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MATH 120-04: Pre-Calculus Mathematics (43416)
JB-138, TuTh 4 - 5:50 PM
SYLLABUS Spring 2013
John Sarli
JB-326
[email protected]
909-537-5374
Office Hours: TuTh 11-1 PM, or by appt.
Text: Swokowski/Cole custom edition for CSUSB
College Algebra & Trig Math
Prerequisites: MATH 110 or satisfactory score on the ELM
This a course in the fundamentals of pre-calculus mathematics. The main emphasis is on the theory and applications of trigonometric
functions, but we will also cover topics in analytic geometry, mathematical induction and the binomial theorem. In particular, we will cover the
following sections of the text in this order: Chapters 6,7,8; Chapter 10, sections 4 and 5. We may cover material from Chapters 11, sections
1,2,3 if time permits.
Grading will be based on two midterm exams, a cumulative final exam, four graded assignments, and a project, weighted as follows: First
Midterm (10%), Second Midterm (20%), Final Exam (40%), Graded Assignments (20%), Project (10%). To reinforce written communication
skills the Graded Assignment solutions should be clearly presented, either in a "bluebook" or sent to me as a .pdf file (do not scan in
handwritten work). The graded assignments will be chosen to encourage understanding of a variety of applications, often within social and
historical contexts, of developments within and related to the natural sciences. To reinforce the importance of clear presentation I will
occasionally take some class time for you to complete an MDTP Written Response item, which I will score on a to scale and return to you
by the next class. These scores do not affect your course grade; they serve to provide you with some feedback on your presentation skills.
The Project must be on some aspect of the mathematics of conic sections, the basics of which are covered in Chapter 11. Details will be given
as we progress but the purpose of the project is to have you present, in your own words, a topic that uses conic sections in some context.
Although there is no attendance requirement for this class, you must complete the CSU/UC Mathematics Diagnostic Testing Project CR test
within the first two weeks of the course (by April 16). Go to mdtp.ucsd.edu and scroll down to MDTP Web Based Tests. Select the CR test.
The items will appear one at a time. You can either print the results or send them to me electronically. The results do not affect your course grade
in any way, but failure to complete this requirement will subject you to disenrollment.
Although you are encouraged to learn the use of a calculator with graphing capability, exams are conducted in class without calculators. A
list of Suggested Exercises will be provided prior to the start of each chapter and the exams will be closely modeled on these. After computing
your total scores weighted according to the percentages above, course grades will be assigned as follows:
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Notes
1) Some important dates:
April 2 - First day of class
April 22 - Last day to submit adds/drops/audits/CBE
April 18 - First exam
May 16 - Second Exam
June 6 - Last day of class
Tuesday, June 11- Final Exam
2) Notes for the lectures and associated topics will appear on my website
www.math.csusb.edu/faculty/sarli/
along with this syllabus.
3) Mid-term exam dates are subject to change. Due dates for the graded exercises will be set as we approach the end of each chapter of the
text. The Project is due on June 6 - no exceptions.
4) This course satisfies the requirement of Basic Skills Category A.3 in the CSUSB General Education Program. Please refer to the Academic
Regulations and Policies section of your current bulletin for information regarding add/drop procedures and consequences of academic
dishonesty.
5) If you are in need of an accommodation for a disability in order to participate in this class, please let me know ASAP and also contact
Services to Students with Disabilities at UH-183, (909)537-5238.
MATH 120-04: Chapter 6 (Swokowski/Cole, CSUSB edition)
.
The following Suggested Exercises are not to be handed in but are a representative sample of techniques required for basic mastery.
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6.1 (pages 375-377):
6.2 (pages 390-393):
6.3 (pages 407-410):
6.4 (pages 416-417):
6.5 (pages 426-429):
6.6 (pages 435-437):
6.7 (pages 443-449):
First Graded Assignment
To reinforce written communication skills the Graded Assignment solutions should be clearly presented in a "bluebook" or provided in .pdf
format. The assignments are chosen to encourage understanding of a variety of applications, often within social and historical contexts, of
developments within and related to the natural sciences. Late papers will not be graded.
First Graded Assignment. Do any one of the following: Due Tuesday, April 30.
6.1:
(Note: "reading of the data occurs at a constant rate" means linear speed remains constant)
6.3:
6.4:
6.5:
6.6:
6.7:
or
or
Review Exercises (pages 449-455):
or
or
Notes on Chapter 6
Central angles
Central angles
An angle whose vertex is at the center of a circle is called central. The angle is often referred to by its measure
circle at points
and
then the circle is divided into two arcs. Normally, when we refer to the arc
along the circle in the counterclockwise direction from
subtends
Let
to
. If the rays of
intersect the
we mean the arc determined by moving
, but context often determines which of the two arcs we mean. We say that the arc
.
be the radius of the circle. If the length of arc
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is equal to
then the measure of
is one radian. Since the circumference of the
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circle is
it follows that
which allows us to convert easily between radian and degree measure. Note that the degree symbol
must always follow degree measure, but,
by convention, we normally suppress the word "radian" when using radian measure. If it is not suppressed, it is often denoted by
. Thus
Arc length and sectors
We will use radian measure unless degree measure is specifically indicated. Again, since the circumference of the circle is
proportionality that if
is the length of an arc that subtends the central angle
By similar reasoning, since the area of the circle is
determined by
, it follows by
then
, it follows by proportionality that if
is the area of the circular sector
then
Angular and linear speed
The formula
for arc length allows us to relate the linear speed of a point
on the circumference of a circular object that is rotating at
a constant rate about its center to the angular speed of the object itself. The linear speed
is the distance that
so it must be measured in units of distance/time, for example, cm/sec. The angular speed
time by a ray from the center through
travels per unit of time
of the object is the angle generated in one unit of
, thus
The letter
is used generically for angular speed though other letters may be used in various contexts. Note that it is measured in units of
angle-measure/time, for example, rad/sec. Angular speed does not depend on the radius of the object but linear speed does; in particular, note that
because
.
Angles in standard position in the plane
An angle with vertex
in the Cartesian plane is in standard position if one of its rays is the positive
-axis (horizontal axis). We adopt
an orientation on such angles by designating the positive -axis ray as the initial ray and the other ray as the terminal ray. (What does it mean if
these rays coincide?) Thus we adopt a dynamic viewpoint: the measure of this angle is determined by how we move from the initial ray to the
terminal ray. By convention, if the movement is counterclockwise the measure is a positive number, whereas it is a negative number if we move
clockwise. We can measure either in degrees or radians, but we will find radians easier to use when we work with functions. Note that neither the
positive nor negative measure of an angle is unique. For example, the angle whose terminal ray is the positive -axis could have measure
, etc., or it could have measure
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, etc., but any measure for this angle is of the form
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where
is an integer.
Fundamental trigonometric functions
The equation
describes the circle of radius
circle. Then we can define real-valued functions of
trigonometric functions. For example, let
centered at
in the Cartesian plane. Let
be a point on this
that are useful in a wide variety of applications. These are known as the fundamental
Since
the functions and satisfy a special property called the Pythagorean identity. We will give them special
names once we interpret their definitions in the more familiar context of functions on the real number line. To accomplish this, first note that if
then
and
real number to the point
let
. Thus it simplifies our work if the circle we use is the unit circle. The functions
. We can associate a real number
to
as follows. Given
as the odometer reading on a car as it drives from
to
each assign a
, determine the terminal ray that passes through
be the radian measure of this standard angle. Note, from the formula for arc length, that
unit circle. Think of
and
is just the distance from
to
along the
, where the reading is positive if the car drives
counterclockwise around the circle and negative if it drives clockwise. For example, if
the smallest positive measure of the standard
angle is
. We have
and now we can write
. Similarly,
. Apparently, the functions and
measuring the oriented horizontal and vertical distances from the center of the circular track as the car drives along it. Note also that
for any integer
. Thus the functions
and
. Now
are just
are not one-to-one; in fact, infinitely many inputs produce the same output.
