YOUNGSTOWN CITY SCHOOLS
MATH: PRECALCULUS
UNIT 4: TRIG FUNCTIONS WITH RADIANS (6 WEEKS) 2013-2014
Synopsis: This unit is comprised of working with angles in radian measure and then extending them to the trig functions.
Students will review trig concepts with angles in terms of both degrees and radians and then apply them to real-life
problems. They will also gain an understanding of the graphs of the trig functions and their inverses.
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 value of sine, cosine, tangent for π/3, π/4 and π/6, and use the
unit circle to express the values of sine, cosines, and tangent for x, π + x, and 2π – x in terms of their values for x, where x is any
real number.
F.TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
F.TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing
allows its inverse to be constructed.
F.TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using
technology, and interpret them in terms of the context.
MATH PRACTICES
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
LITERACY STANDARDS
L.1 Learn to read mathematical text (including textbooks, articles, problems, problem explanations )
L.2 Communicate using correct mathematical terminology
L.4 Listen to and critique peer explanations of reasoning
L.6 Represent and interpret data with an without technology
L.8 Read appropriate text, providing explanation for m0athematical concepts, reason or procedures
MOTIVATION
1. The following web site is an excellent video on ocean waves which connects them to the
TEACHER NOTES
amplitude and period of a sine curve. Click on the video “Ocean Waves – measuring
Amplitude.” http://app.discoveryeducation.com/search?Ntt=ocean+waves
2. Preview expectations for the end of the Unit
3. Have students set both personal and academic goals for this Unit
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TEACHING-LEARNING
Vocabulary:
Radian measure
Periodic
Even function
Arc measure
Coterminal angles
Tangent
Arctangent
TEACHER
NOTES
Standard position
Amplitude
Odd function
Unit circle
Decimal representation
Period
Sin-1
Linear velocity
Frequency
Inverse function
Terminal ray
Sine
Arcsine
Cos-1
Angular velocity
Midline
Reference angles
Initial ray
Cosine
Arccosine
Tan-1
1. Review Radian Measure (F.TF.1, F.TF.2, F.TF.3)
a. start with unit circle to help students see where 0º, 30°, 45°, 60°, 90°, 120 º, 135 º ,150 º, 180, 210 º,
225 º, 240 º, 270 º, 300 º , 315 º , 330 º , and 360 º., have students place them on the unit circle. Then
review the quadrants they are in.
b. from degrees, move to radian measures 0, π/2, π/3, π/4, π/6 and multiples; students can associate
degrees to radians and vice versa. Look at the radian measures of the angles in 1a in relation to the
unit circle.
c. students explain that a radian is an arc measure on a unit circle.
d. discuss positive and negative angles on the unit circle; students draw unit circle and show the location
of angles (e.g., -
)
e. students read the definitions for standard position, terminal ray, initial ray (text page 277) and
reference angles and coterminal angles(angles with same terminal ray - - 5π/4 and -3π/4) (text
page 280 for degrees) in relation to coordinate system and then answer teacher generated questions
and problems about them. Reference angles are between 0 and π/2; and they are measured from
either positive or negative x axis to the terminal ray. Practice problems in Pre-Calc book – Chapter 5
for degrees and Chapter 6 for radians; also, at Kuta website:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Coterminal%20Angles%20and%20Ref
erence%20Angles.pdf
f. discuss decimal representation for radians rather than in terms of π. Show students conversion from
degrees to radian and radians to degrees using a calculator. Students are given angle measure in
degrees (e.g., 35°) and they convert to radians; then they are given radians (e.g., 2.7r), and convert the
radians to degrees.
g. Linear and Angular Velocity: have students read about Linear and Angular Velocity(attached on
pages 7-11) individually and then in groups of two or three, discuss the article and explain the
problems used for examples. Students listen to each other’s explanations and ask questions about key
elements in the problems. Also, Dimensional Analysis problems in textbook on pate 353, #3 and
page 354, #6. Additional problems are also attached on page 12. The following web site may be of
assistance in teaching linear and angular velocity. http://www.intmath.com/trigonometric-functions/8applications-of-radians.php
h. review coordinates of special angles and their multiples using the 30-60-90, 45-45-90 triangles drawn
on the Unit Circle. When students have gained mastery using degrees, then switch to radians and
have them state the coordinates for the angles in radians (0, π/6, π/3, π/4, π, π/2 and their multiples).
