Download Trigonometry - Dinwiddie County Public Schools

Trigonometry
Math Curriculum Guide
Dinwiddie County Public Schools provides each
student the opportunity to become a productive citizen,
engaging the entire community in the educational
needs of our children.
Trigonometry Curriculum Guide
● The DCPS Curriculum Guide contains key concepts and SOL numbers for each week. These skill areas must be cross referenced with the
DOE Enhanced Scope and Sequence and DOE Curriculum Framework.
● Grade Level(s): 10 – 12
● Prerequisite: Algebra II AL (preferred) with a “C” or higher
● Course Description: This course is an academic college preparatory course designed to prepare students for college Algebra and
Trigonometry. The course covers the topics of trigonometric and inverse trigonometric functions, their graphs and equations, and their
applications in fields such as navigation and surveying.
Virginia Department of Education Trigonometry Standards
Virginia Department of Education Trigonometry Curriculum Framework
2
Introduction
The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are
measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists
teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This
Guide delineates in greater specificity the content that all teachers should teach and all students should learn.
The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective.
The Curriculum Guide
​
is divided into
​ sections: ​Curriculum Information, ​Essential
​
Knowledge and Skills, ​Key Vocabulary, ​Essential Questions and Understandings, ​Teacher
​ Notes and Elaborations, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below.
Curriculum Information:
This section includes the objective and SOL Reporting Category, focus or topic, and in some, not all, foundational objectives that are being built upon. (Taken from the Curriculum
Framework)
Essential Knowledge and Skills:
Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that
limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective.
(Taken from the Curriculum Framework)
Cognitive Level:
Blooms Taxonomy: What students must be able to do with what they know.
Key Vocabulary:
This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills.
Essential Questions and Understandings:
This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives.
Teacher Notes and Elaborations:
This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers’ knowledge of the
objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning.
Resources:
3
This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources.
Sample Instructional Strategies and Activities:
This section lists ideas and suggestions that teachers may use when planning instruction.
The following chart is the pacing guide for the Dinwiddie County Public Schools Trigonometry Curriculum. The chart outlines the order in which the objectives should be taught;
provides the suggested number blocks to teach each unit; and organizes the objectives into Units of Study. The Dinwiddie County Public Schools cross-content vocabulary terms
that are in this course are: ​analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize.
The first unit contains review concepts found in Algebra II. For information relating to the Algebra II Standards of Learning, refer to the Algebra II Curriculum Guide for
Dinwiddie County Public Schools.
4
Nine Weeks
Approximate
Number of Days
Topic
Targeted SOL
AII.1 - The student, given rational,
radical, or polynomial
expressions, will
d) factor polynomials completely.
Algebra II Review (Unit Test/Pre-Test)
1​st
10
●
●
●
●
●
Linear/Literal Equations
Quadratic Equations (Factorable Only)
Radical Equations
Rationalizing the denominator
Simplifying Radical Expressions
AII.4 - The student will solve, algebraically and
graphically,
a) absolute value equations and
inequalities;
b) quadratic equations over the set of
complex numbers;
c) equations containing rational algebraic
expressions; and
d) equations containing radical expressions.
Graphing calculators will be used for solving
and for confirming the algebraic solutions.
5
Trig Basics (Unit Test)
1​st
10
●
●
●
●
Degrees ----Radians
DMS ---- DD
Graphing/Terminal/Initial/Co-Terminal
Arc Length/Sector Area
Right Triangle Trig (Unit Test)
1
9
● Pythagorean Theorem Review
● Sine/Cosine/Tangent/Secant/Cosecant/
Cotangent
● Application
T.1 - ​The student, given a point other than the
