Trigonometry
PUHSD Curriculum
2014-2015
Unit 1: Right Triangle Trigonometry
Enduring Understandings:
Essential Questions:
The relationship between the sides and angles of right triangles
How are proportions set up for similar polygons?
leads to the exploration of trigonometric functions.
What are the six trigonometric functions for the acute angles in a right triangle?
By using the special relationships of the 30º-60º-90º triangle and the Why are the trigonometric ratios in similar triangles equal?
45º-45º-90º triangle, the meaning of trigonometric functions
What are the ratios of the sides of a 45º-45º-90º triangle and a 30º-60º-90º
becomes clear, and that knowledge leads to understanding how that triangle?
applies to real-world situations.
How is trigonometry used to solve right triangles, including real-world
applications?
Standard *
Learning Targets
Technology Standards
Recognize
and
represent
proportional
relationships
7.RP.2
● I can use similar figures to write a
● Use a calculator to find the sine,
between quantities.
proportion.
cosine, and tangent of an angle.
● I can write a proportion for similar triangles. ● Use reciprocals on a calculator to
Evaluate the six trigonometric functions of the acute
● Given an acute angle in a right triangle, I can find the cosecant, secant, and
angles of a right triangle.
identify the opposite side, the adjacent side
cotangent of an angle.
and the hypotenuse.
● Use technology to investigate the
Understand that by similarity, side ratios in right triangles
G-SRT.6
● I can evaluate the six trigonometric
special triangles. (Cabri Jr., Geo
are properties of the angles in the triangle, leading to
functions of the acute of a right triangle.
Sketchpad, TI-Nspire, Excel)
definitions of trigonometric ratios for acute angles.
●
I
can
explain
why
two
angles
of
equal
● Use SIN-1, COS-1, and TAN-1 to
Solve a 45º-45º-90º triangle or a 30º-60º-90º given the
measure in different size triangles have the
approximate values of inverse
length of one side.
same trigonometric ratios.
trigonometric functions.
Use special triangles to determine geometrically the
F-TF.3
● I can draw and label a 45º-45º-90º special
values of sine, cosine, and tangent for 30º, 45º, and 60º.
triangle.
Key Vocabulary
Use trigonometric ratios and the Pythagorean Theorem to
G-SRT.8
● I can draw and label a 30º-60º-90º special
proportion
solve right triangles in applied problems.
triangle.
acute angle
● I can solve a special triangle.
right triangle
● I can state the values of the trigonometric
trigonometry
functions of the special angles.
trigonometric ratio
● I can find the missing sides of a right triangle opposite
using the trigonometric ratios.
adjacent
● I can find the acute angles in a right triangle hypotenuse
using an inverse trigonometric function.
trigonometric function
● I can solve a real-world problem that can be sine
Page 1 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
modeled with right triangle trigonometry.
cosine
tangent
secant
cosecant
cotangent
special triangles
Pythagorean Theorem
Page 2 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 2: The Unit Circle
Enduring Understandings:
Essential Questions:
Using special right triangles builds understanding of How are positive and negative angles of all sizes represented on a unit circle?
the relationships in the unit circle.
How are the x and y coordinates of a point related to the angles and their trigonometric functions?
The relationships between the four quadrants and
How is a reference angle used to find trigonometric functions in all quadrants?
the six trigonometric functions make the solving of What is radian measure and how do we convert between degrees and radians?
equations clear.
How are “odd,” “even,” and “periodicity” of trigonometric functions explained using the unit
The equation for the circumference of a circle
circle?
makes the relationship between degrees and
How is arc length and sector area found when the radius and central angle (in degrees or radians)
radians clear.
are given?
Standard *
Learning Targets
Technology Standards
Explain
how
the
unit
circle
in
the
coordinate
plane
F-TF.2
● I can draw an angle in standard position (positive ● Use a calculator to convert
enables the extension of trigonometric functions to all
and negative).
degree measures to radian
real numbers, interpreted as angles (expressed in
● I can determine the measures of angles (positive
measures.
degrees) traversed counterclockwise and clockwise
and negative) which are coterminal with a given
● Use a calculator to convert
around the unit circle.
angle.
radian measures to degree
● I can use trigonometric functions to find the
measures
Find the trigonometric functions of an angle in any
coordinates of a point on the terminal side of an
● Evaluate all six trigonometric
quadrant.
angle in standard position.
functions of angles in radian
●
I
can
determine
the
measure
of
the
angle,
given
measure using the calculator in
Use special triangles to determine geometrically the
F-TF.3
radian mode and in degree
values of sine, cosine, tangent for 30º, 45º, and 60º and the coordinates of a point on the terminal side.
use the unit circle to express the values of sine, cosine,
● I can define the trigonometric functions in terms
measure using the calculator in
and tangent for 180º – x, 180º + x and 360º – x in
of x, y, and r.
degree mode.
terms of their values for x, where x is 30º, 45º, or 60º.
● I can evaluate the trigonometric functions of 0º,
● Determine when it is
90º, 180º, 270º, and 360º.
appropriate to use radian or
Solve basic trigonometric equations involving special
● I can label the unit circle in degrees.
degree mode on a calculator.
triangles, such as 2sin   1  0 , where
● I can determine the reference angle for an angle
0º < θ < 360º.
in any quadrant.
Key Vocabulary
● I can determine the sign of the trigonometric
coordinate plane
Understand radian measure of an angle as the length of
F-TF.1
functions
in
each
quadrant.
quadrants
the arc on the unit circle subtended by the angle.
● I can evaluate the six trigonometric functions of
unit circle
any angle.
clockwise
Explain how the unit circle in the coordinate plane
F-TF.2
● I can state the values of the trigonometric
counterclockwise
enables the extension of trigonometric functions to all
real numbers, interpreted as angles (expressed in
functions of the special angles in degrees.
reference angles
Page 3 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
radians) traversed counterclockwise and clockwise
around the unit circle.
Use special triangles to determine geometrically the
F-TF.3
values of sine, cosine, tangent for

