Download Trigonometry Review and Basic Calculus 2011

Pre-requisite Topic – Trigonometry Review and Basic Calculus
I.
Review of Trigonometric Functions.
a. Angles and Degree of Measure.
i. An angle has three parts: an initial ray, a terminal ray, and a vertex (the point of
intersection of the two rays.
ii. An angle is in its standard position if its initial ray coincides with the positive xaxis and its vertex is at the origin.
iii. It is common practice to use theta, , to represent both an angle and its measure.
iv. Positive angles are measured counterclockwise and negative angles are
measured clockwise.
v. To measure an angle, you must know how the location of the initial and terminal
rays as well as how the terminal ray was revolved.
1. For example. -45o has the same terminal ray as 315o.
2. These angles are coterminal,  and  + n(360).
b.
Radian Measure.
i. To assign a radian measure to an angle , consider  to be a central angle of a
circle of a circle of radius 1.
ii. The radian measure of  is then defined to be the length of the arc of the sector.
iii. Because the circumference of a circle is 2r, the circumference of a unit circle
(of radius 1) is 2.
1. This implies that the radian measure of an angle
measuring 360o is 2.
2. 360o = 2 radians.
iv. Using radian measure for , the length s of a circular arc of
radius r is s = r.
v. You should know conversions of the common angles.
Degrees
30
45
60
90
120
180
270
360
Radians
π/6
π/4
π/3
π/2
2π / 3
π
3π / 2
2π
c.
The Trigonometric Functions.
i. There are two common approaches to the study of trigonometry.
1. In one, the trigonometric functions are defined as ratios of two sides of
a right triangle.
2. In the other, these functions are defined in terms of a point on the
terminal side of an angle in standard position.
r = √(x2 + y2)
y
x
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Pre-requisite Topic – Trigonometry Review and Basic Calculus
ii. We define the six trigonometric functions, sine, cosine, tangent, cotangent,
secant, and cosecant from both viewpoints.
Definition of the Six Trigonometric Functions
Right triangle definitions, where 0 <  , /2
sin  = opp./hyp.
cos  = adj./hyp.
tan  = opp./adj.
csc  = hyp./opp.
sec  = hyp./adj.
cot  = adj./opp.
Circular function definitions, where  is any angle
sin  = y/r
cos  = x/r
tan  = y/x
csc  = r/y
sec  = r/x
cot  = x/y
iii. The following trigonometric identities are direct consequences of the definitions
( is the Greek letter phi.)
Trigonometric Identities [Note that sin2 is used to represent (sin)2.
Pythagorean identities
Reduction formulas
sin2 + cos2 = 1
sin(-) = -sin
sin() = -sin()
tan2 + 1 = sec2
cos(-) = -cos
cos() = -cos()
cot2 + 1 = csc2
tan(-) = -tan
tan() = tan()
Sum or difference of two angles
Half-angle formulas
Double angle formulas
2
sin(± ) = sincos ± cossin
sin  = ½ (1- cos2)
sin 2 = 2 sincos
cos(± ) = coscos -/+ sinsin
cos2 = ½ (1+ cos2)
cos 2 = 2 cos2 - 1
tan(± ) = tan±tan/(1-/+
= 1 - 2 sin2
tantan
= cos2 - sin2
Law of Cosines
Reciprocal formulas
Quotient formulas
a2 = b2 + c2 – 2bc cosA
csc = 1/sin
tan = sin/cos
sec = 1/cos
cot = cos/sin
cot = 1/tan
b
a
A
c
d.
Evaluating Trigonometric Functions.
i. There are two ways of evaluate trigonometric functions.
1. Decimal approximation with a calculator set to the proper mode.
2. Exact evaluations using trigonometric identities and formulas from
geometry.
ii. The quadrant signs of sine, cosine, and tangent functions should be known.
iii. Reference angles may be applied to angles in quadrants other than the first,
using the appropriate quadrant sign.
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Pre-requisite Topic – Trigonometry Review and Basic Calculus
e.
Solving Trigonometric Equations.
i. How would you solve the equation sin = 0?
ii. You know the  = 0 is one solution, but this is not the only solution.
iii. Any one of the following values of  is also a solution.
1. …, -3, -2, -, 0, , 2, 3, …
2. You can write the infinite solution set as {n: n is an integer}.
f.
