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)
The trigonometric graphs are periodic, which means the shape repeats itself exactly after a certain amount of time.
Lets 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 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|.
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|.
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
Anything that has a regular cycle (like the tides, temperatures, rotation of the earth, etc) can be modeled using a sine or cosine curve.
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.
Also, a rough sinusoidal pattern can be seen in plotting average daily or annual temperatures of the year.
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).
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This wave pattern occurs often in nature, including ocean waves, sound waves, and light waves.
Light radiates from a source in waves. Each wave has two parts; an electric part, and a magnetic part. Thats why light is called Electromagnetic Radiation.
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.
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:
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 is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hookes Law. The motion is sinusoidal in time and demonstrates a single resonant frequency.