3
,

4
, and

6
and use the unit circle to express the values of sine,
cosine, and tangent for   x,   x, and 2  x in
terms of their values for x, where x is

3
,

4
, or

6
.
F-TF.4
Use the unit circle to explain symmetry (odd and even)
and periodicity of trigonometric functions.
G-C.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.
● I can evaluate the six trigonometric functions of
an angle in any quadrant whose reference angle is
30º, 45º, or 60º.
● I can use the unit circle to evaluate the
trigonometric functions of the special angles in any
quadrant.
● I can solve basic trigonometric equations for
missing angles, remembering to find all possible
solutions.
● I can define the meaning of radian measure.
● I can convert between degree and radian
measure with and without technology.
● I can represent radian measure in terms of π.
● I can use decimal representation for radian
measures.
● I can evaluate the six trigonometric functions of
angles in radian measure.
● I can draw an angle in standard position (positive
and negative).
● I can determine the measures of angles (positive
and negative) which are coterminal with a given
angle.
● I can use trigonometric functions to find the
coordinates of a point on the terminal side of an
angle in standard position.
● I can determine the measure of the angle, given
the coordinates of a point on the terminal side.
● I can define the trigonometric functions in terms
of x, y, and r.
● I can evaluate the trigonometric functions of 0,
quadrantal angles
radian
arc
arc length
circumference
subtended
central angle
symmetry
even/odd function
period
periodicity
constant of proportionality
sector
3

,,
, and 2 .
2
2
● I can label the unit circle in degrees.
● I can label the unit circle in radians.
Page 4 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
● I can state the values of the trigonometric
functions of the special angles in radian measure –
without converting the angle measures to degrees
first.
● I can evaluate the six trigonometric functions of
an angle in any quadrant whose reference angle is