Graphs of Trigonometric Functions.
i. A function f is periodic if the exists a nonzero number p such that f (x+p) = f(x)
for all x in the domain of f.
ii. The smallest such positive value of  (if it exists) is the period of f.
iii. The sine, cosine, secant, and cosecant functions each have a period of 2.
iv. Tangent and cotangent functions each have a period of .
v. The maximum and minimum values for the sin x and cos x that oscillate between
–a and a is known as the amplitude.
1. For y = a sin bx or y = a cos bx, the period = 2 / | b | and the amplitude
is | a |.
2. The amplitude only applies to sine and cosine.
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Pre-requisite Topic – Trigonometry Review and Basic Calculus
II.
Calculus.
a. Introduction.
i. Famous physicists have often been interviewed for television or written about in
books. In this context they invariably talk about the “beauty” of a particular
physical theory. Though the beauty of a theory of physics is certainly a
subjective notion, it is always linked to the elegance of the mathematics used to
describe the theory. Mathematics is the language of physics. It provides the
precision necessary to make statements that can be tested by experiment.
Mathematics also provides mechanisms that can be used to link concepts in new
and different ways and in so doing it provides predictive power to the theoretical
physicist. Of course, a physicist sitting at a desk does not start manipulating
equations at random. The conceptual content of the laws is foremost in the mind
of the physicist; he or she is always thinking of conceptual interpretations of the
equations as they are manipulated into various forms. This skill is one that can
be developed, and one of the first steps along this path is gaining a from grasp of
the mathematics used to describe physical laws.
As a student in AP Physics C, you will need a solid understanding of vectors and
their algebra, of differential and integral calculus, and also an introduction to
differential equations. These mathematical concepts will be introduced to you
with their physical application always near. There are times when the focus is
primarily on the mathematics, but this will always quickly be followed by
applications that show you how the mathematics actually manifests itself in the
world. At times the instruction exceeds what is required for the AP exam,
because it is felt that this extra level of understanding will provide valuable
insight into much material that is actually relevant to the test. We will work
many problems both together and on your own. Many problems are directly
relevant to the AP curriculum and test. As you master the material, you will
begin to recognize the beauty and elegance displayed in the laws of physics.
ii. During the seventeenth century, European mathematicians were at work on four
major problems. These four problems gave birth to the subject of Calculus. The
problems were the tangent line problem, the velocity and acceleration problem,
the minimum and maximum problem, and the area problem. Each of these four
problems involves the idea of limits.
iii.
The tangent line problem.
1. There is a given function (f), and a point (P} on its graph. The idea of
this problem is to find the equation of the tangent line to the graph at
that point. This problem is equivalent to finding the slope of the tangent
line at that point. This may be approximated by using a line through the
point of tangency and a second point on the curve (Q)—this gives us a
secant line.
2. As point Q approaches point P, the secant line will become a better and
better approximation of the tangent line. This uses the concept of
limits—the limit as Q approaches P will give you the slope of the
tangent line. In other words, choosing points closer and closer to the
point of tangency would give you more accurate approximations. The
derivative of a function gives us the slope of the tangent line to the
function.
3. Although partial solutions to this problem were given by Pierre de
Fermat (1601-1665), Rene Descartes (1596-1650), Christian Huygens
(1629-1695), and Isaac Barrow (1630-1677), credit for the first general
solution is usually given to Sir Isaac Newton (1642-1727) and Gottfried
Leibniz (1646-1716).
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Pre-requisite Topic – Trigonometry Review and Basic Calculus
Figure 2-3 Graph of x versus t for a particle moving in one dimension. Each point on the curve
represents the position x at a particular time t. We have drawn a straight line through points (x1, t1)
and (x2, t2). The displacement Δx = x2 − x1 and the time interval Δt = t2 −t1 between these points
are indicated. The straight line between P1 and P2 is the hypotenuse of the triangle having sides Δx
and Δt, and the ratio Δx/Δt is its slope. In geometric terms, the slope is a measure of the line's
steepness.
iv. The velocity and acceleration problem.
1. The velocity and acceleration of a particle can be found by using
Calculus. This was one of the problems faced by mathematicians in the
seventeenth century.
2. The derivative of a function can not only be used to determine slopes,
but also to determine the rate of change between two variables. This
may be used to describe the motion of an object moving in a straight
line. This is the position function, which, if differentiated (or the
derivative of it is found) gives us the velocity function. In other words,
the velocity function is the derivative of the position function.