origin on the terminal side of the angle, will
use the definitions of the six trigonometric
functions to find the sine, cosine, tangent,
cotangent, secant, and cosecant of the angle in
standard position. Trigonometric functions
defined on the unit circle will be related to
trigonometric functions defined in right
triangles.
T.2 - ​The student, given the value of one
trigonometric function, will find the values of
the other trigonometric functions, using the
definitions and properties of the trigonometric
functions.
T.2 - ​The student, given the value of one
trigonometric function, will find the values of
the other trigonometric functions, using the
definitions and properties of the trigonometric
functions.
T.3 - ​The student will find, without the aid of
a calculator, the values of the trigonometric
functions of the special angles and their
related angles as found in the unit circle. This
will include converting angle measures from
radians to degrees and vice versa.
6
Oblique Triangles (Unit Test)
1​st
10
●
●
●
●
Law of Sines
Law of Cosines
Area of Triangle with Sines
Application
Unit Circle (Unit Test) (Project)
2​nd
12
●
●
●
●
Special Right Triangles
Deriving Unit Circle/Memorizing
Exact Values
Inverse Trig Functions (Tangent Rule)
Graphs of Trig Functions (Assessment)
2​nd
5
● Sine/Cosine/Tangent Graphs
● Domain and Range of Graphs
● Properties of Graphs
T.9 - ​The student will identify, create,
and solve real-world problems involving
triangles. Techniques will include using
the trigonometric functions, the
Pythagorean Theorem, the Law of Sines,
and the Law of Cosines.
T.4 - ​The student will find, with the aid
of a calculator, the value of any
trigonometric function and inverse
trigonometric function.
T.3 - ​The student will find, without the
aid of a calculator, the values of the
trigonometric functions of the special
angles and their related angles as found
in the unit circle. This will include
converting angle measures from radians
to degrees and vice versa.
T.6
a) The student, given one of the six
trigonometric functions in standard
form, will
b) state the domain and the range of
the function;
7
● Translations based on Equations
Trigonometric Equations (Unit Test)
2​nd
7
● Inverse Trig Equations
● Linear Trig Equations
● Quadratic Trig Equations
Rational Review (Assessment)
2​nd
5
● Adding and Subtracting
● Multiplying
● Compound Fractions (Dividing)
c) determine the amplitude, period,
phase shift, vertical shift, and
asymptotes;
d) sketch the graph of the function by
using transformations for at least a
two-period interval; and
e) investigate the effect of changing
the parameters in a trigonometric
function on the graph of the
function.
T.7 - ​The student will identify the
domain and range of the inverse
trigonometric functions and recognize the
graphs of these functions. Restrictions on
the domains of the inverse trigonometric
functions will be included.
T.8 - ​The student will solve trigonometric
equations that include both infinite
solutions and restricted domain solutions
and solve basic trigonometric
inequalities.
AII/T.1
The student, given rational,
radical, or polynomial
expressions, will
a) add, subtract, multiply, divide, and
simplify rational algebraic
expressions;
b) add, subtract, multiply, divide, and
8
simplify radical expressions
containing rational numbers and
variables, and expressions
containing rational exponents;
c) write radical expressions as
expressions containing rational
exponents and vice versa; and
d) factor polynomials completely.
Trigonometric Identities
2​nd
14
● Sum/Difference (Optional)
● sin2 θ + cos2 θ = 1 (Pythagorean
Identities)
● Simplifying Expressions
● Proving Trig Identities
T.5 - ​The student will verify basic
trigonometric identities and make
substitutions, using the basic identities.
9
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Triangular and Circular
Trigonometric Functions
Virginia Standard T.1
The student, given a point other
than the origin on the terminal side
of the angle, will use the
definitions of the six trigonometric
functions to find the sine, cosine,
tangent, cotangent, secant, and
cosecant of the angle in standard
position. Trigonometric functions
defined on the unit circle will be
related to trigonometric functions
defined in right triangles.
Essential Knowledge and Skills
Key Vocabulary
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaboration
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:

Define the six triangular trigonometric
functions of an angle in a right
triangle.

Define the six circular trigonometric
functions of an angle in standard
position.

Make the connection between the
triangular and circular trigonometric
functions.

Recognize and draw an angle in
standard position.

Show how a point on the terminal side
of an angle determines a reference
triangle.
Essential Questions

What is the standard position of an angle?

Given a point on the terminal side of an angle, how are the values of the six trigonometric functions
determined?

What is the relationship between trigonometric and circular functions?
Key Vocabulary
Circular trigonometric function
Degrees
Initial side
Radians
Reference triangle
Terminal side
Triangular trigonometric function
Unit circle
The six trigonometric functions of an angle θ are called sine, cosine, tangent, cotangent, secant and cosecant.
The functions are defined with the angle θ (the Greek letter theta) in standard position.
Essential Understandings

Triangular trigonometric function definitions are related to circular trigonometric function definitions.

Both degrees and radians are units for measure angles.

Drawing an angle in standard position will force the terminal side to lie in a specific quadrant.

A point on the terminal side of an angle determines a reference triangle from which the values of the six
trigonometric functions may be derived.
Teacher Notes and Elaborations
As derived from the Greek language, the word trigonometry means “measurement of triangles”.
An angle is determined by rotating a ray (half-line) about its endpoint. The starting position of the ray is the
initial side of the angle, and the position after rotation is the terminal side.
In the rectangular coordinate system an angle with its vertex at the origin and with its initial side along the
positive x-axis is in standard position. For any point P(x, y) on the terminal side of an angle θ in standard
position, r is defined as the distance from the vertex to 𝑃 (𝑟 = √𝑥 2 + 𝑦 2 ). A point on the terminal side of an
angle determines a reference triangle from which the values of the six trigonometric functions may be derived.
The six triangular trigonometric functions of θ are:
𝑦
𝑟
𝑥
cos 𝜃 =
𝑟
𝑦
tan 𝜃 =
𝑥
𝑟
csc 𝜃 =
𝑦
𝑟
sec 𝜃 =
𝑥
𝑥
cot 𝜃 =
𝑦
Sin 𝜃 =
10
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Triangular and Circular Trigonometric
Functions
Virginia SOL T.1
The student, given a point other than
the origin on the terminal side of the
angle, will use the definitions of the
six trigonometric functions to find the
sine, cosine, tangent, cotangent,
secant, and cosecant of the angle in
standard position. Trigonometric
functions defined on the unit circle
will be related to trigonometric
functions defined in right triangles.
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaborations.
Teacher Notes and Elaborations (continued)
The properties of the trigonometric functions are connected with the circular function definitions by using a unit circle (a circle with the
radius of one). If the terminal side of an angle θ in standard position intersects the unit circle at P(x, y), then the six circular trigonometric
functions are defined as:
sin 𝜃 = 𝑦
𝑥
tan 𝜃 = 𝑦
cos 𝜃 = 𝑥
𝑦
cot 𝜃 = 𝑥
csc 𝜃 = 𝑦
sec 𝜃 = 𝑥
1
1
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. Degrees and radians are
equivalent units for angle measurement. One radian is the measure of a central angle 𝜃 that intercepts an arc s equal in length to the radius r
of the circle.
11
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Triangular and Circular
Trigonometric Functions
Virginia Standard T.2
The student, given the value of one
trigonometric function, will find
the values of the other
trigonometric functions, using the
definitions and properties of the
trigonometric functions.
Dinwiddie County Public Schools
Essential Knowledge and Skills
Key Vocabulary
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:

Given one trigonometric function
value, finds the other five
trigonometric function values.

Develop the unit circle, using both
degrees and radians.

Solve problems, using circular
function definitions and the properties
of the unit circle.

Recognize the connections between
the coordinates of points on a unit
circle and
Coordinate geometry;
Cosine and sine values; and
Lengths of sides of special
right triangles.
Key Vocabulary
Degrees
Pythagorean Identities
Radians
Ratio (Quotient Identities)
Reciprocal identities
Unit circle
Essential Questions and Understandings
Teacher Notes and Elaboration
Essential Questions

What are the Pythagorean, ratio, and reciprocal identities?

