Download Developmental Math – An Open Curriculum Instructor Guide

Developmental Math – An Open Curriculum
Instructor Guide
Unit 19 – Table of Contents and
Learning Objectives
Unit 19: Trigonometry
Unit Table of Contents
Lesson 1: Introduction to Trigonometric Functions
Topic 1: Identifying the Six Trigonometric Functions
Learning Objectives
 Identify the hypotenuse, adjacent side, and opposite sides of an angle in a right




triangle.
Determine the six trigonometric ratios for a given angle in a right triangle.
Recognize the reciprocal relationship between sine/cosecant, cosine/secant, and
tangent/cotangent.
Use a calculator to find the value of the six trigonometric functions for any acute
angle.
Use a calculator to find the measure of an angle given the value of a trigonometric
function.
Topic 2: Right Triangle Trigonometry
Learning Objectives
 Use the Pythagorean Theorem to find the missing lengths of the sides of a right



triangle.
Find the missing lengths and angles of a right triangle.
Find the exact trigonometric function values for angles that measure 30°, 45°, and
60°.
Solve applied problems using right triangle trigonometry.
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Monterey Institute for Technology and Education (MITE) 2012
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1.1
Developmental Math – An Open Curriculum
Instructor Guide
Topic 3: Unit Circle Trigonometry
Learning Objectives
 Understand unit circle, reference angle, terminal side, standard position.
 Find the exact trigonometric function values for angles that measure 30°, 45°, and


60° using the unit circle.
Find the exact trigonometric function values of any angle whose reference angle
measures 30°, 45°, or 60°.
Determine the quadrants where sine, cosine, and tangent are positive and negative.
Lesson 2: Graphing Trigonometric Functions
Topic 1: Degree and Radian Measure
Learning Objectives
 Understand radian measure.
 Convert from degree to radian measure.
 Convert from radian to degree measure.
Topic 2: Graphing the Sine and Cosine Functions
Learning Objectives

Determine the coordinates of points on the unit circle.
Graph the sine function.
Graph the cosine function.
Compare the graphs of the sine and cosine functions.



Topic 3: Amplitude and Period
Learning Objectives
 Understand amplitude and period.
 Graph the sine function with changes in amplitude and period.


