King
K
Fahd Universityy of Petrolleum and Minerals
P
Prep-Year
r Math Prrogram
Math 0002 Term
m 132
Reciitation (5.11)
Question 1:
Lett A and B be the sm
mallest positive annd the larrgest negaative coteerminal anngles
withh
2 B=
873 , then 2A
A) 513
B) 5
7.5
INVERSE TRIGONOMETRIC FUNCTIONS
Inverse Trigonometric Functions
Consider y sin x . The domain if sin x is ( , ). So, it is not one-to-one function
and so has no inverse. However, by looking at the graph of
y sin x on the
interval 2 x 2 . It is very
9.7
PROPERTIES OF MATRICES
Matrices are used as an aid to find solutions of systems of linear equations. In
this section we discuss the algebraic properties of matrices.
An
mn
matrix
A
may
a11 a12
a
a22
21
a31 a32
.
A aij .
.
.
.
.
am1 am 2
a13
a23
10.3
THE HYPERBOLA
Definition of an Hyperbola
A hyperbola is the set of all points in a plane, the difference of whose distances
from two fixed points( the foci) in the plane is a positive constant.
d (V , F ) d (V , F ) (c a ) (c a ) 2a,
So the constant
6.1
Radian Measure
Radian Measure
This is another system to measure angles. To introduce its definition, we need
to define what is meant by a central angle and an arc. The central angle is
the angle formed by the two radii. The portion of the circle betwe
1
7.6
TRIGONOMETRIC EQUATIONS
1
where 0 x 2
2
Solve sin x
Solution
Since sin x > 0 then x is in the first or second quadrant.
So, x
6
(quadrant I) or x
5
(quadrant II).
6
5
Since x is restricted between 0 and 2 , then the solution set is ,
6 6
Now l
1. If the ordered triple (a, b, c) is a solution for the system of
1 2 1 5
2 1 3 0 ,
equations whose augmented matrix is 1 1 1 2 then 3a + 4b + c =
a. 13
b. 7
c. 15
*d. 9
e. 0
2. The linear system
x + y + z = 0
2 x 2 y + z = 1
3 x + y 2 z = 3
a.
b.
c.
4.4
CHANGE OF BASE RULE
CHANGE OF BASE RULE
Change-of-Base Formula
If a and b are positive numbers not equal to 1 and x is
log a x
log b x
positive, then
log a b
log b x
log x
log b
or
log b x
ln x
ln b
EXAMPLE
Write the expression log 2 x 4log8 y as
6.5 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS
FNCTIONS
THE GRAPH OF THE TANGENT FUNCTION
In this section we will discuss how to graph the other trigonometric functions,
namely, the tangent function y a tan bx , the cotangent function y a cot bx ,
the secant
6.3GRAPHSOFTHESINEANDCOSINEFNCTIONS
THE GRAPH OF THE SINE FUNCTION
In this section we will discuss how to graph y a sin bx and y a cos bx
and some of their properties. Let us start with the graph of y sin x .
First, we need to remind our self about some o
6.4 Translations of the Graphs of the Sine and Cosine
Functions
We have two kinds of translation, either vertical translation where the graph
of y f ( x) c is a shift of the graph of y f ( x) vertically c units down or
up. For c 0 , the graph of y f ( x)
7.3
SUM AND DIFFERENCE IDENTITIES
sin sin cos cos sin
cos cos cos sin sin
tan
tan tan
1 tan tan
tan
tan tan
1 tan tan
These identities can be used to find the sum and difference formulas for sec x , csc x
and cot x .
Find cos15
EXAMPLE
Solution
15
5.4
Solving Right Triangles
Tosolveatrianglemeanstofindthemeasuresofalltheanglesandsidesof
thetriangle.
EXAMPLE
SolverighttriangleABC,ifA=300andc=12in.
Find a, then
find b
ApplicationsInvolvingRightTriangles
Inapplicationswemayusethelineofsightasareferenc
10.1
THE PARABOLA
Definition
A parabola is the set of points in a plane that are equidistant from a fixed
line (the directrix) and a fixed point (the focus) not on the directrix. The
line through the focus and perpendicular to the directrix is the axis of
7.1 -7.2
VERIFICATION OF AN IDENTITY
DEFINITION
An identity is an equation that is true for all the values in the domain of the
terms in the equation.
EXAMPLE
What is the domain for which
sin x cos x
sin x is an identity?
cos x
Solution
All terms in the
5.3
EVALUATING TRIGONOMETRIC FUNCTIONS
Trigonometric Functions of Special Angles
Let us start with the 45 angle. If we draw a right triangle with a 45 angle, then
the remaining angle is 180 45 90 = 45.
B
45 a
x = 2a
A
sin 45 =
cot 45 =
a
a 2
=
C
a
1
2
a
2
4.5
EXPONENTIAL AND LOGARITHMIC EQUATIONS
EXAMPLE
Solve the equation 32 x1 8
Solution
If we take the common logarithm of each side we get:
log 32 x1 log8
(2 x 1) log 3 log8
2x 1
2x
log8
log 3
log8
1
log 3
1 log8 1
1
the exact solution is x
or x lo
5.2
TRIGONOMETRIC FUNCTIONS
Trigonometry means triangle measurement. As we will see later in this
section, trigonometry can be used to solve application problems.
