zero and division where zero is the numerator and discuss their reciprocals.
Slide 13: Use the website to practice and then to quiz students on trig ratios.
http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php
Slides 14 - 15: Additional diagrams in case the website does not connect
Slides 16: Introduces a table to practice exact trig values
Slides 18 - 20: Hand tricks help students to remember exact trig ratios (one optional memory aid)
Slide 21: Directions for the Mat Activity (hands on drill to remember exact trig values)
Guided Practice: What activities or exercises will the students complete with teacher guidance?
In the PowerPoint
Slides 7 and 8: provide guided practice of trig ratios when given two sides of a right triangle with answers animated to follow.
Slide 10: allows the teacher to ask the values of the sides of the triangles and then the trig ratios of those angles.
Slide 13: takes the students to a website where they can practice and then quiz themselves on the trig ratios and their signs.
http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php
Slide 17: allows practice using a table after students have practiced and are able to state the trig ratios in less time with answers provided when needed.
Slide 20: provides a finger trick to help the student remember the trig ratios. The PowerPoint may be paused here to allow pair practice.
Slides 22 - 23 begin the homework, but they may be started in class where the teacher can circulate and determine any misconceptions that students may have.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Students will complete a homework assignment reinforcing each of the concepts that they have learned. They may use the website, independently, to check their
solutions.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students will complete a homework assignment reinforcing each of the concepts that they have learned. They may use the website, independently, to check their
solutions.
The teacher will review the strategies used to determine the trig ratios. They should use the interactive white board and the website to call on students to answer all
types of skills learned in this lesson. The website allows the teacher to turn off or on the angles, triangles, and ratios. It also has a quiz function that can be used by the
teacher to practice either as a whole class activity or with groups, pairs, or individuals. The students will begin their homework and the teacher will circulate to monitor
individual progress.
Summative Assessment
At the end of the lesson, students will complete a homework assignment which can be started at the end of class. The teacher can circulate and assess their
understanding of the new concepts. Students will complete the homework which will be evaluated on the next school day. Teachers may select to use this lesson over
page 2 of 3 two class periods and therefore the homework may be assigned as odd problems one day and even problems the next. Slide 23 had a summative assessment that
students can complete and then check using the website to check their solutions.
http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php
Formative Assessment
At the beginning of the lesson, the students will complete questions to assess prior knowledge. These questions can be found on slide 2. The teacher can circulate and
collect data while students complete this learning check. The teacher can embed review within the lesson to remind students of prior learning topics. Also, throughout
the lesson, on slides 5, 13, 15, and 17 students are provided with questions to check their knowledge.
Feedback to Students
Students will have guided practice with an answer check during the lesson. Students will have a website where they can practice concepts and receive immediate
feedback using the quiz feature on the website. Using the interactive whiteboard, students will have an opportunity to take the quiz, on the website, and reset it for
multiple attempts. The teacher will be able to call students to the whiteboard to try the quiz. A new version of the quiz is generated when the quiz is reset; therefore,
the teacher can use the quiz feature until satisfied students understand the concepts. Students may also use this site to practice outside of class.
Practice Exact Trig Values via the website: http://www.dudefree.com/Student_Tools/materials/precalc/unit-circle.php
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Preferential seating is used to make sure that every student is placed in a seat that meets his or her needs.
Seating is structured using Kagan seating so that when students seated in rows use their face partner or shoulder partner the high-learner, medium-high learner, a
medium low-learner and a low-level learner are carefully paired.
The visual example of the interactive unit circle used in a large group setting, accommodate the visual learner.
Student can make and use the Mat Activity.
Using the website at home or in the media center allows the student individual interactivity.
Suggested Technology: Document Camera, Computer for Presenter, Internet Connection, Interactive Whiteboard, LCD Projector, Adobe Flash Player, Microsoft Office
Special Materials Needed:
Individual unit circle dry erase boards and dry erase pens may be used for guided practice (Or the Practice Sheet provided in a sheet protector)
The Mat Activity which can be reproduced on cardstock paper
Further Recommendations: The teacher should practice with the website to see how to use the boxes. The teacher should take several quizzes on the website
to see the challenges the students will face. A model slide is provided in case the website is not available.
Additional Information/Instructions
By Author/Submitter
Important questions are given; however, students may ask and need answers as they inquire for themselves. Teachers may choose homework from their textbook instead of
the problems on the PowerPoint allowing the PowerPoint Homework question to be available for additional guided practice.
SOURCE AND ACCESS INFORMATION
Contributed by: Sharon Linamen
Name of Author/Source: Sharon Linamen
District/Organization of Contributor(s): Seminole
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.F-TF.1.3:
Description
Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the
unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x,
where x is any real number.
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