Sine and Cosine Curves

Dr. Farhana Shaheen

Where do we see CURVES… In our daily life? In Nature? In Engineering? In Science? In Mathematics? https://www.facebook.com/photo.php?v=10

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(Sine wave water-Amazing)

CURVES

Curves in Nature

CURVES in Waves

Curves in Fruits

Curves in Architecture

St. Louis Arch, America

Space age landscape, Spain

Sydney Opera House

Kul Sharif Masjid in Russia

Great Masjid in China

Dome shaped building

Design with Curves

Curves in Nature

Curves in life

Curves in Mathematics

Why study trigonometric graphs?

Right TriangleTrigonometry

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

Trigonometric Graphs

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.

The Sine Curve y = sin x

Graph of Sine Function y = Sinx

Sine curve as WavesThe sine curve occurs naturally when we areexamining waves.

3D representation of Sine curve

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|.  

Amplitude of the graph y=a sinbx

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|.

Period of the graph y=a sinbx

The graph of y = a Sin bxwith amplitude |a| and period 2π/|b|

Sine curves with different amplitudes

y = sinx and y = sinx - 3

Y = a sinx with different values of amplitude a

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

Circular Functions

For unit circle, i.e. of radius 1 x=cos t y=sin t

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

Sine or Cosine Curves

The graph of y = tan x

Trigonometric curves

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.

Temperature Graphs

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

Global Temperature

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).

Graphs in Science and Engineering

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

Analytical Engineer Acoustics Job Trends

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

Wave Patterns

Formation of Waves

Sound Waves

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.

Light Spectrum

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.

Cyclical Behavior

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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:

Voltage Graph

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.

SIMPLE HARMONIC MOTION

SIMPLE HARMONIC MOTION

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.

Motion of a pendulum

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