Graph the cosine function with changes in amplitude and period.
Match a sine or cosine function to its graph and vice versa.
1.2
Developmental Math – An Open Curriculum
Instructor Guide
Unit 19 – Instructor Notes
Unit 19: Trigonometry
Instructor Notes
The Mathematics of Trigonometry
At this point, students are familiar with many types of functions – linear, quadratic,
rational, radical, exponential and logarithmic. This unit introduces yet another type –
trigonometric functions. Students will learn how to identify the six trigonometric
functions as well as find these values on a calculator. Right triangle and unit circle
trigonometry will be used to solve applied problems. The unit concludes with graphing
sine and cosine functions and investigating what happens to these graphs when there
are changes in amplitude and period.
Teaching Tips: Challenges and Approaches
The study of trigonometric functions is one that will be difficult for the average
developmental math student. The material is new and there are a lot of definitions. It is
necessary to spend sufficient time to help your students understand all of what may
seem to them to be disjointed topics in this unit.
One thing that will help your students is to review the Pythagorean Theorem. This is
something that usually is easily understood and can provide a jumping off point to
introduce the need for trigonometry, especially if students need to find the measures of
the angles of a right triangle instead of the length of a missing side.
Trigonometric Functions
It is necessary to stress the definitions of “hypotenuse”, “opposite” and “adjacent”. It is
crucial to explain to your students that each acute angle in a right triangle has an
adjacent side that, along with the hypotenuse, forms the angle and an opposite side that
is “across” from the angle. A number of examples should be given, such as the one
shown below:
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Developmental Math – An Open Curriculum
Instructor Guide
[From Lesson 1, Topic 1, Presentation]
The next step would be to define the six trigonometric functions. Be sure to include a
memory device, such as “sohcahtoa,” to help your students remember the ratios. In
fact, it is an interesting activity for your students to come up with their own way of
remembering the ratios or have them search the internet to find different ones. This
search could also include videos or songs. What is important is that your students
understand what the trigonometric functions are.
Another thing to stress is the fact that there are reciprocal relationships involved.
[From Lesson 1, Topic 1, Topic Text]
Students would not need to remember the definitions for cotangent, secant, and
cosecant if they realize their relationship to the tangent, cosine, and sine functions.
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Developmental Math – An Open Curriculum
Instructor Guide
Stress to your students that it is also important to know this important relationship
between the functions in order to be able to use the calculator to find the cotangent,
secant, and cosecant of an angle.
There will be confusion when students see an inverse trigonometric function such as
1
“sin-1” for the first time. This does not mean
as students might think from their
sin x
study of exponents. It should be stressed that this means “angle whose sine is” and the
use of this key on the calculator should be explained.
When explaining the sine and cosine functions, make sure the students realize that
values can only be between -1 and 1 inclusive. If they calculate one of these ratios and
end up with an answer of 2.5, something is wrong.
The process of using the trigonometric functions, the Pythagorean Theorem, and a
calculator is called “solving the right triangle”. Students should be given enough
problems to become familiar with this process. Have a discussion with your students
about what information is required to be able to solve a right triangle. For example, if
you know the value of one trigonometric ratio, can you find all the lengths of the sides
and measures of the angles of the triangle? What if you knew only the measure of the
angles? Can you determine the lengths of the sides?
Students need to know that there are special angles and that the exact values of their
trigonometric functions can be found without using a calculator. Be sure to show your
students how to figure out the lengths of the sides of 30° - 60° - 90° and 45° - 45° - 90°
triangles. If they understand how to calculate these lengths, there is no need to
memorize what the ratios are!
It is important for your students to understand how to find the values of trigonometric
functions and the measures of angles using right triangle trigonometry before moving on
to unit circle trigonometry. Explain to them that unit circle trigonometry is really an
extension of what has been learned already. The major difference is that when using
the unit circle, one can find the values of trigonometric functions for any size angle, not
just acute angles.
Again, it is necessary to begin this topic with definitions that are crucial for students to
understand. Make sure that unit circle, reference angle, terminal side, and standard
position are all understood by giving many examples.
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Developmental Math – An Open Curriculum
Instructor Guide
Students will accept the concept of angles up to 180°, but anything larger will be a
foreign concept to them. It also will be necessary to explain the concept of a negative
angle.
[From Lesson 1, Topic 3, Topic Text]
Be sure to stress “coterminal” and “reference” angles in order to explain this concept to
your students.
Students will better understand unit circle trigonometry if the similarities are discussed.
[From Lesson 1, Topic 3, Topic Text]
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Developmental Math – An Open Curriculum
Instructor Guide
Once students learn the new terminology, they will be relieved to know that right triangle
and unit circle trigonometry do give the same results.
Let students know that θ is used with unit circle trigonometry. This notation is used to
make the formulas for the trigonometric functions look simpler.
[From Lesson 1, Topic 3, Topic Text]
Stress that values of trigonometric functions are found by using a three-step process:
find the reference angle, determine the value of the trigonometric function of the
reference angle, and then determine if the value of the function is positive or negative.
Help your students learn where the sine, cosine and tangent functions are positive by
using a mnemonic device:
[From Lesson 1, Topic 3, Topic Text]
You can also encourage your students to do an internet search to find other mnemonic
devices to help them remember where these functions are positive.
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Developmental Math – An Open Curriculum