We now have two equivalent ways of describing the functions
and
. We originally defined them as functions whose inputs are points
in the plane, and then noted that these functions are constant on the terminal ray through . This allowed us to define and as
functions on the entire real line by associating that terminal ray with its radian measure. These functions are of fundamental importance. We will
see that they generalize the trigonometry of right triangles. Thus we give them familiar names
and use them to define the other four fundamental trigonometric functions
In practice, we determine the values of the functions for a particular value of
point
on the unit circle
by travelling from
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through the distance . Then
by the following thought process: Given , locate the unique
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Domains, ranges, and periods
Since we now have special names and designations for the fundamental trigonometric functions (cosine, sine, tangent, cotangent, secant,
or if
then the input
cosecant) we can use a generic letter, such as , to stand for any of them in context. Note that if
for
can be any real number, and the output can be any number in the closed interval
sine functions is
and the range is
. Thus the domain of the fundamental cosine or
. These functions are periodic, that is, there is a number
such that
; in fact, the
smallest positive with this property is
, so we say that the period of these functions is
. With just these few observations we can
visualize the graphs of these functions, at least qualitatively. Our goal is to analyze their behavior in more detail, which will then allow us to
analyze the other four functions in detail.
Now suppose
or
. Since the secant function is the reciprocal of the cosine function it is defined everywhere except
where the cosine function is zero. Thus the domain of the secant function is
and its range is
because
. Similarly, the range of the cosecant function is also
for any integer
. Qualitatively we can now visualize the graphs of
, whereas its domain is
and
vertical asymptotes at the values missing from their domains, and by noting that these functions are also periodic with period
by picturing
.
Finally, suppose
or
. These two functions are reciprocals of each other. Since
the domain of the
tangent function will be the same as the domain of the secant function, whereas the domain of the cotangent function will be the same as the
domain of the cosecant function. Since the sine and cosine functions cannot simultaneously be zero and their ranges are bounded, we conclude
that the range of the tangent function is , as is the range of the cotangent function. Again we have a qualitative picture of their graphs, with
vertical asymptotes at the values missing from their domains. These two functions are also periodic, with the same period, but that period is not
. As the terminal ray determined by
sequence
moves through the four quadrants of the plane the signs for the outputs of the cosine function follow the
whereas the signs for the outputs of the sine function follow the sequence
Consequently
and so
Thus
(being the smallest positive number for which periodicity happens) is the period of these two functions. The shape of their
graphs repeats twice as frequently as does the shape of the graphs of the other four fundamental trigonometric functions.
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Pythagorean identities
We have seen that the ordered pair
where, as usual, the symbol
is a point on the unit circle
. This fact gives us the first Pythagorean identity
means that the equation is true for all values of
for which the functions can be evaluated. Since the domain of
the cosine function is , as is the domain of the sine function, this equation is an identity for all values of . Two other Pythagorean identities are
obtained for the other fundamental trigonometric functions
Note that the first equation is an identity for all values of
except multiples of
except odd multiples of
and the second equation is an identity for all values of
. Using the Pythagorean identities we can derive most other identities that are useful in calculus.
Even and odd functions
In a prior course you may have seen the following definitions:
A function is called even provided
for all in its domain.
A function
is called odd provided
for all
in its domain.
Using generic Cartesian variables, note then that the functions
symmetric about the
-axis), whereas
,
and
,
are even functions (their graphs are
, and
are odd functions (their graphs
are symmetric about the origin). To see that these functions are even or odd as claimed, choose any value
in the domain and locate the point on
the unit circle corresponding to this value. Now locate the point corresponding to
, which is antipodal to the first point. If the value of the
function is the same for both points then the function is even, but if the values have opposite signs then the function is odd. As we consider
trigonometric functions that are not fundamental it is important to realize that, in general, they will be neither even nor odd.
At this point you should evaluate your understanding by working Suggested Exercises from sections 6.1 through 6.4.
Functions of the form
and
In practice, trigonometric functions are rarely used in just their fundamental forms. In order to understand general trigonometric functions, we
begin with alterations of the fundamental functions obtained from linear changes to the independent and dependent variables. We illustrate these
for the sine and cosine functions, but the same principles apply to any of the fundamental functions.
Alterations of fundamental functions are easily described in terms of graphs. Linear changes to the dependent variable result in vertical
. we can
changes to the graph. We examine these first since they do not change the domain of the original function. Consider a graph
create a new function by adding a constant
If
to the output of this function, resulting in the graph
this graph is obtained by translating the graph
translating the graph
downward by
units. If
units. This is the simplest of the linear alterations.
The other linear change to the dependent variable is of the form
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upward by
the graph
is obtained by
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where, again,
is a constant. The graph
) vertically. In addition, if
follows. Start with the graph
Next, produce the graph
and then the graph
and finally the graph
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is obtained by stretching the graph
the graph is reflected in the
(if
) or compressing it (if
-axis. As an example, we construct the graph
as
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Note that the domain of
is still
, but the range of this function is
.
Now consider linear changes to the independent variable. These will result in horizontal changes to the graph. For any constant
the graph
Since we have added
if
to the input variable this graph is obtained from the graph
and to the right if
. A good way to remember this is to imagine a particular
, so the graph
if
by translating it horizontally
has an
-intercept at
-intercept
for
, which is to the left of
, consider
units: to the left
. We have
if
and to the right of
. For example,
This graph has its smallest positive
-intercept at
because
has its smallest positive
-intercept at
.
The other linear change to the dependent variable is of the form
for some constant
, and this results in a horizontal stretch or compression of the graph
graph is compressed, whereas the graph is stretched if
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. Note, however, that if
. A good way to remember this is to think of the function
the
, whose graph
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is a straight line of slope
. The function
is a straight line of slope
horizontally. Similarly, the graph of
is reflected in the
, which can be thought of as the line of slope
can be thought of as the line of slope
compressed
stretched horizontally. (Note that if
-axis. For the trigonometric functions we will see that we can always work with the case
the graph
. For example,
, because the fundamental sine function is odd, so its graph is just the reflection of
in the -axis.) It
is important to remember that stretch and compression are in relation to the vertical axis, which is different from dilation about a point. Thus, the
point where a horizontal line intersects
is twice as far from the -axis as the point where this line intersects
.
from the graph
How do we obtain the graph
. Note that the graph
graph
is obtained by translating the graph
is obtained from the graph
illustration, we construct the graph
and then, since
? We have
, where
, as described above. Then the
by stretching or compressing as determined by
, compress this graph to obtain
see what happens to the graph
, when the input variable was unaltered. We let
. To do this, we first undo the translation part
and then undo the stretch/compression part
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. As an
as follows. First obtain the graph
A good way to remember this procedure is to remind yourself that you are making changes to the input variable:
is, we invert the function
and
and proceed to find
. We want to
in terms of
, that
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The actions we take to invert
by
are precisely the actions we apply to
, horizontally: translate by
and then multiply
.
that is useful in applications. Since
There is another approach to obtaining the graph
so now we have
, but this time with
and
obtained from
by stretching or compressing as determined by
the graph
by translating as determined by
. Now the graph
. Then the graph
. Again, we construct the graph
First obtain the graph
which has an
-intercept at
which has an
-intercept at
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, and then translate this graph
. Again,
units to the left to obtain
we can write
is
is obtained from
.
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the actions we take to invert
translate by
are precisely the actions we apply to
, horizontally: multiply by
and then
.
Amplitude, frequency and phase shift
The second procedure, above, for obtaining the graph
introduces an important parameter, called phase shift, used in the
description of harmonic motion. Before we define it precisely we introduce some terminology useful in the description of wave phenomena. The
fundamental sine and cosine functions, as well as their linear alterations, have ranges that are closed intervals. This allows us to define the
amplitude of the wave shapes defined by their graphs. For the functions
we define the amplitude to be the non-negative number
. Note that the amplitude is not the same thing as the range of the function,
though it is related to it. The actual range of the above functions is the closed interval
For example, the range of the function
is the interval
whereas its amplitude is
.
