MAPA INSTITUTE OF TECHNOLOGY
Department of Mathematics
VISION
The Mapua Institute of Technology shall be a global center of excellence in education by providing instructions that are curr ent in
content and state-of-the-art in delivery; by engaging in cut

EXPONENTIAL AND
LOGARITHMIC
FUNCTIONS
EXPONENTIAL FUNCTION
If x and b are real numbers such that b > 0 and
b 1, then f(x) = bx is an exponential function
with base b.
Examples of exponential functions:
a) y = 3x
b) f(x) = 6x
c) y = 2x
Example: Evaluate th

OBLIQUE TRIANGLES
DERIVATION OF LAW OF SINES
Let ABC be an oblique triangle with sides a, b,
and c opposite their respective angles as shown
in the figure below. If an altitude h is drawn to the
base, we can write the following relationship:
C
b
A
h
c
h
s

TRIGONOMETRIC
EQUATIONS
DEFINITION: TRIGONOMETRIC EQUATIONS
Equations that involve the trigonometric
functions are called trigonometric equations.
Solving a trigonometric equation means the
same thing as solving an algebraic equation, like
finding the val

INVERSE
TRIGONOMETRIC
FUNCTIONS
DEFINITION: If f is a one-to-one function with
domain A and range B, then its inverse f 1 is
the function with domain B and range A defined
by
f 1 x y f y x
For a function to have an inverse, it must be
one-to-one. Since th

SPHERICAL
TRIGONOMETRY
DEFINITION OF TERMS:
Spherical Trigonometry is a branch of trigonometry
that concerns with triangles extracted from the surface
of the sphere.
Great Circle is a circle obtained by passing a section
through the center of the sphere

SOLUTIONS OF
RIGHT TRIANGLE
SOLUTION OF RIGHT TRIANGLE
Solving a triangle means determining the measures of
all sides and angles of the triangles. In solving problems
involving right triangle, the following steps may be
considered:
Sketch the required tr

TRIGONOMETRIC
IDENTITIES
TRIGONOMETRIC IDENTITIES
DEFINITION: A trigonometric identity is an
equation involving the trigonometric functions
that holds for all values of the variable.
BASIC TRIGONOMETRIC IDENTITIES
Reciprocal Identities
Pythagorean Identit

GRAPHS OF
TRIGONOMETRIC
FUNCTIONS
GRAPH OF SINE FUNCTION:
(y = sin x)
Properties of Sine Function Graph:
1. The graph crosses the x-axis at the initial point,
endpoint and midpoint of a cycle. These points
are called the zeros of the function.
2. A maximu

TRIGONOMETRIC
FUNCTIONS OF
ANGLES
QUADRANTS
The coordinate axes divide the plane into four parts
called quadrants. For any given angle in standard
position, the measurement boundaries for each
quadrant are summarized as follows:
y
Quadrant I
Quadrant II 0

TRIGONOMETRY
Derived from two Greek words:
trigonon which means triangle and
metron which means to measure.
DEFINITION:
A branch of mathematics which deals
with the measurement of triangles:
three sides and three angles.
CLASSIFICATION OF TRIANGLES
DEFINI

TRIGONOMETRY OF
RIGHT TRIANGLES
TRIGONOMETRIC RATIOS
Consider a right triangle with as one of its acute angles. The
trigonometric ratios are defined as follows .
oppos ite
sin =
hypotenus e
opposite
adjacent
cos =
hypotenus e
hypotenus e
sec =
adjacent
op