Instructor Guide
In the topic text, there is an interesting way to help your students generate and then to
remember the values of the trigonometric functions for the reference angles 0°, 30°, 45°,
60°, and 90° for sine and cosine. Once your students understand this, they can get the
sin
value of tangent from the identity tan 
, and the values of the other three
cos
trigonometric functions using reciprocals.
Graphing Trigonometric Functions
The topic of graphing should be discussed once your students are comfortable with
finding the values for the trigonometric functions of any angle. Before actually graphing,
you will need to introduce measuring angles in radians as opposed to degrees. This
might be troublesome for your students. Tell your students that radians are useful to
find the length of an arc on a circle. This definition leads to the arc length formula:
s  r . Working a few examples of this formula will help your students appreciate the
need for a measure other than degrees.
Work many examples using the conversion formulas from degrees to radians and
radians to degrees.
[From Lesson 2, Topic 1, Worked Example 3]
1.8
Developmental Math – An Open Curriculum
Instructor Guide
The rest of the unit deals with graphing the sine and cosine functions. By now students
have seen a lot of graphs with points of the form (x, y). Stress to your students that
when they graph the sine function, it is normal to use points of the form ( , sin ) and
similarly, the cosine function will use ( , cos ) . It should be noted that  is measured
in radians. The good news is that your students already know how to get the points to
be plotted. Have them construct a table of a few  values in each quadrant and then
have them plot away.
Encourage your students to describe the graphs that they are generating. Mention that
the graphs look like hills and valleys. Start a discussion about what values of  give a
hill for both the sine and cosine.
[From Lesson 2, Topic 2, Topic Text]
Students should be aware of the domain and range of each function. Mention that
graph problems often specify the domain over which the graph is to be constructed.
Have your students compare their graphs of the sine and cosine functions over the
same interval. They should notice that the shapes of the graphs of the two functions
are similar but not identical. Try to get them to notice that the graph of y  cos is
identical to the graph of y  sin but it is shifted.
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Developmental Math – An Open Curriculum
Instructor Guide
Have them notice that because the patterns repeat, you could start with the graph of
either sine or cosine and shift it by different amounts to the right or left to get the graph
of the other function.
Once your students have a good grasp on graphing the sine and cosine functions, it is
time to look at the graphs of y  a sin bx or y  a cos bx , where a and b are constants.
In order to do this, the terms “period” and “cycle” need to be introduced with many
examples.
It is easy to give the formula to find the period of a sine or cosine function. But resist
doing this. Instead, give many examples and see if your students can figure out the
formula. It’s always better for the students to discover rather than just memorize a
formula.
[From Lesson 2, Topic 3, Presentation]
Students should be able to come up with the formula
2
. Be sure to explain that this is
b
almost correct. Since b can be negative, the formula needs to include b in the
denominator. It is always good to illustrate period by starting with the basic graph of
1.10
Developmental Math – An Open Curriculum
Instructor Guide
y  a sin bx or y  a cos bx and then on the same set of axes, either shrinking or
stretching the graph horizontally by changing the value of b.
Similarly, instead of giving the formula for the amplitude of a sine or cosine function,
have your students look at a few examples to see if they can determine the formula.
They might in the beginning say it is just a in y  a sin bx or y  a cos bx . Be sure to
explain why that isn’t correct by giving an example with a negative. It is always good to
illustrate amplitude by starting with the basic graph of y  a sin bx or y  a cos bx and
then on the same set of axes, either shrinking or stretching the graph vertically by
changing the value of a.
At this point, students should be able to graph any sine or cosine function with any
period or amplitude. Be sure to quiz them on this until they are comfortable doing so.
1.11
Developmental Math – An Open Curriculum
Instructor Guide
[From Lesson 2, Topic 3, Topic Text]
Finally, students should be able to look at a graph and tell whether it is a sine or cosine
function, and identify its period and amplitude. There are a few pointers that you can
give. For example, if the graph of the function passes through (0,0), it must be a sine
function as the cosine function has a value of (0,1). Also, remind your students that the
amplitude is easy to figure out because it is the height above (or below) the x-axis. But,
it is necessary to figure out if a is positive or negative. Have your students look at the
1.12
Developmental Math – An Open Curriculum
Instructor Guide
graph – does it look like the basic function? If yes, then a is positive. If no, then it is
negative.
Work through a number of examples until your students feel comfortable with the steps
to determine whether the function is sine or cosine, and its period and amplitude.
[From Lesson 2, Topic 3, Worked Example 4]
Keep in Mind
One of the biggest stumbling blocks in this unit continues to be using the calculator to
find the values of the trigonometric functions as well as angle measures. Students
usually do not know how to use their calculators to do this. Enough practice should be
given so they are comfortable using the calculator to get the correct answer.
Most of the material in this unit is not normally taught to developmental mathematics
students. However, it is appropriate to use this material for enrichment in the
intermediate algebra class.
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Developmental Math – An Open Curriculum
Instructor Guide
Additional Resources
There are a number of sites on the internet that deal with bringing trigonometry to life for
your students. Try searching for “trig function manipulatives” or “interactive trig” or “trig
applets” or something similar and be prepared to marvel at the creativity of others.
Some applets that are particularly useful follow:
For practice with all the concepts in this unit, http://www.intmath.com/trigonometricfunctions/trig-functions-intro.php provides a great interactive review with applets and
review questions.
There is a trigonometric functions calculator at
http://www.analyzemath.com/Calculators/Trigonometry_Cal.html. What is nice about
this site is you can enter the measure of the angle in degrees (e.g. 30°), radians in