The amplitude measures the height of the wave above (or the depth of the wave below) its horizontal center line. For the function above, note
that the period is
of the function
, that is,
for all . For any fundamental trigonometric function
with period
, we define the period
to be the positive number
(This definition agrees with the behavior of the graphs of these functions, and obviously generalizes our use of the word in the case of the
fundamental trigonometric functions.) We define the phase shift of this function to be
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which you can remember as the final step in isolating the input variable when we write
For the graph above, note that
.
At this point you should evaluate your understanding by working Suggested Exercises from sections 6.1 through 6.6. For the first exam,
focus in particular on the following:
Geometric relations among central angle, radius, arc length, and area of sector.
Angular speed and linear speed, in correct units.
Determining values of the fundamental trigonometric functions given a terminal ray in standard position or a point on that ray.
Expressing fundamental trigonometric functions in terms of each other.
Constructing graphs of linear alterations to the dependent variable of fundamental trigonometric functions, and determining:
Domain
Range
Amplitude
Period
Intercepts
Asymptotes
Background for the project: Conics
A circle is an example of a conic. Conics were described in detail prior to Euclid's work (primarily by Menaechmus), but the development was
based on their properties as sections of cones in three dimensions. This approach was reconciled with their descriptions as plane curves (loci) by
later mathematicians such as Apollonius of Perga and, about 600 years after Euclid, by Pappus of Alexandria, who created the foundations of
what would eventually become analytic geometry. Euclid understood the importance of conic curves and asserts the existence of circles in his
postulates as loci of points equidistant from a given point. Numerous manifestations of conics occur throughout history since they are essential to
the understanding of many physical phenomena. Most of these derive from the following locus description that dates from antiquity.
A conic is the locus of a points whose distance from a given point is
constant.
times its distance from a given line, where
is a non-negative
The given point and line are called a focus and its corresponding directrix. The constant is called the eccentricity of the conic. Without
coordinate geometry, mathematicians such as Apollonius used synthetic techniques to describe the conics that would correspond to various values
of . In pre-calculus our approach uses analytic geometry, the geometry of the coordinate plane. Using analytic geometry we can reduce the
lengthy synthetic development to a few basic equations. This shows the power of modern notation. You should make your own drawing to
represent the following argument.
First, let
to
be the given focus and let
, so the condition for
. For any point
let
be the point on the directrix closest
to be on the conic is
Now consider the line through
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be the point on the directrix closest to
parallel to the directrix. Since this line is perpendicular to the line
, by symmetry there must be two
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points,
and
, on this line that are also on the conic. Let
directrix closest to
be the length of segment
(hence also of
). Then, if
is the point on the
we have
Next, for any point
on the conic let
Let
be the measure of angle
But
so
. Then
. Then
We conclude that
an equation that offers a lot of information. For example, we required that
the focus is constant and so the conic is the circle centered at
. If
then
, that is, the distance between
with radius . Where is the directrix in this case? Since
and
the only way for
to hold is for
to be infinitely large. We say that the directrix is the "line at infinity" and use projective geometry to make this idea precise.
Circles, then, are conics with zero eccentricity, and any line through the center is a line of symmetry.
The angle
locates
symmetric about the line
relative to the line
. When
, and we can assume
we have
. Since
, whereas, if
it follows that the conic is
, this line intersects the conic again when
, whereby
. When
the conic is called a parabola. A point where the conic intersects its line of symmetry is called a vertex. Thus, a
parabola has a single vertex and all other conics, except circles, have exactly two vertices. Since a circle does not have a unique line of symmetry
we generally do not think of its points as vertices. Because of this extra symmetry, it was believed for centuries that the motions of celestial
bodies must be perfect circles in all cases.
If
note that
is finite and positive for all values of
is therefore a type of ellipse. If
note that
is positive if
is on the ray through
that is opposite the ray determined by
that such a conic has two separate branches.
Exercise. Let
be such that
is produced by
, so the conic is a closed oval-shaped curve, which we call an ellipse. The circle
. If
. A conic with
. Show that
If
we obtain
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is a negative number; this means that
is called a hyperbola. The fact that
changes sign means
produces one branch of the hyperbola and that the other branch
.
. Let
We can obtain Cartesian equations for conics from the relation
we have
then
and
is the
-coordinate of
. Thus
and let
be on the positive
-axis. With
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as the equation for the parabola.
Exercise. How does the value of
let
If
and let
affect the shape of the parabola?
.
Exercise. Show that the Cartesian equation for the conic is
Application: Harmonic Motion
Functions of the form
describe many physical measurements that vary with time . When
such
functions describe harmonic motion by giving the position of a point moving on a coordinate line relative to the origin. In this context it is
common to write
with
, so the amplitude of the motion is
, the period is
, and the phase shift is
. Since the motion is along a coordinate line
it is also common in practical applications to set the origin so that
, and in case
the phenomenon is referred to as simple harmonic
motion (SHM). The amplitude, then, is the maximum distance from the origin that the point reaches, the period is the time required for one
complete oscillation, and the frequency
, being the reciprocal of the period, is the number of oscillations per unit time. (Note the analogy with
our use of
for angular speed.) SHM applies to many physical systems, such as a mass attached to a spring that is set to move horizontally or
vertically only by the force from an initial stretch or compression of the spring. Generally, the resting position of the mass in the absence of any
stretch or compression of the spring is taken as the origin of the line along which the mass will move. That is, the values of such that
are the times when the mass is at this origin position. Note that origin refers to the physical position of the mass, not the beginning of the motion.
is not necessarily . In fact it will often be some other number: typically positive if the clock starts with the spring in a stretched
That is,
position, negative if it starts in a compressed position.
As an example, consider a mass hanging vertically on a spring attached overhead. Designate the rest position (equilibrium) of the hanging
mass as on the vertical line. Now compress the mass upward through cm and then release it at time
. Assume the future positions of
the mass are described by SHM and measure the time it takes for the mass to return to its initial compress position. Model this position as a
function of
assuming the time measured to complete one oscillation is
To find the function
we interpret the given information:
In order to apply the first condition we must have
The second condition yields
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seconds.
. So the assumption of SHM yields
, for otherwise
. The
implies
.
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With this function we can predict the motion of the mass. In particular, the frequency is
as
cycles/second, and the amplitude is
occurs at times
of a full oscillation per second, sometimes written
cm. If we want to know when the mass will be at the equilibrium position we set
. This
such that
that is,
where
because time is moving forward from
. In particular, the first time the mass reaches the equilibrium position is
seconds after it is released, when the mass is descending. It next reaches this position
the length of time between two successive equilibrium positions is
after it is released, when it is ascending. So, in general,
seconds, precisely half the period.
Since SHM does not model phase shift we need a slightly more sophisticated model to take this phenomenon into account. We now want to
consider functions of the form
We will see that phase shift can be analyzed using some basic identities that we will derive, as usual, from the geometry of the unit circle. This
analysis will explain how phase shift occurs in simple physical systems. The presence of a non-zero phase shift means that the input to the
. Thus it would help to understand how trigonometric functions behave in general
trigonometric function involves a sum of terms, say
when the input is of this form. A natural question is whether the output can be expressed in terms of fundamental trigonometric functions of the
terms individually. This question is the key to the analysis of oscillatory phenomena, from springs to signal analysis. To answer it we return to
the unit circle to develop identities for
unit circle and find the length
of the chord
and
. Locate the points
and
on the
. Using the distance formula, the square of this length is
However, the chord
subtends the central angle
Pythagorean Theorem we find that
, and is the base of the isosceles triangle
, with legs of length
. Using the
Equating these two expressions we find
since
is an even function. Since
Note also that the graph of the function
Therefore,
and so
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is an odd function, replacing
is obtained by translating the graph of
with
in the above identity yields
to the right by
. However,
. Thus
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These identities allow us to interpret phase shift in harmonic motion as follows:
In other words, harmonic motion with amplitude
motions, of equal frequencies, with amplitudes
that exhibits phase shift can be interpreted as the superposition of two simple harmonic
and
such that
This observation has practical applications when we consider the superposition of two SHMs with equal frequencies. For example, let
a sum of two SHMs with different amplitudes but equal frequency
Then
and
. Let
. Since
a single sine wave with phase shift
. Alternatively, let
Then again
, so we can write the function as
but now
we can rewrite this function as
a single cosine wave with phase shift
. The superposition of two fundamental functions has produced a single wave that exhibits
phase shift. The graph below shows this function as a sum of the two fundamental waves (dashed).
