Lesson 4: Trig Ratios: Solving for Angles
D. Peterson
LESSON PLAN
Lesson Title:
Name:
Trig Ratios: Solving for Angles
D. Peterson
Lesson #
Subject:
4
Date:
Pre-Calculus 11 (2008 Curriculum)
Mon, February 17, 2014
Grade(s):
11
Rationale: This lesson builds on the previous lesson, and now works to generalize the notion of the inverse trigonometric functions in
order to solve for angles between 0 – 360 degrees given a legitimate trigonometric ratio. New to students will be the notion that there are
two solutions to equations of the form sin(theta) = a (likewise for cosine and tangent). Such an understanding is necessary in order to
understand the reason why there is an ambiguous case of the sine law. In addition, this lesson sets the stage for Pre-Calculus 12 in which
they look for all real solutions to such equations.
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.
(2.5)
Solve for all values of theta, an equation of the form sin(theta) = a or cos(theta) = a there -1<=a<=1 and an
equation of the form tan(theta)=a where a is a real number.
(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
•
•
•
•
•
•
•
•
•
•
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
Page 1 of 5
Lesson 4: Trig Ratios: Solving for Angles
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.
• Ask them to remind me what we did last week.
• Display the graphic organizer TODO list. Explain
10 min
• Students respond with stuff about standard position,
special triangles, extended defn of sine,cos,... ,
reference angles.
• Students guess
Activity 1: Quiz
• Announce students can use
◦ pencil/eraser/ruler
◦ NO CALCULATORS
• Probably groan about no calculators
10 min
• Work independently.
• Hand out the quiz.
◦ Circulate
◦ Hand back Review assignment while students
write quiz
• Collect quiz
Activity 2: Peer Marking the Quiz
(CHECK WITH ACKERMAN ABOUT THIS,
IF NO GOOD, ADD TIME TO INDEPENDENT
PRACTISE)
• Students respond to the questions.
10 min
• Re-pass out the quiz
• Go through the answers by asking around. Explain
marking scheme.
◦ For “Explain questions” ask if there was
variety, I will judge.
• Collect the quiz.
Page 2 of 5
Lesson 4: Trig Ratios: Solving for Angles
Teacher Activities
D. Peterson
Student Activities
Activity 3: Solving for an Angle given sine, cosine,
tangent value.
(Mostly direct instruction.)
• First an example problem: Solve for theta (to the
nearest degree) given cos(theta) = 4/5
◦ Have the students do this problem and report
back.
Time
10 min
Hopefully the students only give 37 degrees as an
answer
• Are there two different angles with the same cosine
value?
◦ Plot (-4,-3) , (4,3) , (4,-3), (-4,3) and have
students compute the trig ratios.
• Students respond to, and interact with teacher
questions
• So there are two solution! We must state both.
◦ The calculator will only give a reference
angle. You must choose the quadrants.
▪ How do you know which quadrant?
• Hopefully students refer to ASTC (or CAST)
depending on the acronym they remember.
• Followup question: suppose theta is in quadrant III and
cos(theta) = -3/4. Determine the exact value of
sin(theta) and tan(theta).
Activity 4: Student Completed Example
• Ask class to find, to the nearest degree all solutions
theta to the equation tan(theta) = -2
◦ Ask a student volunteer to explain
• Students compute and raise hand when done
5 min
• With assistance, the volunteer answers the class's
questions
Activity 5: Exact Values
• Students respond to question. (I will ask around)
• Ask Prelimary Questions”
◦ What does “exact mean”
▪ not a decimal approximation
◦ Can you use a calculator?
▪ No
◦ What might be a useful thing to remember?
▪ Special triangles.
5 min
• Solve for all solutions to sin(theta) = sqrt(3)/2
◦ Give class 2 minutes, then ask for a volunteer.
• Repeat above for cos(theta) = 1/2
• Show a diagram of the unit circle with special angles.
Page 3 of 5
Lesson 4: Trig Ratios: Solving for Angles
Teacher Activities
D. Peterson
Student Activities
Time
• Students respond to questions.
◦ Their explanations could be interesting to
see.
5 min
• Work independently (through some interaction with
person sitting beside them is OK)
• Ask questions as necessary. Answer questions posed
by circulating teacher.
19 min
Activity 6: Trick Questions.
*Maybe add these to the assignment*
• Ask if it is legitimate to solve a problem of the form
sin(theta) = a where a>1?
◦ No
◦ Why?
▪ Hypotenuse is shorter than the legs
▪ Do a proof by contradiction.
“Suppose such an angle exists, then
one leg is longer than the
hypotenuse, ...”
• Is it legitimate to solve a problem of the form
tan(theta) = a where a>1.
◦ Absolutely
◦ Example
▪ consider tan(theta) = 4/3
Activity 7: Independent Practise
• Assignment from section 2.2 text p. 96- 98
◦ 7 - 14
• Teacher records grades for quiz then hands back
• Teacher circulating.
Closure:
• Have the class summarize the day.
• Develop summary.
5min
• Tell them that tomorrow we will be working with
computers and spaghetti, to do some hands on
exploration of triangles.
• Thank the class for their enthusiasm (hopefully!)
Ask them to hand in outstanding homework from
Thursday & Friday on their way out.
• Hand in homework
• Pack up and leave
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.
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 written in text format, a diagram can be provided. Such students will be shown how to use diagrams to help
overcome symbolic “math block”. Additional concrete examples can easily be done as needed.
For students who are having an easy time, we can easily add in notions of coterminal angles, essentially pushing ahead to content
covered in PreCalc 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 a wide variety of responses.
There are a few individuals who get extremely anxious and embarrassed about being called on in class. These students will not be called
on, rather questions will be asked while circulating during independent practise time.
Behavioural Management Strategies: For the most part I intend to make use of non verbal behaviour management strategies such as
proximity and the “teacher stare”. 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
Page 4 of 5
Lesson 4: Trig Ratios: Solving for Angles
D. Peterson
individually while others are working. To overcome student talkativeness I will announce that I am waiting for students to be quiet and
then wait for the class to quiet down before proceeding (however painful this may be for me).
Assessment and Evaluation: Pre-assessment for this lesson will consist of students' explanation of what we did last week. In addition
the quiz on last week's content will give information about how this lesson will proceed.
Formative assessment of the lesson objectives and through questioning in the middle of the lesson.
Summative assessment will be done through via a small quiz on Monday, February 24 and through the major unit test on Wednesday,
February 26. Questions will be varied and will resemble to sorts of problems tackled in class.
Extensions: The problem of finding solutions to trigonometric equations ties directly to last class in which we extended the definition of
the trigonometric functions. This unit's notion of multiple solutions build background for understanding the ambiguous case of the sine
law which we will be covering Thursday.
Reflections (if necessary, continue on separate sheet):
Page 5 of 5