To define a trigonometric function, we will use the right triangle. If we
have a right trian
Text Section: Chapter 4 C L A S S Q U I Z 2 Monday, March 15, 2010
King Fahd University of Petroleum and Minerals
Prep-Year Mathematics Program
MathOOZ _Term 092
.o
F St. ID: F Name: 5 O L U l l O N Section; _ Serial: "
is increasing.
Q1: Find the int
Text Section: 4.2 4.3 C LA S S Q U I Z I (WWW March 03, 2010
King Fahd University of Petroleum and Minerals
PrepYear Mathematics Program
MathOOZ _Term O92
St. ID: "' Name: 5 O l U T l O N Section:_ _ Serial: '
1
Q1: Find the x and y - intercepts of the
King Fahd University of Petroleum and Minerals
Prep-Year Math Program
Math 002 - Term 132
Recitation (4.1)
Question1:
Decide whether each of the following functions are one-to-one.
f 1 ( x) for those
Find functions that are one to one.
3
2x 1
(a) f ( x) =
9.2
Matrix Solution of Linear Systems
In this section we introduce a powerful method known as Gauss-Jordan
Method which is used to solve systems of linear equations in two or more
variables. First we need to introduce some new terminologies.
THE AUGMENTED
5.1
ANGLES
Definition - Ray
A ray is a half-line that begins at a point, called endpoint, and
extends indefinitely in some direction
Endpoint
EXAMPLE
Definition
- Angle
Two rays that share the same endpoint form an angle. In such a case the
common endpoin
4.3
LOGARITHMIC FUNCTIONS
The function f x b x is a one-one function and therefore has an
inverse function f 1 ,
f 1 x y if and only if x f y
In this case f 1 is called the logarithmic function with base b and
is denoted by log b .
Definition of Logarith
8.3
VECTORS IN THE COORDINATE PLANE
DEFENITION: VECTORS
vectors are equal if they have the same direction and magnitude, then any vector v
in a rectangular coordinate system ( xy plane) is equal to a vector with its initial
point at the origin and its ter
9.1
SYSTEMS OF LINEAR EQUATIONS
A set of equations is called a system of equations. If all the equations in a
system are linear, the system is a system of linear equations.
Examples of systems of linear equations in two variables x and y are:
x 3y 2
3 x 7
9.5
SYSTEMS OF NONLINEAR EQUATIONS
Solving Nonlinear Systems of Equations
A system of equations in which at least an equation is nonlinear is called a
nonlinear system.
2 3 1
y x 2 x y 2 3 x 2 2 y 2 12 y log( x 1)
For example,
,
,
,
,
2
2
3
1
y
3
log(
9.3
Determinant Solution of Linear Systems
Associated with each square matrix A is a real number called the determinant of
A , denoted by A or det A ( reads determinant of A). Another notation for the
determinant of a matrix is to replace the brackets by
6.2
Unit Circle and Circular Measure
The Circular Function
Let us consider a circle given by the equation x 2 y 2 1. We call it the unit
circle. Now, let us draw a vertical line l tangent to the unit circle at (1,0). If
we move t units on the line l , the
9.8
THE INVERSE OF A MATRIX
THE IDENTITY MATRIX I n
The identity matrix I n is the square matrix of order n that has only 1 in each
position on the main diagonal and 0 elsewhere.
Example:
1 0 0 0
1 0 0
0 1 0 0
1 0
, I3 0 1 0 , I 4
I2
0 0 1 0
0 1
0 0
4.2
4.2 EXPONENTIAL AND LOGARITHMIC
FUNCTIONS
Definition of Exponential Functions:
The function f is an exponential function if
f x bf iswhere
b 0 andfunction
b 1, and
The function
an exponential
if x is any real number
x
The number
x
is called the expone
7.4
DOUBLE AND HALF ANGLE IDENTITIES
DOUBEL ANGLE IDENTITIES
Let us first begin by the sine of a double angle:
Since sin sin cos cos sin , then
sin 2 sin sin cos cos sin
sin 2 2sin cos
cos 2 cos cos cos sin sin
cos 2 cos 2 sin 2
cos 2 2cos 2 1 and c
10.2
THE ELLIPSE
Definition
An ellipse is the set of all points in a plane the sum of whose distances from two
fixed points in the plane is constant. Each of the fixed points is called a focus
(plural: foci) of the ellipse. The midpoint of the line throug
Practice Problems Chapter
9.1-9.5
1. If (m,n) is the solution of the system
2 3
x + y = 5
,
3 + 4 = 5
x y
then m-n is equal to:
2
*a. 5
2
b. 5
1
c. 9
1
d.
22
5
e. 16
2
x +
3 +
2. Given the system x
3
= 66
y
,
4
= 91
then x is equal to:
y
1
a. 16
1
b. 3
1