3
,

4
, or

6
.
● I can use the unit circle to evaluate the
trigonometric functions of the special angles in any
quadrant.
● I can determine the relationship between the sin
(cos, tan) of θ and the sin (cos, tan) of –θ.
● I can use the periodicity of trigonometric
functions to label the cos and sin of the angles in
the unit circle.
● I can find the length of an arc given the radius
and the measure of the central angle in degrees or
in radians.
● I can find the area of a sector given the measure
of the central angle in degrees or in radians.
Page 5 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 3: Graphs of Trigonometric Functions
Enduring Understandings:
Essential Questions:
The unit circle and relationships on it can be and
What are amplitude, midline, frequency and period in relation to graphs of trigonometric functions?
are translated to the coordinate plane.
Which of the six trigonometric functions have asymptotes and where do they occur?
The movement and shifts of all graphs can be
What do the key features (max, min, increasing, decreasing, etc.) tell us about trigonometric graphs
shown in trigonometric functions.
and what do they represent in real-world situations modeled by trigonometric functions?
How does changing the parameters affect the graphs of trigonometric functions? (shifts, stretches,…)
What real-world situations are modeled by trigonometric functions?
Standard *
Learning Targets
Technology Standards
F-IF.7e
● Use amplitude, period, and
Graph trigonometric functions, showing period, midline, and
● I can identify the midline,
frequency to determine an
amplitude.
amplitude, and period of a sine and
appropriate window for graphing a
cosine function.
trigonometric function on a
F-IF.4
For a function that models a relationship between two
● I can sketch the graphs of the sine
calculator.
quantities, interpret key features of graphs and tables in
and cosine functions.
● Graph a trigonometric function
terms of the quantities, and sketch graphs showing key
● I can identify the midline, period,
on a calculator in degree mode and
features given a verbal description of the relationship. Key
and asymptotes of a tangent
in radian mode.
features include: intercepts; intervals where the function is
function.
● Verify the solution to a system of
increasing, decreasing, positive, or negative; maximums and
● I can sketch the graph of a tangent
trigonometric functions.
minimums; symmetries; end behavior; and periodicity.
function.
● Use the domain and range to set
● I can sketch the graphs of the
F-BF.3
Identify the effect on the graph by replacing f ( x) with
an appropriate window for
f ( x)  k , kf ( x) , f (kx) , and f ( x  k ) for specific values of secant, cosecant, and cotangent
graphing a trig function on a
functions.
calculator.
k (both positive and negative); find the value of k given the
● I can find the domain and range of
● Use the TRACE and CALCulate
graphs. Experiment with cases and illustrate an explanation of
a trigonometric function.
features to find a specific value, the
the effects on the graph using technology. Include recognizing
● I can find the intercepts of a
intercepts, and relative maxima and
even and odd functions from their graphs and algebraic
trigonometric function.
minima of a trig function.
expressions for them.
● I can identify intervals where a
● Use the TABLE feature of a
ACT
Write the equations of sine and cosine functions given the
trigonometric function is increasing,
calculator to identify key features
Quality
amplitude, period, phase shift, and vertical translation.
decreasing, positive, or negative.
of a trig function.
Core F.3.f
● I can find relative maximums and