Given the value of one trigonometric function, how are the remaining functions determined?
Essential Understandings

If one trigonometric function value is known, then a triangle can be formed to use in finding the other five
trigonometric function values.

Knowledge of the unit circle is a useful tool for find all six trigonometric values for special angles.
Teacher Notes and Elaborations
Given the value of one trigonometric function, a triangle can be formed to use in finding the other give
trigonometric function values of the remaining functions may also be found using one of the following methods:
Definitions of the trigonometric functions are:
𝑦
𝑟
Sin 𝜃 =
csc 𝜃 =
cos 𝜃 =
𝑟
𝑥
sec 𝜃 =
𝑟
𝑦
𝑟
𝑥
𝑦
𝑥
cot 𝜃 =
𝑥
𝑦
Relationships between trigonometric functions are identities.
Reciprocal Identities:
𝑦
𝑟
1
1
Since sin 𝜃 = and csc 𝜃 = , then sin 𝜃 =
and csc 𝜃 =
.
tan 𝜃 =
𝑟
𝑦
csc 𝜃
sin 𝜃
Also, cos 𝜃and the sec 𝜃 are reciprocals as are the tan 𝜃 and cot 𝜃. The reciprocal identities hold for any
angle 𝜃 that does not lead to a zero denominator.
Pythagorean Identities:
sin2 𝜃 + cos2 𝜃 = 1
𝑡𝑎𝑛 𝜃 + 1 = sec 2 𝜃
1 + cot 2 𝜃 = csc 2 𝜃
Ratio of Quotient Identities:
sin 𝜃
tan 𝜃 =
cot 𝜃 =
cos 𝜃
cos 𝜃
sin 𝜃
Degrees and radians are equivalent units for angle measurement. A central angle with sides and intercepted arcs
all the same length measures 1 radian.
A unit circle is one that lies on the x-axis, has origin (0, 0), and a radius of 1.
Knowledge of the unit circle is a useful tool for finding all six trigonometric values for special angles.
12
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Triangular and Circular
Trigonometric Functions
Virginia Standard T.3
The student will find, without the
aid of a calculator, the values of the
trigonometric functions of the
special angles and their related
angles as found in the unit circle.
This will include converting angle
measures from radians to degrees
and vice versa.
Dinwiddie County Public Schools
Essential Knowledge and Skills
Key Vocabulary
Essential Questions and Understandings
Teacher Notes and Elaboration
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:
 Find trigonometric function values of
special angles and their related angles
in both degrees and radians.
 Apply the properties of the unit circle
without using a calculator.
 Use a conversion factor to convert
from radians to degrees and vice versa
without using a calculator.
Essential Questions

What is the relationship between radians and degrees?

What is the relationship between families of coterminal angles?

What is meant by the special angles?
Key Vocabulary
Coterminal angles
Degrees
Quadrantal angles
Radians
Revolution
Unit circle.
Teacher Notes and Elaborations
The two most common units used to measure angles are radians and degrees. The radian measure of an angle in
𝑠
standard position is defined as the length of the corresponding arc divided by the radius of the circle (𝜃 = ).
𝑟
One degree is the result from a rotation 1/360 of a complete revolution about the vertex in the positive direction.
A full revolution (Counterclockwise) corresponds to 360°.
Essential Understandings

Special angles are widely used in mathematics.

Unit circle properties will allow special-angle and related-angle trigonometric values to be found without
the aid of a calculator.

Degrees and radians are units of angle measure.

