New Paltz Central School District Algebra 2 and Trigonometry

New Paltz Central School District Algebra 2 and Trigonometry
New Paltz Central School District
Algebra 2 and Trigonometry
Algebra is more than a set of procedures for manipulating symbols. Algebra provides a way
to explore, analyze, and represent mathematical concepts and ideas. It can describe
relationships that are purely mathematical or ones that arise in real-world phenomena and
are modeled by algebraic expressions. Learning algebra helps students make connections in
varied mathematical representations, mathematics topics, and disciplines that rely on
mathematical relationships. Algebra offers a way to generalize mathematical ideas and
relationships, which apply to a wide variety of mathematical and nonmathematical settings.
NCTM, Guiding Principles for Mathematics Curriculum and Assessment
Our goal is to use varying teaching/learning strategies in order to meet the needs of all
the students and the demands of the content. These strategies include, but are not
limited to, the following:
Give students a new type of problem and have students arrive at solutions individually or in
groups. Then share with group to collect all the different ways to solve a problem.
Present a new problem and think, pair, share.
Give students a new type of problem together with a worked out solution and have students
discover and explain, in writing and verbally, how and why the solution works.
Direct instruction – Typically direct instruction will follow some exploratory time for
students to play around with a new type of problem/situation/scenario. Students’
brainstorming will be the start of direct instruction, with notes and examples and information
that help students make sense of the new problem and place it in the context of prior
knowledge.
Have students analyze a new problem: what about it looks familiar, what about it looks new,
how could they start the problem or, if they can’t start, what might be involved while
attacking the problem. Students share ideas in writing and verbally.
Have students use technology (graphing calculators, Geometer’s Sketchpad, Graphmatica,
etc) to explore functions and mathematical concepts.
Have students reflect on their learning in writing and verbally. A regular class wrap up will
include asking students to write what they learned in the day’s work, what questions they still
have, what it reminds them of from past work, and other associations they have with the new
material.
Expose students to complex problems that involve many concepts and lend themselves to a
variety of solutions and strategies. These could be problems that take anywhere from 15
minutes to an hour to multiple days to solve.
8/09
Instructional goals
Nurture an appreciation for the distinct nature of mathematics as an abstract language system
that is internally consistent and understood through rigorous analytical thinking skills.
Nurture an appreciation for how the analytical thinking and problem solving skills honed in
mathematics is essential for students’ current and future lives regardless of whether they
choose a mathematical or scientific field.
Wherever possible, tie the mathematical content to other fields such as economics, literature,
all the sciences, psychology, politics, etc., so that students can see the relevance and use of
mathematics in other contexts.
Nurture numeracy and statistical savvy so that students may be critical consumers of
statistical information in their current and future lives.
A constant goal is to achieve depth of understanding and connection, despite what we
consider to be a much too full list of topics prescribed by the State of New York.
Nurture mathematical reasoning and analytical skills and the ways to express one’s reasoning,
both verbally and in writing. We want to encourage students to look for and recognize
patterns, internal structure, regularities or irregularities both in “real-world” problems and in
the symbolic language of mathematics. We want students to see when patterns are
meaningful as opposed to when they are by chance or accidental. We want students to justify
their solutions and to see why those solutions make sense.
Assessment
We plan to use both formal and informal assessments to ascertain understanding.
Assessments will also be both formative and summative.
Projects – research and writing projects, statistics projects that involve gathering and
analyzing data, solving and explaining solutions to complex, multi-faceted problems
Tests and quizzes
Group work – group work allows the teacher to circulate and listen in, thus giving the teacher
an idea of student understanding and misconceptions.
Written descriptions of solutions to problems – students will be asked to describe their
process for solving a particular problem in writing, which will give the teacher an insight into
student understanding of the method being assessed.