Attached on page 12 is the unit circle with the radian measures and their corresponding coordinates.
(F.TF.1, MP.2, MP.3, MP.4, MP.5, MP.8, L.1, L.2, L.4, L.8)
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TEACHING-LEARNING
TEACHER
NOTES
2. Review definitions of sine, cosine, and tangent. Have students explain from prior knowledge using
triangles drawn on unit circle (note: make sure they arrive at sin θ= y, cos θ = x, tan θ = y/x). Review
finding values of the trig functions using 0 º, 30º, 45 º, 60 º, 90 º angles and their multiples. Also review
finding values of the remaining trig functions when given the value of one trig function and given an
ordered pair on the terminal ray of an angle find the value of the 6 trig functions. (Text book Study Guide
5-3, page 82 (attached to the unit on page 14); textbook page 294) (F.TF.2, F.TF.3, MP.4, MP.5, MP.6,
MP.7, MP.8, L.2)
3. Have students work on real-life application problems: Textbook: page 297 (#36, 47, and 48), page 303
(#25, 26, 27, 28, 29, and 30), page 309 (#14-b and c). (NOTE TO TEACHER: If you are adding extra
problems, be sure problems ask for sides when given angles and another side). (MP.1, MP.2, MP.4,
MP.5, MP.6, L-2)
GIVE TEST #1 AT THIS POINT ON STANDARDS F.TF.1 THROUGH 3.
4. Graph a trig function such as f(x) = sin x, use this function to discuss period, amplitude, frequency,
phase shift, and midline. Then graph f(x) = 2sinx, f(x) = 2sin(4x), f(x) = 2sin(4(x-π)), and f(x) = 2sin(4(xπ)) + 3 each of these separately, discussing the changes as you proceed through these examples. Have
students apply these concepts to the other trig functions and find the period, amplitude, frequency, phase
shift, and midline for them.(F.TF.5, MP.2, MP.4, MP.7, MP.8, L.2)
5. Work on TI-Nspire activity to reinforce the periodic phenomena, amplitude, phase shift, and midline. (Link
for TI-Nspire activity: (F.TF.5, MP.1, MP.4, MP.5, L.1, L.2)
http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=5025&t=5075&id=17025
6. Sunrise – Sunset Activity (F.TF.5) (attached on pages 15-22); reinforce with applied problems in
textbook in section 6.6 (F.TF.5, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, MP.8, L.1, L.2, L.6)
7. Have students graph f(x) = sin x, f(x) = cos x, and f(x) = tan x by hand, using tables of values with both
radian and degrees, refer to the unit circle in creating the table of values. After the students have
completed graphing the trig functions, discuss with them the definition of odd and even functions using the
unit circle to explain the symmetry needed for odd functions and the symmetry needed for the even
functions. Have them project which trig functions are even and which are odd. Then show video
http://www.onlinemathlearning.com/trig-functions-even-odd.html The video is the first one on the website.
(F.TF.4, MP.2, MP.4, MP.5, MP.7, MP.8, L.2, L.4, L.6)
8. Discuss odd and even functions connecting to the interactive web site:
http://www.geogebratube.org/student/m16325. Then work on attached worksheet on page 23. (F.TF.4,
MP.2, MP.4, MP.5, MP.7, MP.8, L.2, L.4, L.6)
9. Review the concept of inverse function (switch the x and y’s); i.e., graph the inverse function for f(x) = x2
x
f(x)
x
f-1(x)
0
0
0
0
1
1
1
1
2
4
4
2
-2
4
4
-2
Inverse is not a function; discuss restricting the domain on the original function
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TEACHING-LEARNING
TEACHER
NOTES
10. Have students graph the inverse relations for f(x) = sin x, f(x) = cos x, and f(x) = tan x using their tables
from step #7. Students should realize these are not functions and need to restrict the domain. Have
students choose a restricted domain and discuss answers. (See Text, Section 6-8)
a. discuss with students the appearance of the graph of the inverses. Notice the section of the graphs
for the inverses for sine and tangent are increasing and for cosine is decreasing. So the inverses in
the restricted domains form functions as opposed to when a function increases and then decreases
over an interval. In which case the inverse would not be a function.