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Theorem. Let
and
, with
. Then
. Alternatively,
, where
, where
and
.
Note that the choice of
in the above theorem is determined by
. In applications, the value of
is called an initial
condition. As a corollary to this theorem, we can also obtain a general superposition law that allows us to add phase-shifted waves.
, let
and
. Then
Corollary. If
Alternatively,
Exercise. Suppose
and
. What relation between
produces the smallest possible amplitude for
and
produces the largest possible amplitude for
? What relation between
? Give an example of each case.
The following Suggested Exercises are not to be handed in but are a representative sample of techniques required for basic mastery.
Chapter 7 Suggested Exercises (Swokowski/Cole, CSUSB edition)
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7.1 (pages 464-466):
7.2 (pages 476-479):
7.3 (pages 487-490):
7.4 (pages 496-499):
7.5 (pages 503-504):
7.6 (pages 515-519):
Second Graded Assignment: Due May 16
To reinforce written communication skills the Graded Assignment solutions should be clearly presented in a "bluebook" or provided in .pdf
format. Do not send electronic scan of handwritten work. The assignments are chosen to encourage understanding of a variety of applications,
often within social and historical contexts, of developments within and related to the natural sciences. Late papers will not be graded.
Second Graded Assignment. Do any one of the following:
(considered one problem)
Page 489:
Page 490:
(considered one problem)
Page 498:
Page 504:
Page 518:
Page 520:
Trigonometric Equations and Identities
An identity is a mathematical equation which is satisfied by all values in the domains of the functions that appear in the equation. For example,
is an identity for all values of for which
and
values of the variables does equality hold? For example,
is not an identity, so we ask which values of
are defined. Equations that are not identities are actually questions: For what
satisfy this equations. In this example we must have
, and therefore
Finding the solutions of a trigonometric equation usually involves recognizing identities in order to obtain a workable form of the equation.
For example,
can be rewritten
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Thus
must be either
or
, with
. We then conclude
is the complete solution to the original equation.
As with solving any equation, factoring is an important technique. For example, the equation
can be written
and, since
Therefore
, we must have
because
has no real solution.
is the complete solution. In practice, we often look for solutions on a restricted domain. For example, if we want the solution to this equation
with
then we have
Most identities that we will need can be derived from the basic ones we have already verified along with the fundamental definitions. For
example, suppose we want to expand
We have by definition
and so
However, it might be useful in context to express this identity entirely in terms of the tangent function. One way to do this would be to divide
to obtain
numerator and denominator by
As an exercise, you should use this approach to obtain identities for
,
, and
entirely in terms of each
function, respectively. You can now verify the cotrig identities, which generalize right-triangle trigonometry for the trigonometric functions:
This strategy also works for the so-called half-angle identities. Consider, for example, the function
(We have seen that the period of this function is
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.) Suppose we want to write this function entirely in terms of the input
instead of
.
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We reason as follows:
Since this is an identity we deduce
Now we can solve for
in terms of
This identity, however, introduce certain difficulties. For example, if
then the LHS is clearly ; but we cannot evaluate the RHS
without some theorems from calculus. Also, the sign ambiguity in the numerator must be resolved by determining in which quadrants of the unit
circle the angles and
are located. (Which sign would you choose if
? What about
?) It is possible to write the halfangle identity for the tangent without the sign ambiguity. First, we obtain half-angle identities for sine and cosine, using
Using the same reasoning as for the tangent we obtain
Each of these has a sign ambiguity that must be resolved by locating the quadrants for
and
. But now
Notice that the LHS is defined precisely when the RHS is defined, so we can now work with the fraction
It follows that
because
. Finally, if
is in the second or fourth quadrant and so
which we can also write as
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then
because
is in the first or third quadrant. But if
. Thus, in either case, we can write
then
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It follows that we can write our function
as
With just the basic identities you can also verify conversions between sums and products, such as
Inverse Trigonometric Functions
If a function is one-to-one on its domain (
the property that
for all
in the domain of . We say that
for all
in the domain of
unless
and
) then there is a function
. Since each of the fundamental trigonometric functions is many-to-one on its whole domain, it is necessary to
and the full range of
we can find a unique
This function
function by
has
is defined by
then
one-to-one on the
is attained on this interval, so it is a good candidate for a restricted domain. Given any real number
such that
as its domain and
However, because
, in other words,
as its range. Borrowing the generic inverse function notation we often denote this
this notation can be confusing, so we will generally use
This notation reminds us that the trigonometric functions are defined on the unit circle: For example,
is the radian measure in the interval
of an arc on the unit circle whose tangent is
reflecting the graph
:
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with
are inverse functions of each other because it is also true that
restrict the domain in order to define an inverse function. For example, if the function
interval
whose domain is the range of
in the line
. The graph
means that
is obtained by
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Notice that the vertical asymptotes for
have become horizontal asymptotes after reflection in
. The function
is extremely useful in mathematics and its applications. Note that we can restate the superposition theorem for harmonic
motion in terms of this function.
Theorem. Let
, with
. Then
. Alternatively,
.
Inverse functions for the other fundamental trigonometric functions can be defined similarly. The domains are usually restricted so that
because of the identities
Thus,
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, with
(dashed)
which means that we restrict the domain of
on
to
, since that is the range of
, whose graph is reflection of
.
, with
(dashed)
The restricted domains for the other four functions are then obtained so that compositions among the trigonometric functions and their inverses
agree with the properties of right-triangle trigonometry. For
restricted domain is
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:
the restricted domain is
, and for
the
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, with
, with
Then
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(dashed)
(dashed)
has
as its range, which is also the restricted domain for
.
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, with
(dashed)
has
Finally,
, with
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(dashed)
as its range, which is also the restricted domain for
.
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Composition of Trigonometric and Inverse Trigonometric Functions
By definition, the composition of
with
that it is consistent with the definition of
, in either direction, is the identity function, provided we restrict the domain of
so
. This will be important when we form the compositions of the fundamental trigonometric
functions with the inverse functions of the others. Here is a summary of those domains, which are the ranges of the inverse functions
::
A common source of error, particularly when using calculators for applied problems, is failure to observe these domains, particularly because
but
the choice of restricted domain can vary in applications as well as in software design. As a simple example,
; however,
for all
. As we derive formulations for other compositions we will
assume the domains of the trigonometric functions have been declared as in the table above.
Note, however, that the choice of restricted domains can vary from text to text, and, more importantly, from one type of software to
another. In particular, if you have a graphing calculator try graphing
. there is a good chance it will display in the range
instead of
To continue, consider
Thus
composition. Let
.
and
. Then
because of the fundamental cotrig identity. However, we can be more explicit about this
, which we have seen is in
. Then
and so
and so
The sign ambiguity can be resolved by our precise definition of the functions involved. In fact, since
We can now begin to construct a composition table, where we list the function
intersection of row and column records the function
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:
it follows that
in the left column and the function
in the top row; the
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Note that if
is the inverse function of
then
. For example, if
and
This is the result we would expect from right-triangle trigonometry, but now we see that it is true when
when, for example,
is any real number. To evaluate
and
Here again the sign ambiguity is resolved by the fact that sine is an odd function and that the sign of
two more entries in the table because
Exercise. Show that the remaining entries in the table are as follows. Explain, for example, why
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we have
matches the sign of
. We now have
is always positive.