Homework
We hope to train students to make homework a productive, reflective process. Homework is
a time to practice problem solving skills and thinking processes. By providing solutions, we
hope to encourage students to check their own work and work independently to find their
own mistakes and identify any misunderstandings or gaps in knowledge.
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Topics
Unit 1: Relations and Functions
Unit 2: Polynomials and Quadratics
Unit 3: Complex Numbers
Unit 4: Rational Expressions and Equations
Unit 5: Exponents and Radicals
Unit 6: Exponential Functions and Equations
Unit 7: Logarithms
Unit 8: Sequence and Series
Unit 9: Statistics
Unit 10: Probability
Unit 11: Trigonometry – Six Trig Functions
Unit 12: Trigonometry Equations, Identities, and Radians
Unit 13: Trigonometry Graphs
Unit 14: Trigonometry of Non-Right Triangles
Sample problems can be found for each performance indicator in the NYSED
Algebra 2 and Trigonometry Curriculum
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 1: Relations and Functions
Essential Questions:
1. What is a relation and what is a function?
2. How can functions be represented numerically, graphically, algebraically,
and verbally?
3. How can we use different types of functions to model real-world
situations?
4. What effect do transformations have on functions?
Time
Perform
Ind
Content
Sept.
(3 weeks)
A2.A.37
Define a relation and function
A2.A.38
Determine when a relation is a function
A2.A.52
Identify relations and functions, using
graphs
A2.A.51
Determine the domain and range of a
function from its graph
A2.A.39
Determine the domain and range of a
function from its equation
A2.A.40
Write functions in functional notation
A2.A.41
Use functional notation to evaluate
functions for given values in the domain
A2.A.42
Find the composition of functions
A2.A.43
Determine if a function is one-to-one,
onto, or both
A2.A.44
Define the inverse of a function
A2.A.45
Determine the inverse of a function and
use composition to justify the result
A2.A.46
Perform transformations with functions
and relations: f ( x + a ) , f ( x )+ a ,
f (− x) , − f (x ) , af (x)
Lessons
Vocabulary
lesson 1
• introduce and discuss
types of functions, their
graphs, equations, tables,
applications that call for
different functions, etc:
linear, absolute value,
quadratic, power,
exponential, logarithmic,
trigonometric, polynomial
• define relation and
function
• when is a relation a
function
• identify relations and
functions using graphs,
tables, words, mappings,
algebraic expressions
lesson 2
• function notation (f(x) and
set builder notation)
• domain and range (using
graphs, equations, tables,
mappings, words)
• onto
• evaluate functions
lesson 3
• compositions
• inverses
• proving inverses using
composition
• when is an inverse also a
function: one-to-one
lesson 4
• linear functions – direct
variation, applications,
graphs, given f(x) find x.
• solve absolute value
equations and inequalities
relation
function
domain
range
composition
inverse
onto
one to one
linear
absolute value
quadratic
polynomial
power
exponential
logarithmic
trigonometry
transformation
mapping
direct variation
8/09
A2.A-47
Determine the center-radius form for the
equation of a circle in standard form
A2.A.48
Write the equation of a circle, given its
center and a point on the circle
A2.A.49
Write the equation of a circle from its
graph
A2.A.1
Solve absolute value equations and
inequalities involving expressions in one
variable
lesson 5
• transformations of
functions and relations
f ( x + a ) , f ( x )+ a ,
f ( − x) , − f (x ) ,
af (x)
lesson 6
• equation of a circle: use
knowledge of
transformations to justify
equation of a circle
• transformations with
circles
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 2: Polynomials and Quadratics
Essential Questions:
1. What are polynomial equations and quadratic equations and how can we
find roots?
2. How can real-world situations be modeled by quadratics and higher order
polynomials?
3. What does it mean to solve a system of equations?
Time
Sept. – Oct.
