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

Organic Compounds:
Alkanes and Their
Stereochemistry
Based on McMurrys Organic Chemistry, 7th edition
Objectives of discussion
Alkanes are unreactive, but provide useful vehicle to
introduce important ideas about organic compounds
Alkanes will be used t

Chapter 5: Structure and Preparation of Alkenes:
Elimination Reactions
Alkenes (olefins) are hydrocarbons that contain a carbon-carbon
double bond and are said to be "unsaturated."
molecular formula CnH2n
5.1: Alkene Nomenclature (please read and understa

Stereochemistry
Compounds which have the same molecular formula and
the same sequence of covalently bonded atoms, but
different spatial orientations of those atoms.
Conformational stereoisomers those that interconvert
easily at room temperatures through

Chapter 3
EXPERIMENTAL
ERRORS
1
Other Common Analytical Terms
1. Precision refers to reproducibility of repetitive measurements
of equivalent analyte solutions (e.g. aliquots)
Often expressed as (absolute) standard deviation, s,
or relative standard devi

Titrimetric Methods;
Precipitation Titrimetry
Lecture 5
Titration
Based on a reaction between the analyte
and a standard reagent known as the
titrant.
The end point is signaled by an indicator
color change or a change in an
instrumental response.
Widel

Organic Chemistry, 6h Edition
L. G. Wade, Jr.
Chapter 6
Alkyl Halides: Nucleophilic
Substitution and Elimination
Classes of Halides
Alkyl: Halogen, X, is directly bonded to sp3
carbon.
Vinyl: X is bonded to sp2 carbon of alkene.
Aryl: X is bonded to sp

Chapter 5
Aromatic Compounds
Aromatic Compounds
Aromatic compound: A term used to classify
benzene and its derivatives containing 6membered rings with 3 double bonds
Arene: An aromatic hydrocarbon
Aryl group: A group derived from an aromatic
compound b

Chap. 5 part II
Alkenes and Alkynes II:
Addition Reactions
Solomons, Chapter 8
Chapter 5: Structure and Preparation of Alkenes:
Elimination Reactions
Alkenes (olefins) are hydrocarbons that contain a carbon-carbon
double bond and are said to be "unsaturat

Precipitimetry
CHM 112
MAPUA Institute of Technology
11/21/2015
dcapodaca_2Q2012-13
1
Precipitation methods
Are those in which a desired constituent is
precipitated from solution by means of a
standard solution of precipitating agent
May be direct or in

Redox Methods
Oxidimetry and Reductimetry
Titration of an oxidizing agent with a standard
solution of a reducing agent or the titration of a
reducing agent with a standard solution of an
oxidizing agent
Oxidation the increase in the positive direction
o

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 quadra

TRIGONOMETRIC
IDENTITIES
TRIGONOMETRIC
IDENTITIES
DEFINITION: A trigonometric
identity is an equation involving
the trigonometric functions that
BASIC TRIGONOMETRIC IDENTITIES
holds for all values of the
Reciprocal Identities Pythagorean
1
1
Isin + cos =

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

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

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.

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

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 trigonometr

1Trigonometric Identities
Answer the following problems as indicated:
1.) True or False.
a)
b)
c)
d)
A: False
A: False
A: False
A: True
2.) Prove the identity:
A:
3.) Calculate sin 3x, depending on sin x.
A:
S:
References:
1:
http:/www.math.siu.edu/previe

An Introduction to Organic Reactions:
Acids and Bases
Acid-Base Reactions
The Brnsted-Lowry Definitions
In 1923 the Danish chemist Johannes Brnsted (1879-1947) and the
English chemist Thomas Lowry (1874-1936) independently proposed
that acids and bases be