A radian is the measure of the central angle that is determined by an arc whose length is the same as the
radius of the circle.
To convert radians to degrees and vice versa, multiply by the appropriate conversion factor
1° =
𝜋
180°
𝑟𝑎𝑑 and 1 𝑟𝑎𝑑 =
180°
𝜋
Multiples, between 0 and 2𝜋, of first quadrant special angles are found without the aid of a calculator.
Angles that measure greater than 2𝜋 can be formed by adding or subtracting a multiple of 2𝜋 to its co-terminal
angle measuring 0 and 2𝜋.
Two angles in standard position with the same initial and terminal sides are called coterminal angles.
Special angles are widely used in mathematics. The first quadrant special angles of a unit circle (a circle with a
𝜋 𝜋 𝜋
radius of one) are , , . The quadrantal angles (any angle with the terminal side on the x-axis or y-axis) of a
𝜋
6 4 3
3𝜋
unit circle are 0, , 𝜋,
2
2
, 2𝜋.
13
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Inverse Trigonometric Functions
Virginia SOL T.4
The student will find, with the aid
of a calculator, the value of any
trigonometric function and inverse
trigonometric function.
Essential Knowledge and Skills
Key Vocabulary
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:
 Use a calculator to find the
trigonometric function values of any
angle in either degrees or radians.
 Define inverse trigonometric
functions.
 Find angle measures by using the
inverse trigonometric functions when
the trigonometric function values are
given.
Key Vocabulary
Inverse Trigonometric Functions.
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaboration
Essential Questions

What are inverse trigonometric functions?
Essential Understandings

The trigonometric function values of any angle can be found by using a calculator.

The inverse trigonometric functions can be used to find angle measures whose trigonometric function
values are known.

Calculations of inverse trigonometric function values can be related to the triangular definitions of the
trigonometric functions.
Teacher Notes and Elaborations
The values of the trigonometric functions of any angle can be approximated using a calculator. Most values are
approximated to four decimal places. Depending upon the problem, calculators must be in the appropriate
mode, whether radian or degree.
The inverse trigonometric functions can be used to find angle measures whose trigonometric function values are
known. Given the value of any trigonometric function, the angle may be determined by using the appropriate
inverse function key on the calculator. Values of inverse trigonometric functions are always in radians.
Definitions of the Inverse Trigonometric Functions;
14
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Trigonometric Identities
Virginia SOL T.5
The students will verify basic
trigonometric identities and make
substitutions using the basic
identities.
Dinwiddie County Public Schools
Essential Knowledge and Skills
Key Vocabulary
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:
 Use trigonometric identities to make
algebraic substitutions to simplify and
verify trigonometric identities. The
basic trigonometric identities include
-
Reciprocal identities
Pythagorean identities
Sum and difference
identities
Double-angle identities;
and
Key Vocabulary
Identity
Double-angle identities
Pythagorean identities
Reciprocal identities
Sum and difference identities
Trigonometric identities
verify
Essential Questions and Understandings
Teacher Notes and Elaboration
Essential Questions

What is an identity?

What is the difference between solving equations and verifying identities?
Essential Understandings

Trigonometric identities can be used to simplify trigonometric expressions, equations, or identities.

