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 Angle Measures and Trigonometric Functions
Title of Unit
Angle Measures and Trigonometric
Functions
Pre-Calculus
Ryan Duffy
Curriculum Area
Developed By
Grade Level
12
Time Frame
2 weeks (10 days)
Identify Desired Results (Stage 1)
Content Standards
F-TF-1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF-2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle.
F-TF-3. Use special triangles to determine geometrically the values of sine, cosine, and tangent for pi/3, pi/4, and pi/6, and use the unit circle to express the values of
sine, cosine, and tangent for x, pi+x, and 2pi-x, where x is any real number.
Understandings
Essential Questions
Overarching Understanding
Overarching
Students will understand that radian measure of an angle is an alternative to degree measure
that is more useful in mathematical situations
Students will understand that the real numbers can be represented in ways other than on a number
line, and that different representations can be advantageous in different situations.
Students will understand that the relationships between the sides and angles of a right triangle can be
used for larger purposes – particularly for breaking down distances into their vertical and horizontal
components.
Related Misconceptions
Due to its definition, students may have difficulty understanding that a radian is an angle
measure and not just a distance measure. Students may also have trouble remember or
understanding that similar triangles – regardless of size – have the same ratios between their side
lengths. This may lead them to think there’s something inherently special about the unit circle other
than its convenience of radius 1. Finally, students may determine radian values by moving clockwise
(instead of counterclockwise) around a circle. For them, this direction of motion is more familiar.
Knowledge
Students will know…
What pattern is there when
I go around the unit circle? How
can I use this pattern to simplify
problems?
How can I remember this
information without memorizing
it?
How do these new problems
relate to the trigonometry I’ve
done in the past?
In what real-life situations would
this mathematics be useful?
Objectives
Skills
Students will be able to…
Topical
Where is this angle located in
the unit circle?
How do I know if two angles are
coterminal?
How can I convert this degree
measure to a radian measure?
If I know one side length of a 30-6090 or 45-45-90 triangle, how can I
find the others?
If I know one trigonometric ratio for
a given angle, how can I find the
others?
If I know that an angle in standard
position passes through a point
(x,y), how can I find the
trigonometric values of that angle?
the definitions of the six trigonometric functions; that a radian is the angle
measure that measures one arc length along a circle; that there are 2pi radians
and 360 degrees in a circle; the definitions of positive angles, negative angles,
coterminal angles, and standard position; that a unit circle is the circle centered
at the origin with radius 1; the radian equivalent of key degree measures (e.g.,
90 degrees = pi/2).
solve a right triangle; use one trigonometric ratio to find them
all; use a 45-45-90 triangle and 30-60-90 triangle to find trig values of
their angles; convert between radians and degrees; find the values of
several coterminal angles and the trig values of these angles; use a
point to find the trig values of an angle in standard position passing
through that point.
Assessment Evidence (Stage 2)
Performance Task Description
Goal
Role
Audience
Situation
Product/Performance
Standards
Assess ability to convert between radians and degrees; understanding of co-terminal angles; ability to find trig
values of angle given point on terminal side; knowledge of common degree and radian measures around a circle;
ability to find 5 trig values of an angle given only the remaining trig value
Unit test
Myself, mentor teacher (Dee Eberle)
Classroom test; regular class period
Completed test
F-TF-1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
F-TF-2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers,
interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Other Evidence
The third standard (F-TF-3. Use special triangles to determine geometrically the values of sine, cosine, and tangent for pi/3, pi/4, and pi/6, and use the unit
is not explicitly addressed on the unit test. Instead, it
will be assessed with a quiz on Day 3 (see schedule of lessons below).
circle to express the values of sine, cosine, and tangent for x, pi+x, and 2pi-x, where x is any real number.)
Additionally, because of the large number of novel concepts in this unit, I will start class each day with a warm-up problem based on what was taught the
day before. Additionally, the following question will be added to the unit test to ensure students understand that trigonometric functions are ratios, and
that there is nothing inherently ‘special’ about the use of the unit circle:
“Suppose we had used the circle
x 2 + y 2 = 2 to find the sine of the ‘special angles’ we talked about in class. Would the sine of
not? Write a few sentences and draw a diagram to explain your answer.
π
6
change? Why or why
Learning Plan (Stage 3)
Day
in
Unit
Lesson €
Topic
Lesson Learning Objective
Description of how lesson
contributes to unit-level
objectives
€ Assessment activities
1
Degrees, minutes,
seconds & review
how to solve right
triangle
2
Finding trig ratios
given limited
information
- Students will be able to convert
between degrees, minutes, and
seconds.
- Students will be able to define the
ratios of the six trig functions and
solve a right triangle.