(4 weeks)
Perform
Ind
A2.N.3
A2.A.7
A2.A.25
A2.A.24
A2.A.20
A2.A.21
A2.A.4
A2.A.26
A2.A.50
A2.A.3
Content
Lessons
Perform arithmetic operations with
polynomial expressions containing
rational coefficients
Factor polynomial expressions
completely, using any combination of the
following techniques: common factor
extraction, difference of two perfect
squares, quadratic trinomials
Solve quadratic equations, using the
quadratic formula
Know and apply the technique of
completing the square
Determine the sum and product of the
roots of a quadratic equation by examining
its coefficients
Determine the quadratic equation, given
the sum and product of its roots
Solve quadratic inequalities in one and
two variables, algebraically and
graphically
Find the solution to polynomial equations
of higher degree that can be solved using
factoring and/or the quadratic formula
Approximate the solution to polynomial
equations of higher degree by inspecting
the graph
Solve systems of equations involving one
linear equation and one quadratic equation
algebraically Note: This includes rational
equations that result in linear equations
with extraneous roots.
lesson 1
• perform four basic
operations on polynomial
expressions
• factor polynomials
lesson 2
• solve quadratic equations
by factoring and graphing
• completing the square and
its applications (prove
quadratic formula, rewrite
circle equations, solve
quadratics)
• quadratic formula
lesson 3
• quadratic applications and
graphing calculator usage
lesson 4
• write the equation of a
quadratic given roots: sum
and product of roots
lesson 5
• factor and solve
polynomials of higher
degree using factoring and
quadratic formula
• approximate solutions to
polynomials graphically
lesson 6
• quadratic inequalities with
applications
• absolute value inequalities
with applications
lesson 7
• systems of equations:
linear-quadratic
(extraneous solutions)
Vocabulary
monomial
polynomial
roots/zeros
factor
quadratic
radicals
inequalities
systems
rational
irrational
extraneous
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 3: Complex Numbers
Essential Questions:
1. What are imaginary and complex numbers?
2. How can you analyze a quadratic to determine the nature of the roots?
Time
Perform
Ind
Content
Lessons
A2.A.2
Oct. –
Nov.
(2 weeks)
Use the discriminate to determine the
nature of the roots of a quadratic equation
A2.N.6
Write square roots of negative numbers in
terms of i
A2.N.7
Simplify powers of i
A2.N.9
Perform arithmetic operations on complex
numbers and write the answer in the form
a + bi Note: This includes simplifying
expressions with complex denominators.
Determine the conjugate of a complex
number
lesson 1
• determine nature of roots
of quadratic using graphs
and discriminant
• determine missing
coefficient based on
nature of roots or a given
root.
lesson 2
• what is i
• simplify radicals in
terms of i
• solve quadratics with
imaginary roots
lesson 3
• powers of i (cycle of 4)
• basic operations (+, -, x)
with complex numbers
• conjugate
• dividing complex
numbers (rationalize
denominator)
lesson 4
• find quadratic equation
given complex roots
A2.N.8
Vocabulary
discriminant
complex numbers
nature of the roots
conjugate
rationalize
radical
imaginary numbers
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 4: Rational Expressions and Equations
Essential Questions:
1. What are rational expressions and equations and what are the different
ways to solve rational equations?
2. What is inverse variation and what real-world situations can be modeled by
inverse variation?
Time
Nov.
(1-2
weeks)
Perform
Ind
A2.A.16
A2.A.17
Content
Perform arithmetic operations with
rational expressions and rename to
lowest terms
Simplify complex fractional
expressions
A2.A.23
Solve rational equations and
inequalities
A2.A.5
Use direct and inverse variation to
solve for unknown values
Lessons
lesson 1/2
• simplify rational
expressions
• four basic operations
with rational
expressions
• simplify complex
fractional expressions
• solve rational
equations and
inequalities
lesson 3
• inverse variation:
compare to direct
variation, applications,
graphs, solving for
unknown value
Vocabulary
rational expressions
rational equations
complex fractions
inverse variation
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 5: Exponents and Radicals
Essential Questions:
1. How do different numerical exponents affect a base?
2. How can expressions be re-written using exponents or radicals?
3. What situations lend themselves to being expressed by equations with
exponents or radical equations?