Trigonometric identity substitution can help solve trigonometric equations, verify another identity, or
simplify trigonometric expressions.
Teacher Notes and Elaborations
An identity is an equation that is true for all possible replacements of the variables. An identity involving
trigonometric expressions is a trigonometric identity. Trigonometric identities can be used to simplify
trigonometric expressions, equations, or identities. The fundamental trigonometric identities are the following:
Reciprocal identities
Pythagorean identities
Sum and difference identities
Half angle identities, and
Double angle identities.
Reciprocal Identities:
𝑦
𝑟
1
1
Since sin 𝜃 = and csc 𝜃 = , then sin 𝜃 =
and csc 𝜃 =
.
𝑟
𝑦
csc 𝜃
sin 𝜃
Also, cos 𝜃and the sec 𝜃 are reciprocals as are the tan 𝜃 and cot 𝜃. The reciprocal identities hold for any
angle 𝜃 that does not lead to a zero denominator.
Pythagorean Identities:
sin2 𝜃 + cos2 𝜃 = 1
𝑡𝑎𝑛 𝜃 + 1 = sec 2 𝜃
1 + cot 2 𝜃 = csc 2 𝜃
Ratio of Quotient Identities:
sin 𝜃
tan 𝜃 =
cot 𝜃 =
cos 𝜃
cos 𝜃
sin 𝜃
Double Angle Identities
sin 2𝑢 = 2 sin 𝑢 cos 𝑢
cos 2𝑢 = cos2 𝑢 − sin2 𝑢
= 2 cos2 𝑢 − 1
= 1 − 2 sin2 𝑢
tan 2𝑢 =
2 tan 𝑢
1−tan2 𝑢
15
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Trigonometric Identities
Virginia SOL T.5
The students will verify basic
trigonometric identities and make
substitutions using the basic identities.
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaborations.
Teacher Notes and Elaborations (continued)
Sum and Difference Identities: sin(𝑢 ± 𝑣) = sin 𝑢 cos 𝑣 ± cos 𝑢 sin 𝑣
cos(𝑢 ± 𝑣) = cos 𝑢 cos 𝑣 ∓ sin 𝑢 sin 𝑣
tan 𝑢 ± tan 𝑣
tan(𝑢 ± 𝑣) =
1 ∓ tan 𝑢 tan 𝑣
To verify a trigonometric identity, either the left or the right side of the equation may be used to deduce the other
side. Verifying identities is not the same as solving equations. Techniques used in solving equations, such as adding
the same terms to both sides, are not valid when working with identities since the statement to be verified may not be
true. To verify an identity, show that one side of the identity can be simplified so that it is identical to the other side.
Guidelines for Verifying Trigonometric Identities
1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.
2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial
denominator.
3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you
want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents.
4. If the preceding guidelines do not help, try converting all terms to sines and cosines.
5. Always try something. Even making an attempt that leads to a dead end provides insight.
6. Try working backwards from the solution, as it can provide great insight.
16
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Trigonometric Equations, Graphs,
and Practical Problems.
Virginia SOL T.6
The student, given one of the six
trigonometric functions in standard
form, will
a. State the domain and range of
the function.
b. Determine the amplitude,
period, phase shift, vertical
shift, and asymptotes.
c. Sketch the graph of the
function by using
transformations for at least a
two period interval; and
d. Investigate the effect of
changing the parameters in a
trigonometric function on the
graph of the function.
Dinwiddie County Public Schools
Essential Knowledge and Skills
Key Vocabulary
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:
 Determine the amplitude, period,
phase shift, and vertical shift of a
trigonometric function from the
equation of the fun and the graph of
the function.
 Describe the effect of changing A, B,
C, or D in the standard form of the
equation.
 State the domain and range of a
function written in standard form.
 Sketch the graph of a function written
in standard form by using
transformations for at least a two
period interval.
Key Vocabulary
Amplitude
Asymptote
Horizontal phase shift
Period of the function
Periodic function
Range
Vertical phase shift.
Essential Questions and Understandings
Teacher Notes and Elaboration
Essential Questions

What effect does the change in A, B, C, and D in standard form have on the graph of the function?

Why are the terms phase shift, period, amplitude, vertical shift, and asymptote important to curve
sketching?
Essential Understandings

The domain and range of a trigonometric function determine the scales of the axes for the graph of each
function.

The amplitude, period, phase shift, and vertical shift are important characteristics of the graph of a
trigonometric function, and each has a specific purpose in applications using trigonometric equations.