decimal form (.5236), and radians in fractional form ( ).
6
A calculator from degrees to radians and vice versa is available at
http://www.unitconversion.org/angle/degrees-to-radians-conversion.html.
There are good plotting demonstrations of the sine and cosine functions at
http://www.ies.co.jp/math/products/trig/menu.html.
Some trigonometry manipulatives can be found at
http://lgrima.coffeecup.com/apps/trig/trig.htm.
Different mnemonic devices as well as trigonometry videos to help your students
remember some fundamental trigonometric concepts can be found at
http://www.onlinemathlearning.com/mnemonics-for-trigonometry.html.
Summary
After completing this unit, students will understand the six trigonometric functions and
their relationships to each other. They will be able to solve application problems using
either right triangle or unit circle trigonometry.
Once the basics of trigonometry are understood, students will be able to graph the sine
and cosine functions, including changes in amplitude and period.
1.14
Developmental Math – An Open Curriculum
Instructor Guide
Unit 19 – Tutor Simulation
Unit 19: Trigonometry
Instructor Overview
Tutor Simulation: Amusement Park Rides
Purpose
This simulation allows students to demonstrate their understanding of trigonometry.
Students will be asked to apply what they have learned to solve real-world problems
involving:






Trigonometric Functions
Degrees and Radians
Solving Triangles
Amplitude and Period
Matching Graphs to their Functions
Unit Circle
Problem
Students are presented with the following problem:
You have just started working for a travel agency that specializes in planning group trips
– both domestic and international. Thrill ride enthusiasts are frequent clients, and they
are very knowledgeable about the design of their favorite rides. You need to learn about
the more popular rides, so you decide to use your knowledge of trigonometry to better
understand the thrill factor that makes these rides so popular.
Recommendations
Tutor simulations are designed to give students a chance to assess their understanding of unit
material in a personal, risk-free situation. Before directing students to the simulation:


Make sure they have completed all other unit material.
Explain the mechanics of tutor simulations:
1.15
Developmental Math – An Open Curriculum
Instructor Guide
o

