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 c
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
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
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
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 f
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
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
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
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
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
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) vertical
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 s
5.4
Solving Right Triangles
Tosolveatrianglemeanstofindthemeasuresofalltheanglesandsidesof
thetriangle.
EXAMPLE
SolverighttriangleABC,ifA=300andc=12in.
Find a, then
find b
ApplicationsInvolvingRightTr
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
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
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
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
log
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, w
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 S
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:_
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
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
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 en
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
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
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 line
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
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
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