Lesson 11: EXPONENTIAL
AND LOGARITHMIC
FUNCTIONS
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Define Exponential and Logarithmic Functions.
Graph Exponential and Logarithmic Functions.
Apply th
Lesson 4: GRAPHS OF
TRIGONOMETRIC FUNCTIONS
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Develop the properties of trigonometric functions.
Find the amplitude, period, phase shift and vertica
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 t
PLANE AND SPHERICAL
TRIGONOMETRY
TRIGONOMETRY
Derived from the Greek words trigonon which means triangle
and metron which means to measure.
Branch of mathematics which deals with measurement of
triangles (i.e., their sides and angles), or more specificall
LESSON 2
TRIGONOMETRY OF RIGHT
TRIANGLES
THE SIX TRIGONOMETRIC FUNCTIONS
hypotenuse
Adjacent
side
Opposite
side
hypotenuse
Adjacent
side
Opposite
side
Let be an acute angle of a right triangle. The values of the six
trigonometric functions of are
length o
Lesson 2: TRIGONOMETRY OF
RIGHT TRIANGLES
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Define the six trigonometric functions as ratios of the sides of
a right triangle
Evaluate the trigonome
Lesson 6: TRIGONOMETRIC
IDENTITIES
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Review basic identities.
Simplify a trigonometric expression using identities.
Verify a trigonometric identity
Lesson 3: SOLUTIONS OF RIGHT
TRIANGLES
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Solve right triangles
Solve real-world problems using trigonometry
SOLUTION OF RIGHT TRIANGLE
To solve a ri
Lesson 4: TRIGONOMETRIC
FUNCTIONS OF ANGLES
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Define trigonometric functions in the Cartesian plane
Evaluate the trigonometric functions of the quad
Lesson 4: GRAPHS OF
TRIGONOMETRIC FUNCTIONS
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Develop the properties of trigonometric functions.
Find the amplitude, period, phase shift and vertica
TRIGONOMETRY
Math 12
Plane and Spherical Trigonometry
TRIGONOMETRY
Derived from the Greek words trigonon which means
triangle and metron which means to measure.
Branch of mathematics which deals with measurement of
triangles (i.e., their sides and angle
LESSON 3
TRIGONOMETRIC
FUNCTIONS OF ANGLES
COORDINATE PLANE
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
y
follows:
Quadrant
Lesson 6:
TRIGONOMETRIC
IDENTITIES
Math 12
Plane and Spherical Trigonometry
OBJECTIVES
At the end of the lesson the students are expected to:
Review basic identities.
Simplify a trigonometric expression using identities.
Verify a trigonometric identity.
A
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
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
is a one-to-one function
f
1
with domain A and range B, then its inverse
f
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-o
PLANE AND SPHERICAL
TRIGONOMETRY
TRIGONOMETRY
Derived from the Greek words trigonon which means triangle
and metron which means to measure.
Branch of mathematics which deals with measurement of
triangles (i.e., their sides and angles), or more specificall