Students will be given a problem and then guided through its solution by a video
tutor;
o After each answer is chosen, students should wait for tutor feedback before
continuing;
o After the simulation is completed, students will be given an assessment of their
efforts. If areas of concern are found, the students should review unit materials or
seek help from their instructor.
Emphasize that this is an exploration, not an exam.
1.16
Developmental Math – An Open Curriculum
Instructor Guide
Unit 19 – Correlation to Common
Core Standards
Learning Objectives
Unit 19: Trigonometry
Common Core Standards
Unit 19, Lesson 1, Topic 1: Identifying the Six Trigonometric Functions
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
EXPECTATION
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
EXPECTATION
Grade: 9-12 - Adopted 2010
CCSS.Math.Content.HSF
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.A
CCSS.Math.Content.HSFTF.A.3
CCSS.Math.Content.HSF
Extend the domain of trigonometric functions
using the unit circle.
(+) Use special triangles to determine
geometrically the values 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 π-x, π+x, and 2π-x in terms of their values for
x, where x is any real number.
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.B
CCSS.Math.Content.HSFTF.B.6
Model periodic phenomena with trigonometric
functions.
(+) Understand that restricting a trigonometric
function to a domain on which it is always
increasing or always decreasing allows its inverse
to be constructed.
(+) 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.
Geometry
EXPECTATION
CCSS.Math.Content.HSFTF.B.7
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSG
CCSS.Math.Content.HSGSRT
CCSS.Math.Content.HSGSRT.C
Similarity, Right Triangles, and Trigonometry
Define trigonometric ratios and solve problems
involving right triangles
1.17
Developmental Math – An Open Curriculum
Instructor Guide
EXPECTATION
CCSS.Math.Content.HSGSRT.C.6
EXPECTATION
CCSS.Math.Content.HSGSRT.C.7
CCSS.Math.Content.HSGSRT.C.8
EXPECTATION
Understand that by similarity, side ratios in right
triangles are properties of the angles in the
triangle, leading to definitions of trigonometric
ratios for acute angles.
Explain and use the relationship between the sine
and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied
problems.
Unit 19, Lesson 1, Topic 2: Right Triangle Trigonometry
STRAND /
DOMAIN
CATEGORY /
CLUSTER
Grade: 8 - Adopted 2010
CCSS.Math.Content.8.G
Geometry
STANDARD
CCSS.Math.Content.8.G.A.5
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.8.G
Understand congruence and similarity using
physical models, transparencies, or geometry
software.
Use informal arguments to establish facts about
the angle sum and exterior angle of triangles,
about the angles created when parallel lines are
cut by a transversal, and the angle-angle criterion
for similarity of triangles. For example, arrange
three copies of the same triangle so that the sum
of the three angles appears to form a line, and
give an argument in terms of transversals why
this is so.
Geometry
CCSS.Math.Content.8.G.B
Understand and apply the Pythagorean Theorem.
CCSS.Math.Content.8.G.B.7
Apply the Pythagorean Theorem to determine
unknown side lengths in right triangles in realworld and mathematical problems in two and
three dimensions.
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.8.G.A
Grade: 9-12 - Adopted 2010
CCSS.Math.Content.HSF
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.A
Extend the domain of trigonometric functions
using the unit circle.
1.18
Developmental Math – An Open Curriculum
Instructor Guide
EXPECTATION
CCSS.Math.Content.HSFTF.A.3
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSG
(+) Use special triangles to determine
geometrically the values 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 π-x, π+x, and 2π-x in terms of their values for
x, where x is any real number.
Geometry
CCSS.Math.Content.HSG-CO
Congruence
CCSS.Math.Content.HSGCO.C
CCSS.Math.Content.HSGCO.C.10
Prove geometric theorems
EXPECTATION
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
EXPECTATION
EXPECTATION
EXPECTATION
Prove theorems about triangles. Theorems
include: measures of interior angles of a triangle
sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two
sides of a triangle is parallel to the third side and
half the length; the medians of a triangle meet at
a point.
Geometry
CCSS.Math.Content.HSG
CCSS.Math.Content.HSGSRT
CCSS.Math.Content.HSGSRT.C
CCSS.Math.Content.HSGSRT.C.6
Similarity, Right Triangles, and Trigonometry
Define trigonometric ratios and solve problems
involving right triangles
Understand that by similarity, side ratios in right
triangles are properties of the angles in the
triangle, leading to definitions of trigonometric
ratios for acute angles.
Explain and use the relationship between the sine
and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean
Theorem to solve right triangles in applied