Time
Nov. –
Dec.
(2 weeks)
Perform
Ind
A2.A.9
A2.A.8
A2.N.1
A2.A.10
A2.A.11
A2.A.22
Content
Lessons
Vocabulary
Rewrite algebraic expressions that contain
negative exponents using only positive
exponents
Apply the rules of exponents to simplify
expressions involving negative and/or
fractional exponents
Evaluate numerical expressions with
negative and/or fractional exponents,
without the aid of a calculator (when the
answers are rational numbers)
Rewrite algebraic expressions with
fractional exponents as radical expressions
lesson 1
• work with zero,
negative, fractional
exponents: use rule
of exponents, rewrite expressions
with positive
exponents, evaluate
numerical
expressions
without a
calculator, rewrite
expressions with
fractional
exponents as
radicals and vice
versa
lesson 2
• solve equations
with integral
exponents,
fractional
exponents and/or
radicals
lesson 3
• simplify radical
expressions (nth
root)
• basic operations (+,
-, x) with radicals
(simplify nth roots
with variables as
radicands)
• division of radicals
(rationalize
denominator with
conjugates)
base
exponent
radical
radicand
nth root
conjugate
rationalize denominator
irrational number
rational expression
Rewrite algebraic expressions in radical
form as expressions with fractional
exponents
Solve radical equations
A2.N.5
Rationalize a denominator containing a
radical expression
A2.A.15
Rationalize denominators involving
algebraic radical expressions
A2.A.13
Simplify radical expressions
A2.N.4
Perform arithmetic operations on irrational
expressions
A2.A.14
Perform addition, subtraction,
multiplication, and division of radical
expressions
Perform arithmetic operations (addition,
subtraction, multiplication, division) with
expressions containing irrational numbers
in radical form
A2.N.2
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 6: Exponential Functions and Equations
Essential Questions:
1. What is an exponential expression, equation, or function?
2. What situations can be modeled by exponential functions?
3. What is e and how is it useful in modeling natural growth or decay?
Time
Perform
Ind
Content
Lessons
Dec.
(2 weeks)
A2.A.53
Graph exponential functions of the
form y = b x for positive values of b,
including b = e
Solve exponential equations with and
without common bases
lesson 1
• analyze graphs of
exponential functions
and compare to other
functions
• analyze exponential
growth and decay
graphs and the
situations they
describe
lesson 2
• solve exponential
equations using the
method of finding
common bases
lesson 3
• solve application
problems both
algebraically and
graphically
A2.A.27
A2.A.6
Solve an application which results in
an exponential function
Vocabulary
exponential
e
growth
decay
compound interest
percent increase
percent decrease
asymptote
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 7: Logarithms
Essential Questions:
1. What is a logarithm?
2. How can logarithms be used to solve exponential equations?
3. What real world situations can be modeled by logarithmic functions?
Time
Jan.
(2 weeks)
Perform
Ind
Content
Lessons
A2.A.18
Evaluate logarithmic expressions in
any base
A2.A.54
Graph logarithmic functions, using
the inverse of the related exponential
function
Apply the properties of logarithms to
rewrite logarithmic expressions in
equivalent forms
Solve exponential equations with and
without common bases
lesson 1
• graph the inverse of an
exponential function and
discuss the resulting
function and its equation
• what is a log
• log B N = E , B E = N
• solve simple log equations
by rewriting as an
exponential equation
• common logs, natural logs
lesson 2/3
• explore and apply the
properties of logs to rewrite
expressions
• evaluate expressions with
base e
lesson 4
• solve log equations using
the properties of logs
• solve exponential equations
using logs
A2.A.19
A2.A.27
A2.A.28
Solve a logarithmic equation by
rewriting as an exponential equation
A2.A.12
Evaluate exponential expressions,
including those with base e
Vocabulary
logarithm
base
exponent
inverse function
common log
natural log
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 8: Sequence and Series
Essential Questions:
1. What is the difference between a series and a sequence?
2. What is the difference between an arithmetic and geometric
series/sequence?
