Lesson 3: Finding Trig Ratios for Any Angle
D. Peterson
LESSON PLAN
Lesson Title:
Name:
Finding Trig Ratios for Any Angle
D. Peterson
Lesson #
Subject:
3
Date:
Pre-Calculus 11 (2008 Curriculum)
Fri, February 14, 2014
Grade(s):
11
Rationale: This lesson takes the existing notion of the primary trigonometric ratios for angles from 0 – 90 degrees and generalizes it for
angles from 0 – 360 degrees. While a perfectly general algebraic definition is given, the use of reference angles is used to make a strong
connection to the existing definition for 0 – 90 degrees. This is necessary to do general trigonometry, and sets the stage for further
generalization of the trigonometric ratios to the entirety of the real numbers (less certain points for tangent).
Prescribed Learning Outcome(s):
• B2: Students will solve problems using the three primary trigonometric ratios for angles from 0 – 360 degrees in standard
position.
Instructional Objective(s): Students will demonstrate mastery of PLO B2 by:
(2.1)
Determining using the Pythagorean theorem or the distance formula, the distance from the origin to a point
P(x,y) on the terminal arm of an angle.
(2.2)
Calculating the value of sin(theta), cos(theta), or tan(theta), given any point P(x,y) on the terminal arm of
angle theta.
(2.3)
Explaining, without the use of technology, the value of sin(theta), cos(theta), or tan(theta), given any point
P(x,y) on the terminal arm of angle theta where theta = 0, 90, 180, 270, or 360 degrees.
(2.4)
Determining the sign of a given trigonometric ratio for a given angle, without the use of technology, and
providing an explanation.
(2.6)
Calculating the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30, 45 or
60 degrees.
(2.7)
Describing patterns in and among the values of the sine, cosine and tangent ratios for angles from 0 to 360
degrees
(2.8)
Sketching a diagram to represent a problem.
(2.9)
Solving a contextual problem, using trigonometric ratios.
Prerequisite Concepts and Skills: Students should be familiar with angles in standard position, reference angles of angles in standard
position. Also students must be comfortable with finding and naming points in to coordinate plane. Rudimentary problem solving skills
will also be helpful.
Materials and Resources:
Teacher
Students
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Text section 2.2
SmartBoard Notes
GeoGebra Applets + Smartboard
Whiteboard & markers (3+ colours)
Ruler
Text section 2.2
Scientific Calculator
Paper
Pencils/pens (as long as it's readable and erasable)
Ruler/protractor
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Lesson 3: Finding Trig Ratios for Any Angle
D. Peterson
Lesson Activities: Discrepancies in time will be adjusted for by adding/removing additional probing questions. Some activities are
marked optional and can be used as a time buffer.
Teacher Activities
Student Activities
Time
Introduction: The TODO list
• Welcome the class, take attendance, collect
homework from last day. Remind class of
outstanding homework
• Ask them to remind me what we did last day.
• Since we know what angles of say 256 are we ought
to be able to take sine, tan, cosine. Any guesses?
• Display the graphic organizer TODO list. Explain
Activity 1: Connecting with existing definition.
Pair activity
• Pair the students up. Give each pair a point in Q1 and 2
min.
◦ Have the students draw the point in the plane,
bad drawings are okay.
◦ Have the students draw a line from the origin
to the point.
◦ Have the students indicate the angle in
standard position, and call it theta
◦ calculate sine, cosine, and tangent for your
angle.
5 min
• Students respond with stuff about standard position
• Students guess
• Work in pairs to do the requested task. May ask the
teacher for help.
10 min
• Pairs may try to ask teacher. Prompt them to
conjecture. There is no penalty for wrong answers!
• Ask two pairs to share their solutions on the wing
boards..
Activity 2: Expanding the definition
• Ask the class to generalize. Given a point (x,y) how to
we calculate sine, cos tan? Start out by drawing the
point in Q1.
• Responds to the question and takes notes
• Ask if this is a good definition? Plot a point in Q2, Q3,
Q4 and have the class try it out.
◦ Call on 12 voluntolds one for each ratio for
each point.
• Volunteers, or volunteers their friends. Try to get a
variety of students responding.
5 min
• Declare this to be a good definition.
Activity 3: Does the choice of the point matter.
Mostly direct instruction.
• For example, (1,-3) and (2,-6) lay on the same line.
Let's see if the formula gives the same tan, cosine, sine
values. 6 student volunteers for answers. I will scribe
• Take notes, ask question. Volunteers do
computations
10 min
• Ask the students why it doesn't matter.
◦ Remember similar triangles?
• Do the proof that colinear points give the same value.
points (a,b), (A,B)
• Students respond to, and interact with teacher
questions
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Lesson 3: Finding Trig Ratios for Any Angle
D. Peterson
Teacher Activities
Student Activities
Activity 4: ASTC
• Do chart of quadrants, (x,y) signs, and signs of each of
the trig ratios (see attached paper)
• Round robin questioning to get the signs. This must
come from the students.