The graph of a trigonometric function can be used to display information about the periodic behavior of a
real world situation such as wave motion or the motion of a Ferris wheel.
Teacher Notes and Elaborations
Each of the six trigonometric functions is a periodic function whose graph is based on repetition. A periodic
function is a function f such that f (x) = f (x + p) for every real number x in the domain of f and for some
positive real number p. The smallest possible positive value of p is the period of the function. The period of the
sine, cosine, secant, and cosecant function is 2π. The period of the tangent and cotangent function is π.
The amplitude of a function can be interpreted as half the difference between its maximum and minimum
values. The amplitude is half the range (difference between maximum and minimum values). Suggested five
steps to sketch the parent graph of y = A sin Bx or y = A cos Bx, with 0 B > are:
𝜋
1. Determine the period of repeat, 2(|𝐵|). Start at 0 on the x-axis and mark the distance.
2.
3.
4.
5.
Divide the interval into four equivalent parts.
Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum
points, minimum points, and x intercepts.
Plot those points found in step 3 and join them with the curve.
Draw additional cycles to the left and right of the curve.
Transformations to the original graph can be done through phase shifts. The vertical phase shift moves
the horizontal axis of the graph along the y-axis. The horizontal phase shift moves the graph along the
x-axis.
Steps to sketch the graph of y = A sin(Bx – C) + D or y = A cos(Bx – C) + D, with 0 B > are:
1. Determine D the vertical phase shift. This will be the new horizontal axis at y = D.
2. Determine C the horizontal phase shift. This will lie on the x-axis.
3. Follow steps 1 - 5 above.
17
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Trigonometric Equations, Graphs, and
Practical Problems.
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaborations.
Teacher Notes and Elaborations (continued)
The asymptote is a straight line whose perpendicular distance from a curve decreases to zero as the distance from the origin increases
without limit.
Reciprocal identities are used to obtain the graphs of the secant and cosecant functions. The cosecant and secant functions will have vertical
asymptotes. The asymptotes will have equations of the form x = k , where k is the x-intercept of the sine or cosine function.
Virginia SOL T.6
The student, given one of the six
trigonometric functions in standard
form, will
a. State the domain and range of
the function.
b. Determine the amplitude, period,
phase shift, vertical shift, and
asymptotes.
c. Sketch the graph of the function
by using transformations for at
least a two period interval; and
Investigate the effect of changing the
parameters in a trigonometric function
on the graph of the function.
Sketching the graphs of the variations of the tangent and cotangent is similar to sketching the graphs of the transformations of sine and
cosine functions. Key differences are the period of repeat, asymptotes, and the shape of the graph. Tangent and cotangent graphs do not
have amplitude.
The graphing calculator can provide a visual look at how the constants A, B, C, and D affect the graph of a function. Be sure the calculator
is set for radians. Most calculators have a trig window with domain [-2π, 2π], range [-4, 4], xscl = π, and yscl = 1. Other settings may be
preferable for different equations.
Graphs of trigonometric functions model periodic behavior of real world situations such as wave motion, biorhythms, seasonal
temperatures, or the motion of a Ferris wheel.
18
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Inverse Trigonometric Functions
Virginia SOL T.7
The student will identify the
domain and range of the inverse
trigonometric functions and
recognize the graphs of these
functions. Restrictions on the
domain of the inverse
trigonometric functions will be
included.
Essential Knowledge and Skills
Key Vocabulary
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:

Find the domain and range of the
inverse trigonometric function

Use the restrictions on the domains of
the inverse trigonometric functions in
finding the values of the inverse
trigonometric functions.

Identify the graph of the inverse
trigonometric functions.
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaboration
Essential Questions

What are the domains and ranges of the inverse trigonometric functions?

What are the restrictions on the domain of the inverse trigonometric functions?
Essential Understandings

Restrictions on the domains of some inverse trigonometric functions exist.
Teacher Notes and Elaboration
The trigonometric functions are not one to one, so it is necessary to determine the restrictions on domains to
regions that pass the horizontal line test. The inverse trigonometric functions can be denoted in two ways. For
example, the inverse of y = sin x may be written as y = arcsin x or 𝑦 = sin−1 𝑥.
Key Vocabulary
Inverse trigonometric function
Restricitions on domains.
The graphs of the inverse trigonometric functions are obtained by interchanging the x and y coordinates of the
key points of the basic graphs.
.
19
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Trigonometric Equations, Graphs,
and Practical Problems.
Virginia SOL T.8
The student will solve
trigonometric equations that
include both infinite solutions and
restricted domain solutions and
solve basic trigonometric
inequalities.
Essential Knowledge and Skills
Key Vocabulary
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaboration
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:

Solve trigonometric equations with
restricted domains algebraically and
by using a graphing utility.

