Download Unit 6 - Trigonometric Functions

Course Name: Algebra 2/Math III
Unit #6
Unit Title: Trigonometry
Enduring understanding (Big Idea): Students will understand the connection between radian measure, the unit circle, and the values
of trigonometric functions, the relationship between also model periodic phenomena and graphs of trigonometric functions, and the
Pythagorean Identity can be used to find the value of trigonometric functions.

Essential Questions: What is the unit circle? What is radian measure? What is the relationship between the radian measure of
an angle, and a circle’s radius and circumference? What is the relationship between degrees and radians? How are trigonometric
functions related to the coordinate plane using the unit circle? How do you graph sine, cosine, and tangent? How do you
determine the amplitude, frequency, and midline? How do you prove the Pythagorean Identity sin2θ + cos2θ = 1? How do you use
the Pythagorean Identity sin2θ + cos2θ = 1 to find sinθ, cosθ, and tanθ ) given sinθ. cosθ, or tanθ and the quadrant of the angle?
BY THE END OF THIS UNIT:
Students will know…
 The definition of a radian
 The unit circle
 The Pythagorean Identity
Students will be able to:
 define radian measure of an angle in terms of arc length on
the unit circle.
 find the radian and degree measures of an angle given the
measure of an arc and the radius of the circle
 convert radians to degrees and degrees to radians
 explain how trigonometric functions are related to the
coordinate plane using the unit circle.
 find sine, cosine, tangent, secant, cosecant, and cotangent
of an angle measured in radians around the unit circle
 model periodic phenomena, using appropriate trigonometric
functions (with specified amplitude, frequency, and midline).
 prove the Pythagorean Identity sin2θ + cos2θ = 1.
 find sinθ. cosθ, or tanθ using the Pythagorean Identity
(sin2θ + cos2θ = 1) given sinθ. cosθ, or tanθ and the
quadrant of the angle.
Vocabulary: radian, unit circle, sine, cosine, tangent, period,
amplitude, frequency, midline
Unit Resources:
Algebra 2 Textbook
NCDPI Math Wiki
(http://maccss.ncdpi.wikispaces.net/Unpacking+Documents)
Mathematical Practices in Focus:
1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of
others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning
CCSS-M Included:
F.TF.1, F.TF.2, F.TF.5, F.TF.8
Suggested Pacing:
10 days (including one day for review and one day for test)
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 1
Course Name: Algebra 2/Math III
Unit #6
Unit Title: Trigonometry
CORE CONTENT
Cluster Title: Extend the domain of trigonometric functions using the unit circle
Standard: F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Concepts and Skills to Master:
 Definition of a radian
 The relationship between the radian measure of an angle and a circle’s radius and circumference
 Find the radian and degree measures of an angle given the measure of an arc and the radius of the circle
 Convert degrees to radians
 Convert radians to degrees
SUPPORTS FOR TEACHERS
Critical Background Knowledge: radius, circumference, arc, central angle, degree measure of an angle
Academic Vocabulary: radian
Suggested Instructional Strategies:
(from NCDPI Unpacking Documents)
A central angle of a circle intercepts an arc on the same circle.
The length of the arc is some fraction of the circumference of the
circle and is the basis for discovering the radian measure of an
angle. These steps lead the students to the definition of a radian.
A radian is the measure of the central angle of a circle that
intercepts an arc of the same
measure. It is acceptable to let students define 1 radian as about
57o. Through classroom discussion and investigation, (not direct
instruction) the instruction should guide students to the more
precise conversion factors of and 180°= π radians.
Resources:
Algebra 2 Textbook Correlation: 13-3 Radian Measure
NCDPI Math Wiki
(http://maccss.ncdpi.wikispaces.net/Unpacking+Documents)
To understand what a radian is, conduct the following
investigation with your students.
1. Construct three concentric circles with radii of 10, 20, and 30
cm with center O.
2. Extend a ray from the center of the circle through all three
circles. Label the intersections of the ray and the circles as
B1,B2,B3 respectively.
3. Using a piece of string capture the length of the radius OB1.
Use this length to create arc AB1 so that the length of arc
AB1 is equal to the radius OB1.
4. Draw an angle A1OB1 and record its measure in degrees.
5. Repeat steps 3 and 4 for the remaining two circles.
6. Describe the relationship that exists if the intercepted arc and
the radius are the same length. Generalizing your answer for
a circle with center O and radius r. What would be the
approximate degree measure of angle AOB if arc AB is r
units long?