• Label the quadrants with ASTC (all students take
calculus)
◦ A = all positive
◦ C = cosine positive
◦ S = sine positive
◦ T = tan positive
• Test this out with some points. Also have students
name 1 angle from each coordinate to test it out. Write
the results in tables in the wing boards
• Connecting with the reference angles
◦ Let theta_r be the reference angle for theta
◦ For each quadrant write
▪ sin(theta) = <sign> sin(theta_r)
▪ repeat for cos, tan
Time
10 min
• Students calculate ratios. Raise hands when finished
to report back. Given 1 minute
• Ask the students for each of the 12 signs
◦ So we can use special triangle to figure out the
trig ratios for any angle with reference angle 30,
45, or 60.
• GET SLIDE IMAGE FROM MRS. ACKERMAN'S
Activity 5: Strange edge cases
• Draw table and compute sin, cos, tan of 0, 90, 180,
270, 360.
◦ Calculators are forbidden
Activity 6: Applying Reference Angles
• Remix the pairs into groups of 2-3 for a problem
• Pendulum of length 2 cm swings from 210 to 330
degrees. What is the exact vertical displacement?
◦ Ask students to first draw the situation.
◦ Circulate and help each group.
Activity 7: Independent Practise
• Assignment from section 2.2 text p. 96- 98
◦ 1 – 6, 16, 18, 22 (16 is a little tricky)
• Teacher circulating.
• Call up a student scribe for the class
• Work together to solve the problem
◦ Draw diagrams. Ask for help as needed.
◦ Apply a reference angle from a known
special triangle.
◦ At end of time have a few groups explain
their process.
• Work independently (through some interaction with
person sitting beside them is OK)
• Ask questions as necessary. Answer questions posed
by circulating teacher.
5 min
10 min
17 min
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Lesson 3: Finding Trig Ratios for Any Angle
Teacher Activities
Closure:
• Remind students to complete the homework for next
day. If they got it done in class, suggest that students
try some of the other problems in from 7-14. We will
be doing those tomorrow.
D. Peterson
Student Activities
Time
• Listen
7 min
• Thank the class for their enthusiasm (hopefully!)
• Announce Quiz Monday. Remind class that I will be
available in the early morning to help them (I arrange
at 8:00) and at lunch if they need help. Feel free to
email too.
• Ask students to write on a slip of paper 1 piece of
advice to Mr. Peterson to try to make him a better
teacher. (2 min)
• Complete Advice slip
• Pack up to go when directed.
Universal Design for Learning (UDL) and Differentiated Instruction (DI): This lesson aims to make strong connections between the
written/spoken language and the visual representation presented through diagrams. Such an arrangement should appeal to both visual and
oral/aural learners. As much as possible I wish to have students verbalize the language as well as to have written it in order to strengthen
these associations. For kinesthetic learners, I will have students doing and drawing on the board.
There are no students on IEPs in this course. Having said that, some problems may be laddered into, for instance if a student is having
difficulty with a problem involving negative angle measure, have the student first solve a (essentially the same) problem using positive
angle measure. Additional concrete examples can easily be done as needed.
For students who are having an easy time, we can easily bring up the notion of co terminal angles. This leads into the extension of the
definition of sine/cosine made in math 12.
Organizational Strategies: Pair work will be done with students sitting next to each other. By using a copy of my seating plan any
making check marks, I will ensure that I ask every student at least one question during the class. I will also be directing questions to
individual students to ensure variety.
Behavioural Management Strategies: For the most part I intend to make use of non verbal behaviour management strategies such as
proximity and the “teacher look”. For the most part I want to keep a positive atmosphere, so smiling will be my default. Students
disrespecting another students response in a relatively minor way will not be tolerated and this will consist of a verbal reminder, to the
effect of “in this room we will listen with each other respectfully.” Afterwards, depending on the severity, I may speak to the student
individually while others are working. To overcome student talkativeness I will announce that I am waiting for students to be quiet and
then wait (however painful this may be for me).
Assessment and Evaluation: Pre-assessment will be an informal matter consisting of the questioning that occurs during the early
development of definitions. This will cause adaptation to future presentation as needed.
Formative assessment of the lesson objectives and through questioning in the middle of the lesson. It will definitely come up as we work
throught he pendulum problem, and as students work on their homework.
Summative assessment will be done through via a small quiz on Monday, February 17 and through the major unit test on Wednesday,
February 26. Questions will be varied and will resemble to sorts of problems tackled in class. Eg. for definitions give examples/nonexamples, state the definition, and draw a picture, apply knowledge of reference angles to find ratio, etc.
Extensions: The pendulum problem and similar connect directly with last day's review of the FMPC 10 trigonometry unit. This unit
builds the definitions for the trigonometric functions that form the basis for the rest of the unit. It will be practised constantly.
Reflections (if necessary, continue on separate sheet):
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