problems.
CCSS.Math.Content.HSGSRT.C.7
CCSS.Math.Content.HSGSRT.C.8
Unit 19, Lesson 1, Topic 3: Unit Circle Trigonometry
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
Grade: 9-12 - Adopted 2010
CCSS.Math.Content.HSF
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.A
Extend the domain of trigonometric functions
using the unit circle.
1.19
Developmental Math – An Open Curriculum
Instructor Guide
EXPECTATION
CCSS.Math.Content.HSFTF.A.2
EXPECTATION
CCSS.Math.Content.HSFTF.A.3
EXPECTATION
CCSS.Math.Content.HSFTF.A.4
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSG
EXPECTATION
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.
(+) Use special triangles to determine
geometrically the values 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 π-x, π+x, and 2π-x in terms of their values for
x, where x is any real number.
(+) Use the unit circle to explain symmetry (odd
and even) and periodicity of trigonometric
functions.
Geometry
CCSS.Math.Content.HSGSRT
CCSS.Math.Content.HSGSRT.C
CCSS.Math.Content.HSGSRT.C.7
Similarity, Right Triangles, and Trigonometry
Define trigonometric ratios and solve problems
involving right triangles
Explain and use the relationship between the sine
and cosine of complementary angles.
Unit 19, Lesson 2, Topic 1: Degree and Radian Measure
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
Grade: 9-12 - Adopted 2010
CCSS.Math.Content.HSF
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.A
CCSS.Math.Content.HSFTF.A.1
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSG
Extend the domain of trigonometric functions
using the unit circle.
Understand radian measure of an angle as the
length of the arc on the unit circle subtended by
the angle.
Geometry
CCSS.Math.Content.HSG-C
Circles
CCSS.Math.Content.HSG-C.B
Find arc lengths and areas of sectors of circles
EXPECTATION
CCSS.Math.Content.HSGC.B.5
Derive using similarity the fact that the length of
the arc intercepted by an angle is proportional to
the radius, and define the radian measure of the
angle as the constant of proportionality; derive
the formula for the area of a sector.
EXPECTATION
1.20
Developmental Math – An Open Curriculum
Instructor Guide
Unit 19, Lesson 2, Topic 2: Graphing the Sine and Cosine Functions
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
Grade: 9-12 - Adopted 2010
CCSS.Math.Content.HSF
Functions
CCSS.Math.Content.HSF-IF
Interpreting Functions
CCSS.Math.Content.HSF-IF.C
Analyze functions using different representations.
EXPECTATION
CCSS.Math.Content.HSFIF.C.7
GRADE
EXPECTATION
CCSS.Math.Content.HSFIF.C.7e
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSF
Graph functions expressed symbolically and show
key features of the graph, by hand in simple cases
and using technology for more complicated
cases.
Graph exponential and logarithmic functions,
showing intercepts and end behavior, and
trigonometric functions, showing period, midline,
and amplitude.
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.A
CCSS.Math.Content.HSFTF.A.2
EXPECTATION
CCSS.Math.Content.HSFTF.A.3
EXPECTATION
CCSS.Math.Content.HSFTF.A.4
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSF
Extend the domain of trigonometric functions
using the unit circle.
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.
(+) Use special triangles to determine
geometrically the values 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 π-x, π+x, and 2π-x in terms of their values for
x, where x is any real number.
(+) Use the unit circle to explain symmetry (odd
and even) and periodicity of trigonometric
functions.
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.B
CCSS.Math.Content.HSFTF.B.5
Model periodic phenomena with trigonometric
functions.
Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency,
and midline.
EXPECTATION
EXPECTATION
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Developmental Math – An Open Curriculum
Instructor Guide
Unit 19, Lesson 2, Topic 3: Amplitude and Period
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
Grade: 9-12 - Adopted 2010
CCSS.Math.Content.HSF
Functions
CCSS.Math.Content.HSF-IF
Interpreting Functions
CCSS.Math.Content.HSF-IF.C
Analyze functions using different representations.
EXPECTATION
CCSS.Math.Content.HSFIF.C.7
GRADE
EXPECTATION
CCSS.Math.Content.HSFIF.C.7e
STRAND /
DOMAIN
CATEGORY /
CLUSTER
STANDARD
CCSS.Math.Content.HSF
Graph functions expressed symbolically and show
key features of the graph, by hand in simple cases
and using technology for more complicated
cases.
Graph exponential and logarithmic functions,
showing intercepts and end behavior, and
trigonometric functions, showing period, midline,
and amplitude.
Functions
CCSS.Math.Content.HSF-TF
Trigonometric Functions
CCSS.Math.Content.HSFTF.B
CCSS.Math.Content.HSFTF.B.5
Model periodic phenomena with trigonometric
functions.
Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency,
and midline.
EXPECTATION
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1.22