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Special Topic:
What about
? Not all of these compositions are possible. For example, if
composition is only defined at the isolated points where
domain of
range of
. Which of the other
is
and
is
. Similarly,
is also
then the
is only defined when the range of
compositions are not possible? On the other hand, if
and the domain of
Note, however, that the range of
, because
is only within the
and
then the
, so the composition is possible. To evaluate it, note that
so this evaluation is only valid for
in
, which is precisely the restricted domain of
, and, by the same reasoning,
and
. The other compositions are not so easily obtained. The attempt to understand them was an
important development in calculus. For example, is there any way to simplify
range of
is
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and the graph
on the interval
? Since the range of
looks like
is
the
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with tangent line for
Further,
with
and
in terms of related angles
Let
and
. Here is such a construction using the unit circle:
and suppose the terminal ray corresponding to
horizontal line through
intersect the tangent line to the circle at
, that is,
changes from
to
changing at a constant rate from
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, and let
be the angle
. Let
be the point where the
. Then
and
to
. How would we describe the rate at which the angle
? It is certainly not a constant rate of change. In fact, it can be shown with basic calculus that its rate of
change is given by the function
on the interval
intersects the unit circle at
. This is a geometric definition of the function graphed above. One way to understand its
properties is to imagine the angle
The graph of
. Mathematicians constructed geometric figures to express this equation
looks like
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gives the slope of the tangent line at the point
on the graph
and the slope of the tangent line is
. For example, if
then
. What is the slope of the tangent line to the graph
at the origin?
The following eight compositions are obtained similarly, and their graphs are constructed from their rates of change as given by
each
in the domain.
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for
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,
,
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;
;
,
,
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,
;
,
;
,
,
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Eight more compositions are possible and can be obtained from those graphed above. For example,
.
The function
that gives the rate of change of a function
for all values of
If
is the derivative of
and
is the derivative of
then the derivative of the function
then
, and the slope when
is a straight line with slope
then
, that is, the function has a constant rate of change. The derivative has the following important property:
For example, if
is
is called the derivative of . If the graph of
is
is
.
, so the slope of the tangent line to the graph of
when
. Use these properties together with the above graphs in the following assignment:
Extra Credit Assignment: Due June 11
A point
on the unit circle moves at a constant rate of
and point
,where
is the foot of the perpendicular through
to the tangent line to the circle at
to
. Let
. At time
be the angle
is the origin.
a) Make a precise sketch that shows the relation between
b) Determine
radian per second, counterclockwise from
when
and when
c) The graph below shows the rate of change
Express the function
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explicitly in terms of .
. Express
of
and
. Express
as a trigonometric function of .
explicitly in terms of
at each time . The range of
and sketch the graph of this function.
is negative because
decreases as
increases.
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d) Find the rate of change of
e) For what value of
in radians per second when
will the rate of change of
be
.
radians per second?
Second Midterm Exam: Areas of Focus
Trigonometric equations:
Finding all solutions
Finding solutions within a specific interval
Fundamental identities - be able to apply these when solving equations and when evaluating trig functions given values of other trig functions:
Pythagorean
Cofunction
Addition
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Half-Angle
Harmonic motion:
Given
, rewrite as
or as
Determine amplitude, period, frequency, phase shift
Find values of such that
Inverse trigonometric functions:
Domains and ranges
Algebraic form of compositions
Cofunction identities
, where
is an inverse trigonometric function and
is a fundamental trigonometric function
Chapter 8 Suggested Exercises (Swokowski/Cole, CSUSB edition)
8.1 (pages 532-534):
8.2 (pages 541-544):
8.3 (pages 554-557):
8.4 (pages 565--567):
8.5 (pages 572-574):
8.6 (page 579):
Third Graded Assignment: Due May 30
To reinforce written communication skills the Graded Assignment solutions should be clearly presented in a "bluebook" or provided in .pdf
format. Do not send electronic scan of handwritten work. The assignments are chosen to encourage understanding of a variety of applications,
often within social and historical contexts, of developments within and related to the natural sciences. Late papers will not be graded.
120.htm[6/6/2013 1:02:31 PM]
120.htm
Do any one of the following:
page 533:
page 543:
page 544
page 555:
page 567:
page 574:
page 579:
page 582:
Law of Sines
Let
be any triangle. In the Euclidean plane there is a unique circle through the vertices, called the circumcircle of the triangle. Let
the interior angle at
Then
where
where
Sines.
and let
be the length of side
is a right triangle and the angle at
is also
. Choosing either
or
be
, let the diameter through this vertex intersect the circle at
since it is subtended by arc
, which also subtends the angle at
.
. Thus
is the radius of the circle. Similarly,
is the angle at
Corollary. Let
with opposite side
be the area of triangle
and
is the angle at
with opposite side
. This is the proper statement of the Law of
. Then
Law of Cosines
The Law of Cosines computes the length of a side of a triangle in terms of the other two sides and the opposite angle. Euclid realized this as a
generalization of the Pythagorean Theorem and provides its proof in Book II of the Elements, Propositions 12 and 13. The sine and cosine
functions were not explicitly defined by Euclid, but the fact that two propositions were used to state this law (one for the case where the opposite
angle is obtuse and the other for the case where the opposite angle is acute) is a geometric interpretation of the fact that the cosine of an acute
angle is positive whereas the cosine of an obtuse angle is negative (Note that the sine of any angle in a triangle is positive.) Here is Euclid's proof
of Proposition 12, the obtuse angle case, using modern algebraic notation:
Let
be the obtuse angle and let
outside the triangle. Let
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be the altitude from
be the length of segment
to side
. Since
is obtuse this altitude meets the line
. By the Pythagorean Theorem we have
at point
,
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Thus, Euclid concludes that the square on the side opposite the obtuse angle is greater than the sum of squares on the other two sides by
the area of the rectangle on the side to which the altitude is drawn whose height is equal to the extension of that side to the foot of the
altitude. The awkward language is typical of the more technical results from ancient mathematics. In modern language we would say
that
Then
because
and we have the Law of Cosines:
Proposition 13 is proved similarly, but this time Euclid shows that the square on the side opposite an acute angle
is less than the sum of
squares on the other two sides by the area of a rectangle. Using the cosine notation we do not need a separate law, because now
so
and
. This is an example of how modern mathematics simplifies cases by introducing the language of functions.
Heron's Formula for the Area of a Triangle:
Since
we have
and from the Law of Sines
thus
Now, using the Law of Cosines
where
, the semi-perimeter of the triangle. The square of the area of the triangle can now be written as a simple formula
using only the lengths of the sides:
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Corollary: Radius of the circumcircle.
Vectors
Representations and complex numbers
A vector is a quantity that is characterized by magnitude and direction. Vectors are defined in any dimension but we will only study them here
in the plane. The segment
has a length
determined by the distance formula, but no
in two dimensions. Consider two points and
direction. If
we can specify a direction, for example, from
represents a vector. (If
. The directed line segment
, sometimes called an arrow,
the segment reduces to a single point and represents the zero vector, whose magnitude is
undefined.) Any directed line segment parallel to
,
direction. For example, if
or complex numbers
represents the same vector, as long as the arrow points in the same
and
,
then
and
and
represent the same vector. We say that
. We will interchangeably represent points in the plane by either ordered pairs
. Thus, every vector can be represented by an arrow from
, for unique values of
and whose direction is
with length
the arrows are equivalent; in particular, both arrows have length
complex number
to
to
, equivalently by a
.