● Describe the period, domain,
F-TF.5
Choose trigonometric functions to model periodic phenomena
minimums of a trigonometric
range and asymptotes of a
with specified amplitude, frequency, and midline.
function.
trigonometric function graphed on
● I can identify any symmetry in the
a calculator.
graph of a trigonometric function.
● Use a graphing calculator to
● I can use vertical and horizontal
experiment with changing the
Page 6 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
shifts to sketch graphs of
trigonometric functions.
● I can use vertical and horizontal
“stretching” to sketch graphs of
trigonometric functions.
● I can use reflections to sketch
graphs of trigonometric functions.
● I can classify the transformation by
comparing two given graphs.
● I can determine whether a
trigonometric function is even or
odd.
● I can use the midline, amplitude,
period, and phase shift to write the
equation of a graph of a sine and
cosine function.
● I can write the equation of a
sinusoid as a sine function and as a
cosine function.
● I can apply the meaning of the key
features of a trigonometric function
to a real-world situation.
● I can solve real-world problems
that can be modeled with
trigonometric functions.
parameters of an equation of a
trigonometric function and
determine their effect on the graph.
● Use a calculator to translate sine,
cosine and tangent graphs.
● Use a calculator to illustrate how
the graphs of sine, cosine and
tangent are affected by translations.
● Graph a trigonometric model of a
real-world problem and use the
various features of the calculator to
solve the problem.
Key Vocabulary
midline
amplitude
frequency
phase shift
vertical translation
domain
range
intercepts
asymptote
interval
increasing/decreasing
maximum/minimum
symmetry
end behavior
periodicity
parameter
even/odd function
Page 7 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
Enduring Understandings:
A firm understanding of domain and range and the
inverse of functions is applied to trigonometric
functions.
2014-2015
Unit 4: Inverse Trigonometric Functions
Essential Questions:
Since the trigonometric functions are not one-to-one, how can the domain be restricted to graph
the inverse functions?
How are inverse trigonometric functions used to find angles in real-world problems?
Standard *
Read values of an inverse function from a graph or a table,
F-BF.4c
F-TF.6
F-BF.4b
G-SRT.7
Learning Targets
● I can determine the coordinates of the
given that the function has an inverse.
points on an inverse trigonometric function
from a table of values.
Understand that restricting a trigonometric function to a
● I can explain why the domain of the sine,
domain on which it is always increasing or always decreasing cosine and tangent functions must be
allows its inverse to be constructed.
restricted in order to have an inverse.
Verify by composition that one function is the inverse of
● I can determine the domain for the inverse
another.
sine, inverse cosine, and inverse tangent
Evaluate expressions containing inverse trigonometric
functions.
functions.
● I can graph the arcsin, arccos, and arctan
functions.
Explain and use the relationship between the sine and
cosine (and other functions) of complementary angles.
● I can use composition of functions to solve
problems such as arcsin(sin  ) and
sin(arcsin x) .
● I can evaluate expressions such as
Technology Standards
● Choose an appropriate window
on a calculator to graph the arcsin,
arccos, and arctan functions.
● Use the calculator to evaluate
expressions such as
tan  arcsin 0.35  .
Key Vocabulary
inverse
vertical line test
horizontal line test
one-to-one
invertible/non-invertible
composition
arcsine
arccosine
1

