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
EXAMPLES
7
2
A
LESSON 5
INVERSE TRIGONOMETRIC
FUNCTIONS
INVERSE TRIGONOMETRIC FUNCTION
Definition: If f is a one-to-one function with domain A and
1
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
LESSON 6
SOLUTIONS OF OBLIQUE
TRIANGLES
OBLIQUE TRIANGLES
Solving a triangle involves finding the lengths of all sides and the
measures of all angles in the triangle. In this lesson, we will
discuss the solutions of oblique triangles, which are triangles
LESSON 7
TRIGONOMETRIC IDENTITIES
TRIGONOMETRIC IDENTITIES
A trigonometric identity is an equation involving
trigonometric functions that hold for all values of the
argument, typically chosen to be . In other words, an
identity is an equation that is true
LESSON 8
TRIGONOMETRIC EQUATIONS
TRIGONOMETRIC EQUATIONS
DEFINITION:
Equations that involve the trigonometric functions are called
trigonometric equations.
Solving a trigonometric equation means the same thing as solving
an algebraic equation, like findin
LESSON 9
SPHERICAL TRIGONOMETRY
TERMS and DEFINITIONS
Spherical Trigonometry is a branch of trigonometry that
concerns with triangles extracted from the surface of the sphere.
Sphere is a three the dimensional surface, all points of which
are equidistan
LESSON 11
EXPONENTIAL and LOGARITHMIC
EQUATIONS
Exponential Equations
If a variable appears in the exponent of a term of an
equation, the equation is called exponential equation.
Equality of Exponents Theorem
If bx = by , then x = y, provided b > 0 and b
Engr. Ericson D. Dimaunahan
Mapua Institute of Technology
Chapter 1
Points, Lines, Planes, and
Angles
Exercise
Definition of Terms
Point
a zero-dimensional
mathematical object that has
position only and has no length, no
width nor thickness
P (2,
1)
1
1
POLYHEDRA
Solid
A solid is any limited portion of space bounded by
surfaces or plane figures.
Parts of a Solid
A right section is a section of the solid that is
perpendicular to one of its lateral edges.
Edges are the intersections of the bounding
planes.
PRISM AND CYLINDERS
PRISM
A prism is defined as a polyhedron with
two congruent bases that lie in parallel
planes, and whose every section that is
parallel to a base has the same area as
that of the base.
By definition, the cubes (square prisms)
and rec
TRIANGLES
Morris Martin M. Jaballas
Triangles
Just like any polygon, triangle is one of the
most popular geometric figures in
Mathematics. It is the simplest three-sided
polygon with various topics and practical
applications in the field of mathematics an
LESSON 4
GRAPHS OF TRIGONOMETRIC
FUNCTIONS
GRAPHS OF THE TRIGONOMETRIC FUNCTIONS
The trigonometric functions can be graphed on a
rectangular coordinate system by plotting the points
whose coordinates belong to the function. But the
easier way to graph tri
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
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 angles)
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
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
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
INVERSE
TRIGONOMETRIC
FUNCTIONS
DEFINITION: If f is a one-to-one function with
1
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-one. Since t
Lesson 5: 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 vertical
LESSON 2
TRIGONOMETRY OF RIGHT
TRIANGLES
THE SIX TRIGONOMETRIC FUNCTIONS
hypotenuse
Opposite
side
Adjacent
side
hypotenuse
Adjacent
side
Opposite
side
Let be an acute angle of a right triangle. The values of the six
trigonometric functions of are
l engtho
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
PYRAMIDS AND CONES
PYRAMIDS
A pyramid is a polyhedron containing
triangular lateral faces with a common
vertex and a base which is a polygon.
A pyramid is a right pyramid if the line
joining the vertex and the center of base is
perpendicular to the plane