Unit vectors and the unit circle
The magnitude of a vector is easy to compute as a non-negative number. Since the vector can be represented by
corresponds to the ray with vertex
The arrow
has length
that contains
. This ray intersects the unit circle at
. A vector whose magnitude is
elementary texts this complex number is abbreviated by the acronym
where
is the natural log base (
is denoted
The unit vectors corresponding to
is
. In many
), so we will use this notation. In the abstract, it is common to notate vectors with bold letters.
and
are often denoted
and direction
. If
it follows that the direction of
is the unit vector
and , respectively. As complex numbers we have
. It follows that the vector is represented by the complex number
which is called the polar form. In the polar form of a complex number, the angle
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.
. However, it is shown in calculus that
, which in context may be abbreviated to
In general, suppose a vector has magnitude
Addition and scalar multiplication
for some
is called a unit vector. Thus, the direction of any non-zero vector can be described
by a unit vector; equivalently, by a point on the unit circle. As a complex number, the point
The magnitude of vector
its direction
is called its argument.
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Vectors were developed historically in two related contexts: physics and geometry. These two contexts are naturally related by the need to
represent translation through a distance and physical quantities that depend on translations, such as velocity and force. For example, to describe
the velocity of an object it is necessary to represent both its speed (distance/time) and its direction of motion. Even before the invention of
calculus it was discovered that translations, forces and velocities add according to vector rules: If
and
are represented by arrows
and
then
is represented by the arrow
, where is the fourth vertex of the parallelogram
parallelogram law of addition. Notice that this law is equivalent to the coordinate addition law:
If
and
For example, if
then
, where
and
. This was known as the
.
then
, using
are
the complex number representation. To summarize, the other standard representations of
where
and
The
representation is sometimes called the component form of the vector. Another notation for the
. This notation is used to avoid confusion between the point
and the vector
representation is
. Thus
where
. Context determines
.
which of these equivalent notations is used. Note that
There is a natural way to change the magnitude of a vector. If is a real number then
The multiplication of a vector by a real
number is called scalar multiplication and it is applied to any of the standard representations, above, by multiplying each component or coordinate
by . Consequently,
. If
then
is the zero vector. If
then the direction of
is opposite the direction
of
. In particular,
is the vector whose unit vector is
parallelogram law says that if
and
if the unit vector for
then
is
. Also, since
is represented by the arrow
the
in the parallelogram.
Dot product, Angles and Law of Cosines
While complex numbers can be used to represent vectors in the plane (because the real and imaginary parts are the coordinates of a point) they
also behave like the real numbers that they generalize: they can be added multiplied and divided. The addition of complex numbers corresponds
to their addition as vectors, and the multiplication of a complex number by a real number corresponds to scalar multiplication. However, the
multiplication of two complex numbers generally does not directly correspond to a vector operation (physically, the multiplication of two forces
or two velocities is not a natural phenomenon). There is another operation with two vectors that does have important applications: the scalar
product, sometimes called the dot product:
where is the angle between and , measured by representing the vectors as arrows
not matter which angle we use. From the definition, the dot product is commutative
Suppose we use the component representation
and
and
. Since
it does
. Then
. By the Law
of Cosines, since
and so
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is the length of side
in the triangle
, we have
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Therefore, when
and
are written in component form,
from which it is easy to see that
and
For example, if
and
are represented by the complex numbers
and
as above, then
, and the angle between these vectors is found by
Now
, so
Normally we state the angle
, as expected.
between two vectors so that
.
The dot product offers a lot of flexibility in describing the geometry of vectors. Note that
and that
(Be careful to distinguish between the vector
and the scalar
.) Whenever
we say that the two vectors are orthogonal. If
neither is the zero vector then they are orthogonal precisely when the angle between them is
. If
or
we say the vectors are
parallel; these two cases maximize and minimize the scalar product.
Projection and Reflection
Consider a vector written in component form
. We say that
is the component in the direction
and
is the component in
the direction . Note that
where
non-zero vector
is the unit vector for
. Let
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. We can generalize this idea by defining the component of any vector
relative to a given
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the dot product of
and the unit vector for
If
then
is parallel to
directions. Note that
that is sometimes denoted
Exercise. If
and
if
if
and
. If this unit vector is
and
and the unit vector for
have the same direction whereas
are orthogonal. When
is
if
then
and
is multiplied against the unit vector for
are a pair of non-zero vectors that are orthogonal, and
have opposite
we obtain a vector
is any vector, then
A vector transformation that is important in many applications is the reflection of a vector in a given line through the origin. For example, the
in the horizontal axis results in the vector
. Any line through the origin has Cartesian equation
reflection of
for a unique
. One of the two unit vectors perpendicular to this line is
parallel to the line. Now let
The vector
because the vector
is
and define
has the same magnitude as
but its arrow is in the opposite half-plane relative to the line. Specifically,
Exercise. Use the dot product to show that the angle from
to the line of reflection is the same as the angle from the line to
.
Force, velocity
Even without calculus some basic applications can be described with vectors. One of the earliest discoveries described the resultant force
acting on an object as the result of multiple constant forces acting on that object. Each of these forces is represented by a vector whose magnitude
measures the amount of force and whose direction specifies the direction in which the force is applied. The resultant force is just the vector sum
of the individual forces. Force is defined by Newton's Second Law in terms of mass, distance, and time:
If mass is in kilograms, distance in meters, and time in seconds then force is in
where the symbol
A force of
stands for Newtons. Example:
acts on a body on a planar surface in the direction
body in the direction
The resultant force is
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. Simultaneously, as second force of
acts on the
. What is the resultant force? If no other forces act on the body, in what direction will it move?
. The magnitude of this force is
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, which is
. The direction the body will move is the direction of the resultant force,
which is given by the unit vector
, where
.
.
Thus
Velocity also has both magnitude and direction. The magnitude of a velocity is the speed. Velocities add as vectors, which is why aircraft need
to adjust velocity for the velocity of the wind so that the sum of the two results in the intended velocity.
The dot product has a natural interpretation as the work done by a force acting on a body. If a force
through a given distance in a given direction then
where
is the vector representing the displacement. Thus, if the force
straight line from the origin to the point
The total distance the body moves is
acts on a body so as to displace it
in the above example acts on a body that is constrained to move in a
the work done is
and the work done by the force is
Note that in this case
, a negative number. This means that
the force vector and the displacement vector is obtuse (
, where the symbol
stands for Joules.
, equivalently, the angle between
in this case). Note that
Complex Number Arithmetic
DeMoivre's Theorem
We have seen that complex numbers add and subtract like vectors. This is because they have a real part and an imaginary part: We write
and
. Since
we can also multiply complex numbers:
Any complex number other than
Thus we can perform division:
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has a reciprocal:
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It helps to introduce the notation
where
if
. The number
is called the complex conjugate of , the reflection of
in the horizontal axis. Note that
The multiplicative arithmetic of complex numbers is simplified when we use the polar representation
where
and
. If
and
then
This allows us to interpret multiplication of complex numbers geometrically:
The product of two complex numbers is the complex number whose magnitude is the product of the two magnitudes and whose argument
is the sum of the two arguments.
Since
, it follows that
This observation provides a simple method for computing integer powers. If
for any integer
. Let
To evaluate
with
then, just as for non-zero real numbers,
and
. Thus, for any integer
we first notice that
and that
. We then make a conjecture:
We cannot prove this conjecture by simply verifying it for larger and larger values of
. Instead, we use a principle from mathematical logic
called the Principle of Mathematical Induction, which states that if a conjecture has been verified for some integer such as
hypothesis that it is true for arbitrary
allows us to deduce that it is true for
have already shown that our conjecture is trivially true for
where
non-zero:
, and the
, then the conjecture is true for all positive integers. We
(and we even established it for
), so now take as our hypothesis
is arbitrarily chosen but unspecified. Since our hypothesis takes the form of an equation we can multiply both sides by anything
The LHS is just
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whereas the RHS expands to
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Therefore, we have deduced
from the hypothesis
This establishes De Moivre's Theorem:
,
If
, then for every integer
,
Note that De Moivre's Theorem holds for negative integers without further proof because if
Powers and roots
then
with
.