tan  arcsin  .
arctangent
2

sin–1
–1



● I can explain why the sin   cos     . cos–1
2
 tan


  .
2



● I can explain why the tan   cot     .
2

● I can explain why the sec   csc 
Page 8 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 5: Trigonometric Identities
Enduring Understandings:
Essential Questions:
The basis of trigonometric identities comes from
How can the Pythagorean, reciprocal, co-function, even/odd, and quotient trigonometric identities
both the unit circle and the Pythagorean Theorem. be developed from the unit circle?
Manipulation, substitution, and reduction can
How can algebraic operations be used to simplify trigonometric expressions and verify
simplify complex trigonometric expressions.
trigonometric identities?
How can trigonometric identities for sum, difference, double angle and half angle simplify
evaluating expressions and solving equations?
Standard *
Learning Targets
Technology Standards
Use the unit circle to explain symmetry (odd and even) and
F-TF.4
● I can determine the relationship
● Verify trigonometric identities by
periodicity of trigonometric functions.
between the sin (cos, tan) of θ and the sin graphing.
(cos, tan) of –θ.
● Verify the solution to a system of
2
2
F-TF.8
● I can use the unit circle to illustrate the
trigonometric functions.
Prove the Pythagorean identity sin   cos   1 and use
2
2
Pythagorean identity sin   cos   1 .
it to find tan  , sec , csc and cot  .
Key Vocabulary
● I can use the identity
Apply the fundamental trigonometric identities to simplify and
ACT
2
2
trigonometric identities
sin   cos   1 to develop the other
evaluate trigonometric expressions and prove trigonometric
Quality
quotient identities
Pythagorean identities and the quotient
Core F.3.i identities.
Pythagorean identities
identities.
Identify the sum and difference identities for the sine, cosine,
ACT
● I can use the fundamental trigonometric sum/difference formulas
and tangent functions; apply the identities to solve
Quality
double-angle formulas
identities to evaluate trigonometric
Core F.3.g mathematical problems.
half-angle formulas
functions and simplify trigonometric
Derive, identify, and apply double-angle and half-angle
ACT
co-functions
expressions.
formulas to solve mathematical problems.
Quality
even/odd identities
● I can verify trigonometric identities
Core F.3.h
using the quotient, reciprocal, co-function, simplify
Apply the fundamental trigonometric identities, the doubleACT
evaluate
even/odd, and Pythagorean identities.
angle and half-angle identities, and the sum and difference
Quality
● I can state the identities for the sine and verify
Core F.3.i identities to simplify and evaluate trigonometric expressions
solve
cosine of the sum and difference of two
and prove trigonometric identities.
angles.
● I can use the sum and difference
formulas to evaluate trigonometric
functions without using a calculator.
● I can use the identities for the sine and
cosine of sums and differences to develop
the identities for the tangent of sums and
Page 9 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
differences.
● I can use the identities for the sine and
cosine of sums and differences to develop
the double-angle and half-angle identities.
● I can use the sum and difference
formulas to evaluate trigonometric
functions, verify trigonometric identities,
and solve trigonometric equations.
● I can use multiple-angle formulas to
rewrite and evaluate trigonometric
functions.
● I can use half-angle formulas to rewrite
and evaluate trigonometric functions.
Page 10 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 6: Trigonometric Equations
Enduring Understandings:
Essential Questions:
The rules for solving all types of equations are integrated with How are algebraic operations used for solving trigonometric equations (including those
the identities of trigonometry in order to solve trigonometric in quadratic form)?
equations.
What substitutions involving trigonometric identities need to be used for solving some
The equations are solved using the understanding of inverse
trigonometric equations?
trigonometric functions in conjunction with the unit circle.
How can we represent and solve real-world problems with trigonometric equations?
Standard *
Learning Targets
Technology Standards
Use inverse functions to solve trigonometric equations.
F-TF.7
● I can use inverse functions to solve
● Use SIN-1, COS-1, and TAN-1 to
trigonometric equations.
approximate values of inverse
● I can use and apply the Pythagorean and
trigonometric functions.
2
A-REI.4b
● Use the zero or root feature to
Solve quadratic equations by inspection (e.g., for x  49 ), quotient identities to simplify and solve
trigonometric equations.
approximate x-intercepts of
taking square roots, completing the square, the quadratic
● I can solve trigonometric equations in
trigonometric functions.
formula and factoring, as appropriate to the initial form of
the equation.
quadratic form by taking square roots, by
● Use a graphing calculator to solve
Understand that the graph of an equation in two variables is completing the square, by using the
A-REI.10