b. Look at the notations below for the inverse functions and have students become familiar with them.
sin -1 (x) = arcsin x
cos -1 (x) = arcos x
tan -1 (x) = arctan x
c. When restricting the domain for f(x) = cos x to 0  x  π, f(x) = sin x and f(x) = tan x to –π/2  x  π/2,
this is referred to as the principal values and is denoted with a capital letter (i.e., Sin x) to let the
problem solver know only to work with these values. So, if f(x) = Tan x, we know only to use the
interval –π/2  x  π/2 for the values of x. NOTE: this is the domain that is used in the calculator (e.g.,
solve cos x =
/ 2, the calculator will give x = π/4, however, there are infinitely many answers to this
equation: x = 2 πk - and x = 2 πk +
MP.2, MP.4, MP.5, MP.7, MP.8, L.2 )
for k > 0. (Refer to section 7-5 in the textbook) (F.TF.6,
Look at sine function and restrict from –π/2 to π/2
Look at cosine function and restrict from 0 to π
Look at tangent function and restrict from –π/2 to π/2
11. Students are to solve applied problems involving finding angles using trig functions and the inverse trig
functions. Refer to the textbook for problems, page 310-11 (#46 – 50). Additional problems at:
http://www.youtube.com/watch?v=jhakDu-YTEg and
http://campuses.fortbendisd.com/campuses/documents/Teacher/2008%5Cteacher_20081001_1018.pdf
pages 350-351 and problems 91-97, also attached on pages 24-26. (F.TF.7, MP.1, MP.2, MP.4, MP.5,
MP.6, L.1, L.2, L.6)
TEST #2 GIVEN HERE ON STANDARDS F.TF.4 THROUGH 7
TEACHER NOTES
TRADITIONAL ASSESSMENT
1. Unit Tests: Multiple-Choice Questions (20% of grade)
Test #1 after standards 1, 2, and 3 are taught
Test #2 after standards 4, 5, 6, and 7 are taught
TEACHER CLASSROOM ASSESSMENT
1. Teacher Classroom Assessments: 50% comes from grades the teacher takes.
2. Smaller authentic assessments as you go along
TEACHER NOTES
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TEACHER
NOTES
AUTHENTIC ASSESSMENT (30% of grade)
1. Have students evaluate goals for the unit.
2. Students work the following: Below is a table of gas and electric utility usage in CCF’s (one
hundred cubic feet) for gas and kWh (kilowatt hours) for electric, from February,2012
through February,2013.
Choose one of the rows - - EITHER gas OR electric for one of the households:
a. Plot the points on the coordinate system.
b. Label each axis and the units on the axes.
c. Explain what the units represent.
d. Connect the dots and write an equation of a sine curve that fits the data. You can use the
sine regression on the TI calculator.
e. State the amplitude and what it represents in relation to the problem.
f. State the period and what it represents in relation to the problem.
g. State the frequency and what it represents in relation to the problem.
h. Find the midline and state what it represents in relation to the problem.