We now have a convenient theorem for computing integer powers of complex numbers that also allows us to solve equations of the form
where
is a given complex number. Apparently the solutions
just
is a square root of
so if
. For example, the solutions of
are
let
. Then
, that is,
we have
We conclude that
and
and
in the interval
does
solutions are
is chosen so that
. Certainly
since then
Since
works (it is the smallest value of
. The next largest value of
would be
the two solutions are
square roots are
number
roots of
, but we need to be clear what we mean by the square root of a complex number. We use De Moivre's Theorem to clarify this
concept. If
with
are all possible
. The radical notation
. There is no convention for non-real
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but this is not in
. For example, if
then
. So the two
,
, and the two
was originally defined to be the non-negative square root of the non-negative real
which of the two roots we call
The procedure for finding square roots works for
that solves the equation), but so
as opposed to
roots in general: First determine
.
, the unique non-negative
root of the real
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number
of
; then divide all representations of the argument of
by
, until the quotient is no longer in
. Example: Find the cube roots
. Here we want to solve
Since
and
the roots are
We can summarize this procedure in a theorem that is actually a special case of the Fundamental Theorem of Algebra:
If
there are exactly
solutions of the equation
,
Mathematical Induction
The proof of De Moivre's Theorem was a typical application of the Principle of Mathematical Induction. This principle is used frequently in
calculus to simplify sums that depend on an integer index, such as those that occur in the theory of integration. One such sum is probably familiar
The sum on the LHS depends on the index
. It has been reduced to a closed formula on the RHS which only requires substitution of the
index itself to evaluate. To prove that this formula is valid for any
we must first establish that it is true for at least one value of
the Basis Step of the induction proof. We try to find the smallest value of
for which the claim is true. Here,
. This is called
works because
Although the Basis Step is often trivial to verify, it must be carried out in precise form. Next is the Induction Step, where we hypothesize that
the formula holds for an arbitrary but unspecified value of the index, and then deduce from the hypothesis that the formula holds for the next
value of the index. Thus, from
we want to deduce
If the conjecture takes the form of an equation, as this one does, the deduction is usually carried out by altering each side of the equation to
produce an equivalent equation. Here we make the LHS of the hypothesis look like the LHS of what we want to deduce by adding
must add
. So we
to the RHS of the hypothesis as well:
If our conjecture is valid it should now be possible to rewrite the LHS of this equation so that it looks exactly like the RHS of what we want to
deduce:
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We have deduced the equation when the index is
theorem.
from the hypothesis, when the index is
. Therefore, the closed formula is actually a
Binomial Theorem
Historically, one of the first applications of mathematical induction was the expansion of binomials raised to integer powers. (This is the idea
behind Pascal's Triangle. Newton generalized these binomial expansions to the case when the power is any real number by using infinite series
techniques.) If is a non-negative integer the expansion of
can be collected into
terms of the form
Here we assume that any expression raised to the
power is formally equal to
, the idea being that no expansion would be necessary if any
term were zero to start with. The content of the Binomial Theorem is the determination of the coefficients
coefficients, but only by construction from the previous value of
. Pascal's Triangle provides these
. The Binomial Theorem provides them in closed form:
The symbol
denotes the number of subsets of size in a set of size
, that is, the number of ways to select objects from
distinct objects. With this interpretation it is reasonable that the Binomial Theorem should take this form: Expanding
by the distributive law involves choosing of the terms (and thus
of the terms) from the factors in
order to create the term
.
Summation notation is used for expressions of the type that occur in the Binomial Theorem:
This notation represents a sum of terms indexed by , indicating there will be one term for each value of
prove the Binomial Theorem we need a way to evaluate
where
, for any positive integer
exactly one way to choose no objects from
(there is exactly one subset of size
objects unchosen. The formula for
Basis Step. Let
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up to and including
. To
explicitly. We claim that
. We define
based on the set selection interpretation of
objects, that is, there is exactly one empty subset. It follows that
). More generally,
from
, since choosing
; there is
, and also
of the objects is equivalent to leaving
of the
makes this symmetry obvious. We can establish this formula by Mathematical Induction:
be any non-negative integer. Then
because there is exactly one empty subset of the set of
objects. But
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, so the formula holds for
Induction Step. Assume
.
for an arbitrary but unspecified value of
. We want to deduce from this
hypothesis that
We need to do something to
to change it to
. Remember that
new member to each subset. There are
collections of size
, and then do the same thing to
is the number of subsets of size
. We can make all subsets of size
objects to choose from for each subset of size
. But we have counted each of these collections exactly
members it was counted when we augmented the resulting collection of size
we need to divide
by
. Thus
.
Pascal's Lemma.
Proof.
The Binomial Theorem can now be stated as follows:
For any non-negative integer ,
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, so there are now
. So to get the exact number of subsets of size
This completes the induction step, so
Pascal's Triangle displays the binomial coefficients
by adding one
times, because if we remove each of its
From our hypothesis we have the equation
holds for every
and show this changes it into
because of the following result:
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Example. Find the terms in the expansion of
Let
and
that contains the factors
. We want the terms
and
and
and
.
. Since
these terms are
.
The Binomial Theorem shows how Pascal's Triangle works. If we assume
for some particular but unspecified value of
right side by
, then we would obtain
after multiplying both sides by
produces
whereas multiplying the right side by
produces
Then, after adding like terms, we have
The right side is now
plus terms of the form
However,
as we saw by Pascal's Lemma. Since
thus establishing the Binomial Theorem by mathematical induction.
Chapter 10 Suggested Exercises
10.4 (pages 715-716):
10.5 (pages 723-724):
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we have
. Multiplying the
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Fourth Graded Assignment: Due June 6
Do any one of the following:
page 715:
page 724:
Final Exam: Areas of Focus
The exam is cumulative but some material from the first two exams will be worked in. Review in particular the following ideas from
those exams:
Geometric relations among central angle, radius, arc length, and area of sector; Angular speed and linear speed, in correct units.
Constructing graphs of linear alterations of fundamental trigonometric functions:
asymptotes)
Trigonometric equations:
Finding all solutions or only solutions within a specific interval
Use of factoring and fundamental identities
Harmonic motion:
Given
(domain, range, intercepts,
, rewrite as
or as
Determine amplitude, period, frequency, phase shift (in terms of
Inverse trigonometric functions, particularly compositions
trigonometric function
)
, where
is an inverse trigonometric function and
Material since the first two exams covers the following ideas:
Law of Cosines
Vectors
Representation by arrows
Component form and vector addition
Unit vectors and scalar multiplication
Dot product and angle between vectors
Force, displacement, work
Complex numbers
Rectangular and Polar form
Arithmetic:
De Moivre's Theorem and roots
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is a fundamental
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Mathematical Induction
Study the suggested exercises from this section
Binomial Theorem
Find terms in expansion of
with given characteristics
Conics in Polar Form
Curves in the plane that are described by Cartesian equations of degree two correspond to conic sections and so are often called conics. A
polynomial in and of degree two takes the form
where
are constant coefficients with
the conic. For example,
not all zero. The pairs
which is just the unit circle. The affine type of the conic is determined by
following theorem:
Let
Examples. The curve
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that make this polynomial zero are the points on
gives
, the coefficients of the quadratic terms, according to the
. Then the conic is
has
, so
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Figure 1. Parabola:
The curve
has
, so
Figure 2. Hyperbola:
The curve
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has
, so
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Figure 3. Ellipse:
The three affine types correspond to the ways a cone can intersect a plane. Each type has degenerate cases associated with it: If the plane
intersects the cone only at its vertex we get a single point, a degenerate ellipse; if the plane is tangent to the cone we get a single line, a
degenerate parabola; if the plane passes through the vertex but is not tangent we get two lines, a degenerate hyperbola.