Sample Assessment Tasks
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 2
Course Name: Algebra 2/Math III
Unit #6
Skill-based task:
(From NCDPI Unpacking Documents)
Ex. What is a radian? Describe the relationship between the
radian measure of an angle, and a circle’s radius and
circumference.
Ex. In a circle with a radius of 5cm and is centered at point O
angle AOB intercepts arc AB. Arc AB has a length of 10cm. What
are the radian and degree measures of angle AOB?
Ex. In a circle with a radius of 8cm and is centered at point O
angle AOB intercepts arc AB. Arc AB has a length of 16cm. What
are the radian and degree measures of angle AOB?
(Level III)
Unit Title: Trigonometry
Problem Task:
(From NCDPI Unpacking Documents)
Ex. Two students are discussing moving between degrees and
radians. Critique and use their arguments in the questions that
follow their discussion.
James
I remember that 360o = 2Pi radians
In finding the radian measure of 120o I divided 120 by 360 to get
1/3. Since my angle was 1/3 of the whole circle and that there are
2Pi radians in a circle I think that the angle will be 1/3 of 2Pi which
is 2Pi/3
Jim
If 360 degrees equals 2Pi radians then if I divide both sides 360,
then 1 degree is equal to Pi/180. There 120 degrees times Pi/180
would be radian measure, which reduces to 2Pi/3.
a. Using James’ method, find the radian measure of 30 degrees.
b. Using Jim’s method, find the radian measure of 30 degrees.
c. Which method is your preferred method? Why?
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 3
Course Name: Algebra 2/Math III
Unit #6
Unit Title: Trigonometry
CORE CONTENT
Cluster Title: Extend the domain of trigonometric functions using the unit circle
Standard: 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 measure of angles traversed counterclockwise around the unit circle.
Concepts and Skills to Master:
Find sine, cosine, tangent, secant, cosecant, and cotangent of an angle measured in radians around the unit circle
SUPPORTS FOR TEACHERS
Critical Background Knowledge: Pythagorean Theorem, 30-60-90 Triangle, 45-45-90 Triangle, sine, cosine, tangent, radian measure
Academic Vocabulary: unit circle sine, cosine, tangent, secant, cosecant, cotangent
Suggested Instructional Strategies:
“A Trip Around the Unit Circle” flipchart
Resources:
Algebra 2 Textbook Correlation: 13-2 Angles and the Unit Circle
“A Trip Around the Unit Circle” flipchart
NCDPI Math Wiki
(http://maccss.ncdpi.wikispaces.net/Unpacking+Documents)
Sample Assessment Tasks
Skill-based task:
Find the value.
1. sin
7
4
3
7. sec
4
4. sin
2. cos

2
7
5. cos
6
8. csc2
3
2
7
6. tan
4
5
9. cot
3
3. tan
Problem Task:
Sectors of Circles
http://map.mathshell.org/materials/lessons.php?taskid=427&subpage=concept
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 4
Course Name: Algebra 2/Math III
Unit #6
Unit Title: Trigonometry
CORE CONTENT
Cluster Title: Model periodic phenomena with trigonometric functions.
Standard: F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Concepts and Skills to Master:
 Graph sine and cosine functions; find the amplitude, frequency, and midline.
 Use sine and cosine to model periodic phenomena
 Given the amplitude, frequency, and midline in situations or graphs, determine a trigonometric function used to model the situation.
SUPPORTS FOR TEACHERS
Critical Background Knowledge: The Unit Circle
Academic Vocabulary: amplitude, frequency, midline, period
Suggested Instructional Strategies:
NOTE: frequency = 1/period.
Resources:
Algebra 2 Textbook Correlation: 13-1, 13-7, 13-8
Use sine and cosine to model periodic phenomena the rotation of
a Ferris wheel from F.TF.2 (in NCDPI Unpacking Documents):
This standard should be approached by students by rotating a
circular object (ie Ferris Wheel) around and recording the angle
formed with the origin, the height off the ground, and the
horizontal height from the center of the circle. Below are brief
instructions of how to set this up.
Construct a coordinate axes on a large sheet of paper. From the
origin mark off “famous” unit circle angles. (This will create a large
asterisk like drawing.) On a separate piece of a paper, construct a
large “unit” circle. Attach the center of the circle to the origin of the
axes with a brad for rotational purpose. Students should draw or
place a sticker of a “rider” at (1,0). While not imperative, this
would work better if the grid was placed on a 10mm graph paper
and the circle constructed with a transparency.