3. How can we derive the formula for any series?
4. What real-world situations can be modeled by a sequence or series?
Time
Perform
Ind
Feb.
(2 weeks)
A2.N.10
Know and apply sigma notation
A2.A.34
Represent the sum of a series, using sigma
notation
A2.A.29
Identify an arithmetic or geometric
sequence and find the
formula for its nth term
Determine the common difference in an
arithmetic sequence
A2.A.30
Content
A2.A.31
Determine the common ratio in a
geometric sequence
A2.A.33
Specify terms of a sequence, given its
recursive definition
A2.A.32
Determine a specified term of an
arithmetic or geometric sequence
A2.A.35
Determine the sum of the first n terms of
an arithmetic or geometric series
Lessons
Vocabulary
lesson 1
• what are sequences and
series
• sigma notation (forwards and
backwards)
• identify sequences as being
arithmetic or geometric
• find formula for nth term of
sequence
lesson 2
• find common difference and
common ratio
• determine a specified term of
a sequence
lesson 3
• recursive definition
• find sum of series
series
sequence
sigma
arithmetic
geometric
recursion
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 9: Statistics
Essential Questions:
1. What are the different statistical tools that can be used to collect and
analyze data?
2. What are some valid ways to use statistics and what are some non-valid
ways to use statistics?
3. How is the normal distribution curve used as a predictor of outcomes?
Time
Feb. –
March
(2 weeks)
Perform
Ind
A2.S.1
A2.S.2
A2.S.6
A2.S.8
A2.S.7
A2.S.3
A2.S.4
A2.S.5
A2.S.16
Content
Lessons
Understand the differences among
various kinds of
studies (e.g., survey, observation,
controlled experiment)
Determine factors which may
affect the outcome of a survey
lesson 1
• discuss and experience
data collection methods
(one-variable stats)
• calculate measures of
central tendency with
data sets including
frequency distributions
• discuss and calculate
measures of dispersion
(range, variance,
standard deviation for
populations and
samples).
lesson 2
• discuss normal
distribution and
determine whether data
is normally distributed
• analyze normal
distributions using the
bell curve.
lesson 3
• create and analyze
scatterplots for twovariable statistics
datasets
• determine the
regressions for all types
of two-variable datasets
• for linear regressions,
determine strength of
the relationship using
the correlation
coefficient
• use regressions to
interpolate and
extrapolate
Determine from a scatter plot
whether a linear, logarithmic,
exponential, or power regression
model is most appropriate
Interpret within the linear
regression model the value of the
correlation coefficient as a measure
of the strength of the relationship
Determine the function for the
regression model, using
appropriate technology, and use
the regression function to
interpolate and extrapolate from
the data
Calculate measures of central
tendency with group
frequency distributions
Calculate measures of dispersion
(range, quartiles, interquartile
range, standard deviation,
variance) for both samples and
populations
Know and apply the characteristics
of the normal distribution
Use the normal distribution as an
approximation for binomial
probabilities
Vocabulary
correlation
correlations coefficient
normal distribution
standard deviation
variance
regression
central tendency
mean, median, mode
scatterplot
interpolate
extrapolate
measures of dispersion
frequency
outlier
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 10: Probability
Essential Questions:
1. What is the difference between empirical probability and theoretical
probability?
2. What is binomial probability and for what situations is the binomial
probability formula useful?
3. How can the number of elements in a sample space be generated using
permutations, combinations, and the Fundamental Principle of Counting?