Exercise. Find coefficients that produce degenerate conics of all three types.
It is often preferable to express conics in polar form by using the representation
Note that this representation is equivalent to viewing each point
as the complex number
often simplifies the description of the conic. For example, the unit circle
which we can simply write as
equation
In general,
since
. In other words, the unit circle consists of all complex numbers
such that satisfy the
becomes
This expression suggests a "standard form" for conics: The conic passes through the origin if and only if
removes the constant term. Then
. This
becomes
satisfies the equation, and if
, so apply a rigid motion that
then
For example,
is the ellipse in Figure 3.
It is easy to show that the denominator in this expression can be zero only if
hyperbola. The values of such that
intersect the curve a second time. For example, consider the conic
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, that is, only when the conic is a parabola or
correspond to directions of lines through the origin that do not
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Figure 4.
Since
this is a hyperbola. The denominator is zero for
. The lines through the origin for these
directions are
and
, as shown in Figure 4. It also appears that the line
, the -axis, does not intersect the curve a second
time. But this line is tangent to the curve at the origin. In analytic geometry we say the tangent intersects the curve twice at the point of tangency.
By the same reasoning, the parabola
Figure 5.
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is intersected by the
(corresponding to
-axis exactly once. Every other line through the origin intersects the curve a second time, including the line
, which makes
), the tangent to the curve at the origin, as shown in Figure 5.
is not degenerate then the tangent line at the origin is
Theorem. If the conic
where
.
For example, the tangent line at the origin to the ellipse in Figure 3 is the
? Here
-axis because
yields
. What about the conic
. What is the Cartesian equation of the tangent at the origin, shown in Figure 6?
Figure 6. The ellipse
If the conic is degenerate the it is possible for the numerator and denominator of
is undefined for
degenerate conic is
because
Note, however, that cancelling the common factor
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to be zero for the same value of
. For example,
. The Cartesian form of this
in the polar form results in
, which eliminates the line
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and obscures the fact that this conic is a degenerate hyperbola.
Figure 7. Degenerate hyperbola
Special Topic: Polar Conics and Inverse Trigonometric Functions
Polar representation of curves is best understood in terms of vectors. A given input
is
, equivalently, the complex number
the point
on the curve. As
direction; if
produces a value , and the resulting point on the curve
. Now consider the vector represented by the arrow with tail at the origin and tip at
changes we can picture the length of this arrow changing. The length is
the arrow points in the
calculus we compute the derivative of
direction; if
as a function of
the arrow represents the zero vector. How is
in order to find the this rate of change, assuming
that we introduced this idea when we studied the inverse trigonometric functions. For example, if
we noted that the derivative is
The polar curve
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, so if
is an ellipse. Its Cartesian form is
the arrow points in the
changing as
changes? In
changes at a constant rate. Recall
with
,
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Figure 8.
The vertical axis is the tangent to this curve at the origin. We can generalize this idea to produce a standard polar form for any non-degenerate
conic: We want the curve to pass through the origin and have its tangent at the origin be the vertical axis. These conditions imply that
; in particular, there is no
term in the denominator. Suppose, now, that the conic is a non-degenerate ellipse, so
we can represent it as
with
and
. Then
after setting
and
(since
). Note that this ellipse is similar to the ellipse
because multiplication by
is a dilation about the origin. We say that the two ellipses have the same eccentricity, which we define to be
In Figure 8 we have
, so
and
they have the same eccentricity, so we can assume
eccentricity
is a circle. Writing
. Then the ellipse becomes
and its eccentricity is
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. Thus the eccentricity of this ellipse is
. Two ellipses are similar if and only if
to represent all possible eccentricities. Then
, let
where
and an ellipse with
since we are assuming
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Using calculus we can show that
Thus, setting
is the derivative of
we obtain the ellipse with eccentricity
Question: For the ellipse in standard form,
horizontal axis?
as the rate of change of this inverse trigonometric function.
locates the point furthest from the vertical axis. Which points are furthest from the
By symmetry, these are the two points whose
-coordinate is
. Since
we want to solve
equivalently
Since
the two points are
, and so the maximum vertical width of the ellipse is
. (The intersection
of the line through these two points with the horizontal axis is called the center of the ellipse.)Note that, for
, where
Exercise. Let
be its distance to
,
is the eccentricity.
and
. For any point
on the hyperbola let
be its distance to
and let
. Use the Law of Cosines to show
Then show that
. Thus: There are two points (foci) inside any ellipse, the sum of whose distances to any point on the ellipse is
constant. If the ellipse is a circle then the two foci coincide (at the center of the circle).
The origin and the point
are called the vertices of the ellipse. This verifies the general principle that the eccentricity is the distance
between the foci divided by the distance between the vertices.
Hyperbolas can also be described in terms of inverse trigonometric functions. Using
it is possible to express
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as the logarithmic function
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from which it follows, by substituting
Thus,
, that
is actually a real-valued function of
which describes a hyperbola provided
Figure 9.
Setting
for
and the same methods from calculus show that its derivative is
. For example, if
with asymptotes
for
we can write
and the eccentricity of the hyperbola is
The vertices of the hyperbola are the intersections with the horizontal axis so, as with the ellipse, the distance between the vertices is . We
can determine foci for the hyperbola by finding two points on the horizontal axis symmetric about the vertical line in Figure 9 that is half way
between the vertices. The intersection of this vertical line and the horizontal axis is called the center of the hyperbola. Again, the eccentricity is
the distance between the foci divided by the distance between the vertices:
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so the foci are the points
Exercise. Let
and let
and
be its distance to
. For any point
on the hyperbola let
be its distance to
. Show that
Thus: There are two foci for any hyperbola, the absolute difference of whose distances to any point on the hyperbola is constant.
and
are functions of
The distances
hyperbola with directions given by
that have
in their denominators. The two lines through the center of the
are called the asymptotes of the hyperbola. These are shown in Figure 9 for the hyperbola with
.
Parabolas
The horizontal line
Consider the triangle
hypotenuse
, and so
in polar form is
with right angle at
. The rate of change of
,
on the horizontal line, and
measures how fast the hypotenuse is changing assuming
parabola, with the horizontal axis as a line of symmetry. A larger value of
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as a function of
is given by
. Then
measures the
changes at a constant rate. As a polar curve
results in a "wider" parabola, as in Figure 10.
is a
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Figure 10.
and
However, all of these parabolas are similar as figures in the plane since
every parabola to have the same eccentricity,
single focus, on the line of symmetry. Let
distance between
and
and, since
is just a dilation constant relative to the origin. Thus we expect
, which agrees with the formula
be the point
and let
since
be any point on the parabola. By the Law of Cosines, if
is the
then
,
However, the distance between
and the vertical axis is
and
Thus the distance between and
is always equal to the distance between and the vertical line
parabola and this vertical line its directrix. For any parabola there is a focus and a directrix with this property.
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. A parabola has a
. We call
the focus of the
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Figure 11. Distance to focus equals distance to directrix
Exercise. Let
triangle
be the vertex of a parabola and let
be its focus. Let
be a point on the parabola such that
is equilateral.
Project: Conics
Suggested project items:
Page 768:
Page 780:
Page 781:
Page 792:
Page 827:
Page 827:
(refer to
on page 768)
Due date: June 6
Format: Any standard word or math processing system may be used, but the following format must be observed.
Title Page: Page 1 should contain only the following information:
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. Show that
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Introduction: Starting on Page 2, explain in your own words (using complete sentences) the context of the problem you will solve. Define any
terms you use and explain how the mathematics of conics is involved. Keep this introduction to a single page.
Solution: Starting on Page 3, present your solution. Be clear and complete, and use your notation consistently. Separate out equations from the
text. For example, most word processing software allows you to
equations. If you cite references other than the course textbook then list them at the end of your solution. For example:
Hermann Bondi, Relativity and Common Sense, Dover, 1986.
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