Students will be able to use one trig
ratio to find the other 5
Students will be able to use 30-60-90
and 45-45-90 triangles to find the
trig values of 30, 60, and 45-degree
angles.
This lesson ensures students know
the definition of the six trig functions
and that they have a concept of
degree measurements – crucial for
the development of further lessons.
•
•
•
•
Meets following unit objective: Students will be able to use special triangles to determine geometrically the values of the six standard trigonometric functions. Additionally, this lesson will make sure students are familiar with special triangles before using them along the unit circle. Warm-up activity: Using the
words “opposite,” “adjacent,” and
“hypotenuse,” write the definition
of sin, cos, tan, cot, csc, and sec.
Warm-up activity: Solve the
following right triangle: A=90,
B=40, c=25. Then, find the
sin(B), cos(B), tan(B), cot(B),
csc(B), and sec(B)
3
Definition of a
radian
Students will be able to convert
between degrees and radians and be
able to explain why there are 2pi
radians in a circle.
Students will be able to calculate the
arc length along a circle of radius r
subtended by an angle theta.
Meets following unit objectives: •
• Students will understand that a radian is the angle that measures an arc length of one radius along a circle. • Students will be able to •
convert between degree and radian angle measurements. Additionally, this lesson introduces students to radian measures that will be essential for later lessons in the unit. Quiz: Use the triangles we talked
about yesterday to find sin(30)
without using a calculator. Be sure
to clearly label any diagrams you
may draw, and be specific in your
writing.
Exit slip: Explain in your own
words a) what a radian is and b)
why there are 2pi radians in a
circle.
4
Coterminal angles
Students will be able to construct
several coterminal angles – both
positive and negative – given an
angle theta. They will also
understand that the number line can
be continuously wound around a
Meets the following unit objective:
Warm-up activity: What is the
length around a circle of radius 2
Students will understand that coterminal angles are angles with different measures that share both an initial and •
subtended by an angle of
•
π
?
3
Exit slip: Write two angles that are
coterminal with 3 radians.
€
5
Find the trig values
of an angle in
standard position
passing through a
given point (x,y)
6
Introduction to the
unit circle
circle, and use this information to
explain why sine and cosine have a
period of 2pi.
terminal side. They will be able to use this information to explain why the sine and cosine functions have a period of 2pi. Students will understand that given a
point (x,y), they can find the distance
between that point and the origin
using the distance formula. They can
then define the six trig functions in
terms of x, y, and r.
Students will understand that by
using right triangles with a
hypotenuse of 1, the sine and cosine
of an angle in standard position
passing through a point (x,y) can be
interpreted as y and x respectively.
This lesson allows students to begin
to see sin as a vertical distance and x
as a horizontal distance. It establishes
the basic understanding necessary for
using the unit circle in the next
lesson.
Students will be able to use the unit
circle and their knowledge of special
triangles to calculate the trig
functions for the various “special
angles.”
7
Finding trig values
of pi+x and 2pi-x
8
Practice with
special
angles/Review
Students will understand how to use
the unit circle and the trig values of
an angle x to quickly find the trig
values of the angles 2pi-x and pi+x.
Students will be able to label sine
and cosine as odd and even functions
respectively and explain why.
Students will have time to look for
patterns around the unit circle,
create flashcards to memorize
values, or ensure that they can
quickly find the trig values of special
angles. This day will be a “catch-up”
•
•
•
•
•
Warm-up activity: Given a point
(x,y) in the coordinate plane,
what is the distance between that
point and the origin?
• Warm-up activity: Find the sine,
Meets following unit cosine, and tangent of an angle in
objective: Students will be standard position passing through
able to use a point on the unit 2 2
circle to calculate the point (
, ).
2 2
trigonometric functions of the angle in standard position whose terminal sides is incident with that €
point. Additionally, this lesson allows
students to visualize the trig
values of an angle x in such a way
that will help them find the trig
values of 2pi-x and pi+x in the
following lesson.
Helps students to use unit circle
to quickly find the trig values of
more complex or non-standard
angles.
Through individualized attention
and guidance, I will ensure that
all students are on track to
demonstrate their understanding
of the unit objectives on next
day’s test.
•
Warm-up activity: What is the
sin(pi/6)? Cos (pi/2)? Sin(pi)?
•
Warm-up activity: Without using
a calculator, what is cos(-pi/3)?
sin(pi+7pi/4)?
Explain, using either words or a
diagram, why sin(x) is
considered an odd function.
•
9
Review
10
Test
day of sorts. Students who are
comfortable with the material and do
not have any questions will be given
the homework assignment (review
packet) and allowed to work on it in
class.
- Go over and discuss the questions
from review packet
Test
•
Review
Test
None
Test