Time
March
(2 weeks)
Perform
Ind
Content
Lessons
Vocabulary
A2.S.14
Calculate empirical probabilities
A2.S.13
Calculate theoretical probabilities,
including geometric applications
Calculate the number of possible
permutations ( n Pr ) of n items
taken r at a time
Calculate the number of possible
combinations ( n C r ) of n items
taken r at a time
Differentiate between situations
requiring permutations and those
requiring combinations
Use permutations, combinations,
and the Fundamental Principle of
Counting to determine the number
of elements in a sample space and
a specific subset (event)
Know and apply the binomial
probability formula to events
involving the terms exactly, at
least, and at most
Apply the binomial theorem to
expand a binomial and determine
a specific term of a binomial
expansion
lesson 1
• determine the number of
elements in a sample space
using combinations,
permutations, and the
Fundamental Principle of
Counting.
• compare and contrast
empirical probability and
theoretical probability
• calculate single-event
empirical probabilities and
theoretical probabilities
lesson 2
• discuss binomial probability
and generate/derive the
binomial probability
formula
• solve exactly, at least, and
at most problems using
binomial probability
formula
lesson 3
• investigate expanding
binomials and
generate/derive the
binomial theorem using
Pascal’s Triangle and
combinations
• discuss relationship
between binomial theorem
and binomial probability
empirical probability
theoretical probability
combinations
permutations
binomial probability
binomial expansion
Pascal’s Triangle
sample space
at least, at most
A2.S.10
A2.S.11
A2.S.9
A2.S.12
A2.S.15
A2.A.36
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 11: Trigonometry – Six Trig Functions
Essential Questions:
1. What are the historical and current uses of trigonometry?
2. How are angles and trig ratios represented in the x-y coordinate plane?
3. How can we use our knowledge of special triangles to find exact values of
trig functions?
Time
MarchApril
(2 weeks)
Perform
Ind
Content
Lessons
A2.A.66
Determine the trigonometric functions
of any angle, using technology
Sketch the unit circle and represent
angles in standard position
Find the value of trigonometric
functions, if given a point on the
terminal side of angle θ
Sketch and use the reference angle for
angles in standard position
Know the exact and approximate
values of the sine, cosine, and tangent
of 0º, 30º, 45º, 60º, 90º, 180º, and 270º
angles
Express and apply the six trigonometric
functions as ratios of the sides of a
right triangle
Know and apply the co-function and
reciprocal relationships between
trigonometric ratios
Use the reciprocal and co-function
relationships to find the value of the
secant, cosecant, and cotangent of 0º,
30º, 45º, 60º, 90º, 180º, and 270º angles
lesson 1
• angles as rotations,
terminal side, coterminal
angles
• standard position and
reference triangles
• revisit SOHCAHTOA in
the context of the first
quadrant
• find value of cosine,
sine, tangent given a
point on the terminal
side in any quadrants
lesson 2
• find exact values of sine,
cosine, tangent using
special triangles or unit
circle
lesson 3
• introduce reciprocal trig
functions
• find exact values of six
trig functions
• explore the co-function
relationships
A2.A.60
A2.A.62
A2.A.57
A2.A.56
A2.A.55
A2.A.58
A2.A.59
Vocabulary
cosine
sine
tangent
cosecant
secant
cotangent
standard position
terminal side
coterminal angle
reference angle
reference triangle
unit circle
complementary
co-function
reciprocal function
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 12: Trigonometry Equations, Identities, and Radians
Essential Questions:
1. What are radians and why are they used in mathematics and science?
2. Which situations call for trigonometric equations and how are these
equations solved?