Solve trigonometric equations with
infinite solutions algebraically and by
using a graphing utility.

Check for reasonableness of results,
and verify algebraic solutions using a
graphing utility.
Essential Questions

Do trigonometric equations have unique solutions? Why or why not?

What is the relationship of the domain and range to the solution of trigonometric equations?
Key Vocabulary
Trigonometric equation
Trigonometric identities
To solve a trigonometric equation, use standard algebraic techniques and fundamental trigonometric identities.
The fundamental trigonometric identities are the following:
- reciprocal identities,
- Pythagorean identities,
- sum and difference identities,
- half angle identities, and
- double angle identities.
Essential Understandings

Solutions for trigonometric equations will depend on the domains.

A calculator can be used to find the solution of a trigonometric equation as the points of intersection of the
graphs when one side of the equation is entered in the calculator as 𝑌1 and the other side is entered as 𝑌2 .
Teacher Notes and Elaboration
Trigonometric equations, like most algebraic equations, are true for some, but not for all values of the variable.
Trigonometric equations do not have unique solutions. Solutions for trigonometric equations will depend on the
domains. They have infinitely many solutions, differing by the period of the function. If the domain of the
equations is restricted to one revolution then only those solutions between 0 and 2π will be determined.
Standard algebraic techniques are used to solve trigonometric inequalities.
20
Trigonometry CURRICULUM GUIDE (Revised August 2016)
Curriculum Information
Topic
Data Analysis
Virginia SOL T.9
The student will identify, create,
and solve real world problems
involving triangles. Techniques
will include using the
trigonometric functions, the
Pythagorean Theorem, Law of
Sines, and The Law of Cosines.
Essential Knowledge and Skills
Key Vocabulary
The student will use problem solving,
mathematical communication,
mathematical reasoning, connections,
and representations to:

Write a real world problem involving
triangles.

Solve real world problems involving
triangles.

Use the trigonometric functions,
Pythagorean Theorem, Law of Sines,
and Law of Cosines to solve real world
problems.

Use the trigonometric functions to
model real world situations.

Identify a solution technique that could
be used with a given problem.
Key Vocabulary
Directed Line Segment
Law of Cosines
Law of Sines
Magnitudes
Oblique
Pythagorean Theorem
Scalar
Vector
Vector Quantity
Dinwiddie County Public Schools
Essential Questions and Understandings
Teacher Notes and Elaboration
Essential Questions

How are practical problems involving triangles and vectors solved?

What is the relationship of a vector to right triangles and trigonometric functions?

What is meant by an ambiguous case when determining parts of a triangle?
Essential Understandings

A real world problem may be solved by using one of a variety of techniques associated with triangles.
Teacher Notes and Elaboration
Practical problems involving right triangles can be solved by applying the right triangle definitions of
trigonometric functions and the Pythagorean Theorem. Problems involving oblique (non-right) triangles are
solved using the Law of Sines or the Law of Cosines depending upon the given information.
The Law of Sines states that for any triangle with angles of measures A, B, an C, and sides of lengths a, b, and c
(a opposite of angle A, b opposite of angle B, and c opposite of angle C).
sin 𝐴 sin 𝐵 sin 𝐶
=
=
𝒂
𝒃
𝒄
The Law of Cosine states that for any triangle with sides of lengths a, b, and c then
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
To solve an oblique triangle, the measure of at least one side and any two other parts of a triangle need to be
known. This breaks down the following into cases.
Heron’s area formula is used if the lengths of the sides of the triangle are known. If two sides of a triangle and
1
the angle between the two sides are known then the area formula 𝐴 = 𝑏𝑐 sin 𝐴 is used.
2
Many quantities in mathematics involve magnitudes. These quantities are called scalar. Other quantities called
vector quantities, involve both magnitude and direction. A vector quantity is often represented with a directed
line segment, which is called a vector. The length of the vector represents the magnitude of the vector quantity.
Each vector has a horizontal and vertical component. Vectors may be added and subtracted.
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