NCDPI Math Wiki
(http://maccss.ncdpi.wikispaces.net/Unpacking+Documents)
Foxes and Rabbits 2
http://www.illustrativemathematics.org/illustrations/816
Foxes and Rabbits 3
http://www.illustrativemathematics.org/illustrations/817
As the World Turns
http://www.illustrativemathematics.org/illustrations/595
Ex. With your rotational circle, record the following data points:
Angle Formed by extending ray from center of circle to “rider.”
Height of “rider” from x-axis (ground)
Horizontal distance “rider” is from origin.
Pi/6
Pi/4
a. On a coordinate axes, graph the following points (angle formed,
height)
b. On a separate axes, graph the following points (angle formed,
horizontal distance)
c. For each angle, find the ratio of the height to the horizontal
distance.
d. On yet another axes, graph the following points (angle formed,
ratio in part (c))
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 5
Course Name: Algebra 2/Math III
Sample Assessment Tasks
Skill-based task:
Graph. State the amplitude, frequency, and midline.
1. y = sinθ
2. y = cosθ
3. y = tanθ
4. y = 2sinθ
5. y = -3cosθ
6. y =-2tanθ
7. y = sin3θ
8. y = cos2θ
9. y = tan0.25θ
10. y = sinθ + 3
11. y = cosθ - 1
12. y = tanθ + 2
13. y = -sin2θ - 3
14. y = 2cos3θ + 1
15. y = -0.5tan2θ -3
Unit #6
Unit Title: Trigonometry
Problem Task:
(from NCDPI Unpacking Documents)
At midnight the water at a particular beach is at high tide. At the
same time a gauge at the end of a pier reads 10 feet. Low tide is
reached at 6 AM when the gauge reads 4ft.
a. Choose which trig function would be the best fit for this model
(assuming midnight is t=0). Justify your choice using specific
characteristics of trigonometric function graphs.
b. Determine the midline, amplitude and frequency using the
above tidal information. You must show all computations and
explain why you performed each computation.
c. Write a function based on parts one and two to represent the
above tidal information.
d. If the times for high and low tides are reversed what (if
anything) would change in the equation from part (c)? Justify your
conclusion.
e. If you were instructed to let t=0 represent 9pm, would your
function in part (a) still be the most convenient choice? Why or
why not? If not, convince you teacher what a better choice
would be.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 6
Course Name: Algebra 2/Math III
Unit #6
Unit Title: Trigonometry
CORE CONTENT
Cluster Title: Prove and apply trigonometric identities.
Standard: F.TF.8 Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to calculate trigonometric ratios.
Concepts and Skills to Master:
Given the value of sinθ, cosθ, or tanθ and its quadrant, find the remaining trigonometric ratios using the Pythagorean Identity.
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Pythagorean Theorem, solving square root equations, sine, cosine, tangent , unit circle, quadrant
Academic Vocabulary: Pythagorean Identity
Suggested Instructional Strategies:
Introduce sin2(θ) + cos2(θ) = 1. Review cosθ = x, sinθ = y, and
tanθ = sinθ/cosθ = y/x in the unit circle. Model problems 1 and 2
(see Skill-based task). Then discuss how to transform
sin2(θ) + cos2(θ) = 1
to
tan2(θ) + 1 = 1/ cos2(θ)
by dividing both sides by cos2(θ). Model problem 3 (see Skillbased task).
Sample Assessment Tasks
Skill-based task:
Using the Pythagorean Identity:
1. Given sinθ = ½ where θ is in quadrant II, find cosθ and tanθ.
2. Given cosθ = - ½ where θ is in quadrant II, find sinθ and tanθ.
3. Given tanθ = ½ where θ is in quadrant III, find sinθ and cosθ.
Resources:
Algebra 2 Textbook Correlation: 14-1
NCDPI Math Wiki
(http://maccss.ncdpi.wikispaces.net/Unpacking+Documents)
Problem Task:
(From NCDPI Unpacking Documents)
a. Verify that at every height and horizontal value on the circle the
radius is constant at 1. (Students should either construct the
triangle within the circle and apply the Pythagorean theorem)
b. What trigonometric function represents the height of the rider at
any angle? How do you know?
c. What trigonometric function represents the horizontal distance
of the rider at any angle? How do you know?
d. Rather than writing x and y to determine the radius at any
angle, use the results from (b) and (c) to rewrite the Pythagorean
Theorem relationship in terms of trigonometric functions.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 7