a trigonometric equation.
the set of all its solutions plotted in the coordinate plane,
quadratic formula, and by factoring.
● Use a calculator to solve a
often forming a curve (which could be a line).
● I can solve trigonometric equations by
trigonometric equation
Use sum/difference, multiple-angle and half-angle formulas
graphing.
to solve trigonometric equations.
● I can use the sum and difference formulas Key Vocabulary
Solve advanced trigonometric equations.
to solve trigonometric equations.
trigonometric equation
● I can use multiple-angle formulas to solve
quadratic
Use inverse functions to solve trigonometric equations that
F-TF.7
trigonometric equations.
completing the square
arise in modeling contexts; evaluate the solutions using
● I can use half-angle formulas to solve
quadratic formula
technology, and interpret them in terms of the context.
trigonometric equations.
factoring
● I can solve advanced trigonometric
zeros of a function
equations in quadratic form, including those domain
which involve a substitution.
● I can solve advanced trigonometric
equations involving multiple angles.
● I can solve real-world problems that can
be modeled with trigonometric equations.
Page 11 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 7: Additional Topics in Trigonometry
Enduring Understandings:
Essential Questions:
Trigonometry is expanded beyond the right
How is the area of a triangle found when two sides and the included angles are given?
triangle.
How are oblique triangles solved using the Law of Sines and the Law of Cosines?
The sides and angles of all triangles can be found,
In real-world situations, such as navigation, surveying, etc., how can the Law of Sines of the Law of
providing solutions to real-world problems.
Cosines be used?
Standard *
Learning Targets
Technology Standards
G-SRT.9
● I can derive the formula
1
Derive the formula A  ab sin C for the area of a triangle by
Key Vocabulary
1
2
A  ab sin C .
auxiliary line
drawing an auxiliary line from a vertex perpendicular to the
2
perpendicular
opposite side.
● I can find the area of a triangle
oblique triangle
Use various methods to find the area of a triangle (e.g., given the
ACT
given two sides and the included
Law of Sines
length of two sides and the included angle).
Quality
angle.
Law of Cosines
Core F.3.a
● I can use the Law of Sines to solve
G-SRT.10 Prove the Laws of Sines and Cosines and use them to solve
an oblique triangle.
problems.
● I can use the Law of Cosines to
solve an oblique triangle.
Understand
and
apply
the
Law
of
Sines
and
the
Law
of
Cosines
to
G-SRT.11
find unknown measurements in right and non-right triangles (e.g.,
● I can use the Law of Sines and Law
surveying problems, resultant forces).
of Cosines to solve real-world
problems.
Page 12 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 8: Vectors
Enduring Understandings:
Essential Questions:
By combining the properties of linear equations,
What notation is used to represent vectors?
directionality, and trigonometry, vectors bring new Given the initial and terminal point on a vector, what is the component form?
dimensions to mathematics.
What is velocity and how are vectors used to represent it?
How can operations on vectors (+, -, x, ÷) be performed graphically and algebraically?
Standard *
Learning Targets
Technology Standards
Recognize vector quantities as having both magnitude and
N-VM.1
● I can represent a vector as a directed
● Use inverse trigonometric
direction. Represent vector quantities by directed line
line segment.
functions to find angle measures.
segments, and use appropriate symbols for vectors and their
● I can use appropriate symbols for
magnitudes (e.g., v, |v|, ||v||, v).
vectors and their magnitudes.
Key Vocabulary
Find the components of a vector by subtracting the coordinates
N-VM.2
● I can represent a vector in component
vector
of an initial point from the coordinates of a terminal point.
form.
magnitude
●
I
can
represent
a
vector
in
linear
form.
directed line segment
Solve problems involving velocity and other quantities that can
N-VM.3
● I can calculate the magnitude of a
component form
be represented by vectors.
vector.
parallelogram rule
Add vectors end-to-end, component-wise, and by the
N-VM.4a
●
I
can
use
the
Law
of
Sines
and
the
Law
initial
parallelogram rule. Understand that the magnitude of a sum of
of Cosines to solve real-world problems.
terminal
two vectors is typically not the sum of the magnitudes.
●
I
can
solve
real-world
problems
velocity
Given two vectors in magnitude and direction form, determine
N-VM.4b
involving
quantities,
including
velocity,
resultant
the magnitude and direction of their sum.
force, and work, that can be represented scalar
Understand vector subtraction v – w as v + (–w), where –w is
N-VM.4c
by vectors.
the additive inverse of w, with the same magnitude as –w and
pointing in the opposite direction. Represent vector subtraction ● I can add vectors geometrically by using
graphically by connecting the tips in the appropriate order, and the parallelogram rule.
perform vector subtraction component-wise.
● I can add vectors in component form