3. How are the trigonometric identities useful?
Time
Perform
Ind
Content
Lessons
Vocabulary
April
(3 weeks)
A2.A.64
Use inverse functions to find the
measure of an angle, given its sine,
cosine, or tangent
Solve trigonometric equations for all
values of the variable from 0º to 360º
lesson 1
• revisit when and how to use
inverse trig functions
• define what a trig equation
is and explore how to solve
linear trig equations
lesson 2
• solve quadratic trig
equations, with the same
trig function
• introduce and prove
Pythagorean identities
• solve quadratic trig
equations algebraically and
graphically using
Pythagorean identities and
double angle identities
lesson 3
• application problems with
mixed trig equations
• become familiar with trig
identities: sum and
difference of angles, half
angles, double angles (from
reference sheet)
lesson 4
• introduce radians
• convert between radians
and degrees
• explore the relationship
between the length of an
arc of a circle, its radius,
and its central angle
• application problems with
radian
identities
inverse
arc length
sector
double angle
half angle
sum of angles
difference of angles
A2.A.68
A2.A.67
Justify the Pythagorean identities
A2.M.1
Define radian measure
A2.M.2
Convert between radian and degree
measures
Determine the length of an arc of a
circle, given its radius and the
measure of its central angle
Apply the angle sum and difference
formulas for trigonometric functions
Apply the double-angle and halfangle formulas for trigonometric
functions
A2.A.61
A2.A.76
A2.A.77
S = θr
lesson 5
• solve mixed trig equations
and applications using
radians
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 13: Trigonometry Graphs
Essential Questions for this unit:
1. What is unique about sinusoidal and trigonometric curves?
2. How does changing the equation of a trig function affect the graph of the
function?
3. What situations can be modeled using trigonometric graphs and
functions?
Time
May
(2 weeks)
Perform
Ind
A2.A.70
A2.A.69
A2.A.72
A2.A.65
A2.A.63
A2.A.71
Content
Lessons
Vocabulary
Sketch and recognize one cycle of a
function of the form y = A sin Bx
or y = A cos Bx
Determine amplitude, period,
frequency, and phase shift, given the
graph or equation of a periodic
function
Write the trigonometric function that
is represented by a given periodic
graph
Sketch the graph of the inverses of
the sine, cosine, and tangent
functions
Restrict the domain of the sine,
cosine, and tangent
functions to ensure the existence of
an inverse function
Sketch and recognize the graphs of
the functions y = sec(x ) ,
y = csc(x) , y = tan(x ) , and
lesson 1
• explore sine and
cosine curves and
discover how
changes in the
equation affect the
graph (amplitude,
frequency, vertical
shift, phase shift)
• unwrap the unit
circle to create a sine
and cosine curve
• find equation given
graph, sketch graph
given equation
lesson 2
• explore tangent
curve by unwrapping
the unit circle
• explore reciprocal
trig graphs
• find equation given
graph, sketch graph
given equation
lesson 3
• application
problems, solved
graphically and
algebraically
lesson 4
• graphs of inverse trig
functions
• explore how to
restrict domains to
make an inverse
relation a function
amplitude
frequency
phase shift
period
sinusoidal
restricted domain
y = cot(x)
8/09
New Paltz Central School District
Algebra 2 and Trigonometry
Unit 14: Trigonometry of Non-Right Triangles
Essential Questions:
1. How can you use trigonometry in non-right triangles?
2. What situations can be described by non-right triangles?
3. What is ambiguous about the “ambiguous case”?
Time
May –
June
(2 weeks)
Perform
Ind
A2.A.73
A2.A.74
A2.A.75
Content
Lessons
Solve for an unknown side or angle,
using the Law of Sines or the Law of
Cosines
Determine the area of a triangle or a
parallelogram, given the measure of
two sides and the included angle
Determine the solution(s) from the
SSA situation (ambiguous case)
lesson 1
• discuss the
applications of trig
in non-right
triangles
• use law of sines to
find missing sides
and angles
• use law of cosines
to find missing
sides and angles
lesson 2
• applications of law
of sines and law of
cosines
lesson 3
• explore and prove
the area of a
triangle formula
• find area of
triangles and
parallelograms
• applications
lesson 4
• the ambiguous
case
Vocabulary
law of sines
law of cosines
ambiguous case
area
8/09
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