i. Predict the usage for May 2013 using your model
Rubric on page 6 of unit plan
TABLE OF GAS AND ELECTRIC USAGE FOR HOUSEHOLDS
FEBRUARY 2012 THROUGH FEBRUARY 2013
Household
#1
#2
#3
#4
Utility
Feb
2012
Mar
2012
Apr
2012
May
2012
June
2012
July
2012
Aug
2012
Sept
2012
Oct
2012
Nov2
012
Dec
2012
Jan
2013
Feb
2013
Gas
143
87
58
24
38
7
27
17
29
142
238
191
133
Electric
1025
1209
1211
1478
1855
1689
1698
1528
1435
1376
1275
1352
1598
Gas
200
153
119
58
71
44
65
54
67
186
278
236
177
Electric
2225
2413
2420
2680
3005
2803
2813
2740
2600
2578
2567
2635
2884
Gas
178
124
113
49
72
43
67
54
68
189
293
256
194
Electric
3656
3760
3859
3920
4270
4067
4078
4002
3507
3845
3943
4015
4268
Gas
245
198
156
107
124
98
116
108
130
255
346
300
244
Electric
705
768
803
876
1228
1014
1026
954
449
789
756
712
746
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RUBRIC for AUTHENTIC ASSESSMENT ACTIVITY WITH GAS AND ELECTRIC (22 points possible)
ELEMENTS OF THE
PROJECT
Plot the points on the
coordinate system
Label each axis and the
units on the axes
Explain what the units
represent
Write an equation of a
sine curve that fits the
data.
State the amplitude and
what it represents in
relation to the problem
State the period and
what it represents in
relation to the problem
State the frequency and
what it represents in
relation to the problem
Find the midline and
state what it represents
in relation to the
problem
Predict usage for May
0
Did not
attempt
Did not
attempt
Did not
attempt
Did not
attempt
1
2
3
Plotted 4 points correctly
Plotted 9 points correctly
Plotted 14 points correctly
Labeled only one axis
with units
Explanation of units on
one axis
Wrote an equation of sine
curve that does not fit the
data
Stated amplitude only or
said what it represents in
relation to problem
Stated period only or
said what it represents in
relation to problem
Stated frequency only or
said what it represents in
relation to problem
Labeled both axes with
units on one axis
Labeled both axes and
units on each
Explained what the units on
both axes represent
Wrote an equation of sine
curve that does not fit the
data
NA
Stated frequency correctly
and how it relates to the
NA
Did not
attempt
Stated midline only
correctly or said what it
represents in relation to
problem
Stated midline correctly and
what it represented
NA
Did not
attempt
Made prediction that is
not realistic according to
the model
Made prediction that was
appropriate according to the
model
NA
Did not
attempt
Did not
attempt
Did not
attempt
Wrote an equation of sine
curve, but made an error
Stated amplitude correctly
and how it relates to the
problem
Stated period correctly and NA
how it relates to the problem
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PRACTICE PROBLEMS for T-L #1g
A pottery wheel rotating at a speed of 755 radians per minute is turned off. It slows at a
constant angular deceleration and comes to a stop after 21 minutes. What is the
approximate angular deceleration?
A
radians per square minute
A.
radians per square minute
B.
radians per square minute
C.
radians per square minute
It takes 90 seconds for a Ferris wheel to complete one revolution. What is its angular
speed?
A.
A.
B.
C.
The wheel of a cart starts at rest and accelerates at 2.5 radians per second squared. What is
the approximate angular speed of the wheel after 15 seconds?
A.
6.0 radians per second
B.
11.9 radians per second
C.
17.5 radians per second
D.
37.5 radians per second
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T-L #1h
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T-L #8 WORKSHEET:
Are the functions even or odd, explain why.
1. F(x) = x3 sin (x)
2. F(x) = sec (x) * tan (x)
3. F(x) = x4 sin (x) * cos2(x)
4. F(x) = cos (x) + sin (x)
5. F(x) = cos (x) * sin (x)
6. F(x) = tan (x) * sin (x)
7. F(x) = x2 cos2(x)
8. F(x) = x + sin (x)
____________________________________________________________________________________
Answers:
1. Even;
2. Odd
3. Odd
4. Neither
5. Odd
6. Even
7. Even
8. Odd
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Applied Problems for T-L #11 Problems 91 - 97
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Answers to Problems on pages 24-25
91. A.
92. A.
B. 0.1253, 0.2527
, B. 0.3805, 1.1022
93. A. graph, B. 1.96 ft. c. asymptote is y axis, After a maximum of 1.96 ft. is reached, as the camera
is moved farther away the angle approaches 0
94. A.0.5743, B. 12.94 ft.
95. A. 0.4538 B. 24.39 ft.
96. A.
97. A.
, B. 0.7086, 1.4056
, B. 0.2450, 0.5404
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