3.
You may also find the acceleration function by finding the derivative of
the velocity function. So the velocity and acceleration problem helped
in the development of Calculus.
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Pre-requisite Topic – Trigonometry Review and Basic Calculus
v. The minimum and maximum problem.
1. What if we want to examine a function by finding where it is
increasing? Where it is decreasing? What is the behavior of its
concavity? When does it have a maximum point? Where does it have a
minimum point? All of these questions were answered with the
development of Calculus. The minimum or maximum of the function
must occur at a critical point, or a critical number. If we find the
derivative of a function, its zeros are called critical numbers.
2. Now, we must analyze the behavior of the function. The values over
which the derivative is positive equates into the actual function
increasing. When the derivative is negative, the function is decreasing.
If the function is increasing, and then changes to decreasing, that point
is a relative maximum of the function. Similarly, if the function is
decreasing, and then changes to increasing, that point is a relative
minimum. An easier way to analyze the minimum and maximum
problem is to graph the derivative. If the point to the left of the critical
number is a negative, and the point to the right of it is a positive, then
the critical number is a minimum of the function. Similarly, if the point
to the left of the critical point is a positive, and the point to the right is a
negative, the point is a maximum of the function.
3. We may also analyze concavity. If the second derivative of the function
is positive over a given interval, then the function is concave up over
that given interval. If the second derivative is negative, then the
function is concave down.
vi. The area problem.
1. This classic Calculus problem is used to find the area of a plane region
that is bounded by the graphs of functions. Like the tangent line
problem, the limit concept is applied here. To approximate the area of
the plane region underneath the graph, one may break the region up
into several rectangles, and sum up the values of the rectangles. This
method is a form of the Riemann Sums. This would give an
approximation of the area of the graph.
2. Now, if the amount of rectangles is increased, the approximation will
become more and more precise. The area will therefore be the sum of
the areas of the rectangles as the number of rectangles increases
without bound. In other words, the limit as the number of rectangles
approaches infinity, will give you the area of the region. This
eventually leads into the idea of integration.
Figure 2-12 Graph of a general vx(t)versus-t curve. The total displacement from
t1 to t2 is the area under the curve for this
interval, which can be approximated by
summing the areas of the rectangles.
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Pre-requisite Topic – Trigonometry Review and Basic Calculus
b.
c.
Basic Differentiation Rules/Methods to Get Us Started.
i. d/dx (xn) = nxn-1
ii. d/dx (ex) = ex
iii. d/dx (ln x) = 1/x
iv. d/dx (sin x) = cos x
v. d/dx (cosx) = -sin x
vi. df/dx = df/du ∙ du/dx (Chain Rule and u-substitution)
vii. Future rules and techniques we will use: product rule, quotient rule, “u”
substitution.
Basic Integration Rules/Methods to Get Us Started.
i. Integration is the “inverse” of differentiation.
ii. The operation of finding all solution of an equation is called anti-differentiation
(or indefinite integration) and is denoted by an integral sign ∫.
iii. y = ∫ f(x)dx = F(x) + C, where C is the constant of integration.
iv. Formulas.
1. ∫ xndx = xn+1 / n+1 + C, n ≠ 1 (Power Rule)
2. ∫ cos x dx = sin x + C.
3. ∫ sin x dx = -cos x + C
4. ∫ dx/x = ln |x| + C.
5. ∫ ex dx = ex + C.
v. The Fundamental Theorem of Calculus.
1. If a function f is continuous on the closed interval [a, b] and F is the
antiderivative of f on the interval [a, b], then:
b
∫a f(x) dx =F(b) – F(a).
2.
This is used to define a definite integral. Notice that the integration
constant C is eliminated.
3. This theorem helped solve the “area problem.”
vi. The process of finding the area under a curve on the graph illustrates integration.
1. The total area under some stretch of a curve is found by summing all
the area elements it covers and taking the limit as each ti approaches
zero.
2. ∫ f dt = areai = lim  fiti Refer to Figure 2-12 on the previous page.
ti 0 i
Citations
Mooney, James. Physics - Calculus of AP* Physics C And Beyond. Peoples Pub Group, 2005. Print.
Tipler, Paul Allen, and Gene Mosca. Physics for Scientists and Engineers. New York, NY: W.H. Freeman, 2008. Print.
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