Represent scalar multiplication graphically by scaling vectors
N-VM.5a
and in linear form.
and possibly reversing their direction; perform scalar
● I can find the magnitude of a sum of
multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
two vectors.
N-VM. 5b Compute the magnitude of a scalar multiple cv using ||cv|| =
● I can find the sum of the magnitudes of
|c|v. Compute the direction of cv knowing that when |c|v  0, two vectors.
the direction of cv is either along v (for c > 0) or against v (for c
● I can show that the magnitude of a sum
< 0).
of two vectors is not the same as the sum
Apply the properties of vectors to finding angle measures.
of the magnitudes of two vectors.
Page 13 of 16
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*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
N-CN.6
Calculate the distance between numbers in the complex plane
as the modulus of the difference, and the midpoint of a
segment as the average of the numbers at its endpoints.
2014-2015
● I can represent a vector in magnitude
and direction form.
● I can find the magnitude and the
direction of the sum of two vectors.
● I can represent vector subtraction
geometrically.
● I can subtract vectors in component
form and in linear form.
● I can represent scalar multiplication
geometrically.
● I can multiply a vector in component
form and in linear form by a scalar.
● I can calculate the magnitude of a
scalar multiple of a vector.
● I can determine the direction angle for
a vector.
● I can find the dot product of two
vectors and use the properties of the dot
product.
● I can find the angle between two
vectors.
● I can find the distance between two
numbers in the complex plane.
● I can find the midpoint of segment
joining two numbers in the complex
plane.
Page 14 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
2014-2015
Unit 9: Complex Numbers
Enduring Understandings:
Essential Questions:
With the use of imaginary numbers solutions are no longer restricted to
How are complex numbers written in polar form?
the tangible world.
How can we perform computations with complex numbers in polar form?
Complex numbers are often converted into polar form to simplify the
procedure of performing some mathematical operations.
Standard *
Learning Targets
Technology Standards
2
N-CN.1
●
I
can
write
any
complex
number
in
Know there is a complex number i such that i  1 , and every
standard form.
Key Vocabulary
complex number has the form a + bi with a and b real.
● I can add, subtract, and multiply
imaginary number
2
N-CN.2
complex
numbers
in
standard
form.
complex number
Use the relation i  1 and the commutative, associative, and
● I can determine the conjugate of a i
distributive properties to add, subtract, and multiply complex
complex number.
standard form
numbers.
● I can divide complex numbers in
complex plane
Find the conjugate of a complex number; use conjugates to find
N-CN.3
standard form.
commutative property
moduli and quotients of complex numbers.
● I can find the absolute value of a
associative property
complex number.
distributive property
Represent complex numbers on the complex plane in rectangular
N-CN.4
● I can graph a complex number in
conjugate
and polar form (including real and imaginary numbers), and explain
rectangular form.
modulus (magnitude)
why the rectangular and polar forms of a given complex number
●
I
can
graph
a
complex
number
in
argument
represent the same number.
polar form.
rectangular form
●
I
can
convert
from
rectangular
polar form
Represent multiplication, and conjugation of complex numbers
N-CN.5
coordinates
to
polar
coordinates.
r cis θ
geometrically on the complex plane; use properties of this
3
● I can convert from polar
De Moivre’s Theorem
representation for computation. For example, 1  3i  8
coordinates to rectangular
coordinates.
because 1  3i has modulus 2 and argument 300°.
● I can multiply and divide complex
numbers written in trigonometric
form.
● I can use DeMoivre’s Theorem to
find powers of complex numbers.




Page 15 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment
Trigonometry
PUHSD Curriculum
Enduring Understanding:
Polar graphs add aesthetics to geometric designs.
Standard *
Graph and analyze polar equations.
2014-2015
Unit 10: Polar Equations and Graphs
Essential Question:
Which parameters and which functions change the families of polar curves?
Learning Targets
● I can plot points and find multiple
representations of points in the polar
coordinate system.
● I can convert points between
rectangular and polar coordinates.
● I can convert equations between
rectangular and polar form.
● I can graph polar equations by point
plotting.
● I can use symmetry as a sketching
aid for polar graphs.
● I can recognize special polar
graphs.
Technology Standards
● Use a calculator in POLar mode to
graph polar equations.
● Use the table feature to verify
points on a polar graph.
Key Vocabulary
polar equation
families of polar curves
Page 16 of 16
th
*4 year math curriculum standards are aligned to community college course descriptions and some standards do not have a specific ACCR-M standard alignment