Brief History
Hipparchus of Nicaea the founder of trigonometry, begun to use trigonometric methods in a
systematic way to study astronomy and hence was able to approximate the radius of the earth.
Trigonometry remained a tool for astronomy until sometime
Circular Motion
The motion of a point along a circle is described in two ways:
1. Linear Velocity
2. Angular Velocity
Linear Velocity
the rate at which the distance traveled is changing (distance traveled divided by the time elapsed).
s
is the measure o
ARCLENGTH, SECTOR, LINEAR VELOCITY AND ANGULAR VELOCITY
ARCLENGTH
1. Length of a Circular Arc:
An angle whose radian measure is is subtended by an arc whose length is the fraction (/2) of
the circumference of the circle C.
s = (/2) C
since C = 2r
therefor
AREA OF A SMALLER SEGMENT
A Smaller Segment is the portion within the sector but outside the triangle formed by the radaii and the
cord connecting the endpoints of the radaii.
Example:
1. Find the area of a smaller segment of a circle with radius 12 cm an
AREA OF A CIRCULAR SECTOR
Area of a Circular Sector: As
Circular sector A sector may be defined as a portion of a circle bounded by two radii and their
intercepted arc.
Let As = area of a sector, then by ratio and proportion,
area of a sector central angl
TRIANGLE
DEFINITION
a polygon of three sides, three angles and three vertices.
A closed curve in a plane formed by three non-collinear points.
The three points are called the vertices of a triangle.
The three line segments joining the vertices are cal
GRAPHS
Discrete Mathematical Structures
<professor>
INTRODUCTION TO GRAPHS
Definition: A simple graph G = (V, E) consists of
V, a nonempty set of vertices, and E, a set of
unordered pairs of distinct elements of V called
edges.
A simple graph is just like
RELATIONS
Discrete Mathematical Structures
<professor>
RELATIONS
To describe a relationship between elements of two sets A
and B, it can use ordered pairs with the first element
taken from A and the second element taken from B.
Since this is a relation be
TREES
Discrete Mathematical Structures
<professor>
Computer Science Department
TREES
Definition: A tree is a connected undirected
graph with no simple circuits.
Since a tree cannot have a simple circuit, a tree
cannot contain multiple edges or loops.
Ther
EQUIVALENCE
RELATIONS
Discrete Mathematical Structures
<professor>
Computer Science Department
EQUIVALENCE RELATIONS
Def 1. A relation R on a set A is called an
equivalence relation if it is reflexive, symmetric,
and transitive.
Example 1.
Let R be the re
BOOLEAN
ALGEBRA
Discrete Mathematical Structures
<professor>
Computer Science Department
BOOLEAN ALGEBRA
Boolean algebra provides the operations and
the rules for working with the set cfw_0, 1.
The three operations are the following:
Boolean complementati