This presentation is about Sine and cosine curves and its applications. This also shows where we see curves in nature, architecture, science, etc.

1. Dr. Farhana Shaheen

2.  Where do we see
 CURVES…
 In our daily life?
 In Nature?
 In Engineering?
 In Science?
 In Mathematics?
 https://www.facebook.com/photo.php?v=10200
818061237502&set=vb.307551552600363&type
=2&theater
 (Sine wave water-Amazing)

22.  The trigonometric graphs are periodic, which
means the shape repeats itself exactly after a
certain amount of time.

23.  Let's investigate the shape of the curve
y = sin x
 The scale for this is radians. Remember that
π radians is 180°, so in the graph, the value
of 3.14 on the x-axis represents 180° and
6.28 is equivalent to 360°.
 Note that Angle is positive in
the anti-clockwise direction.

27.  The a in the expression
y = a sin bx
represents the amplitude of the graph. It is
an indication of how much energy the wave
contains.
 Amplitude is always a positive quantity. We
could write this using absolute value signs.
For the curves y = a sin bx,
Amplitude = |a|.

28.  The period of a function is the smallest
positive value p such that
f(x + p) = f(x).
That is, the value when the graph of the
function f(x) completes one cycle.
 The period of
y = a sinbx is
P=2π/|b|.

33.  The measure of an angle is based upon a unit
circle (a circle of radius 1 and center at the
origin).
The coordinates of any point on the unit circle
are (cos ø, sin ø). The relationship between
right triangle trigonometry and the circular
function concept is illustrated below.
 x = cos ø
 y = sin ø

34.  x=cos t
 y=sin t

35.  Anything that has a regular cycle (like the
tides, temperatures, rotation of the earth, etc)
can be modeled using a sine or cosine curve.

38.  Temperature and sunlight (solar radiation)
play an important role in the chemical
reactions that occur in the atmosphere to
form photochemical smog from other
pollutants.

39.  Also, a rough sinusoidal pattern can be seen
in plotting average daily or annual
temperatures of the year.

40.  The graphs that we are discussing are
probably the most commonly used in all
areas of science and engineering. They are
used for modeling many different natural and
mechanical phenomena (populations, waves,
engines, acoustics, electronics, UV intensity,
growth of plants and animals, etc).

41.  This graph displays the percentage of jobs
with your search terms anywhere in the job
listing.

42.  This wave pattern occurs often in nature,
including ocean waves, sound waves, and
light waves.

46.  Light radiates from a source in waves. Each
wave has two parts; an electric part, and a
magnetic part. That's why light is called
Electromagnetic Radiation.

47.  We model cyclical behavior using the sine and
cosine functions. An easy way to describe
these functions is as follows.
 Imagine a bicycle’s wheel whose radius is
one unit, with a marker attached to the rim of
the rear wheel, as shown in the following
figure.

51. THANK YOU

52.  The typical voltage V supplied by an electrical
outlet in the U.S. is a sinusoidal function that
oscillates between 165 volts and +165 volts
with a frequency of 60 cycles per second.
 The graph for the voltage as a function of
time t is:

54.  Any object moving with constant angular
velocity or moving up and down with a
regular motion can be described in terms of
SIMPLE HARMONIC MOTION.
 The displacement, d, of an object moving
with SHM, is given by:
 d = R sin ωt or y = A sin ωt
 where R is the radius of the rotating object
and ω is the angular velocity of the object.

56.  Simple harmonic motion is typified by the
motion of a mass on a spring when it is
subject to the linear elastic restoring force
given by Hooke's Law. The motion is
sinusoidal in time and demonstrates a single
resonant frequency.