Modeling real-world periodic phenomena with sinusoidal functions

Modeling Real-World Data with Sinusoidal Functions

The sinusoid which is sometimes referred to as the sine wave referrers to a function of mathematics describing a smooth oscillation that is also repetitive. It usually takes place in pure mathematics and also in physics, electrical engineering and signal processing besides numerous other fields. Its form as a function of time (t) is:

A, which is the amplitude. It is the peak deviation of the function from its center position.

, which is the angular frequency, specifies the number of oscillations occurring in a unit time interval, in radians per second, which is the phase, gives specifications where in its cycle the oscillation begins at t = 0.

The swinging of an undamped spring-mass organization round the equilibrium is referred to as a sine wave. The sine wave is of great use in physics as it regains its wave shape when it is added to a different sine wave having similar frequency and also arbitrary phase. Sine wave is the only periodic waveform having this feature. This feature will result into its significance in the Fourier analysis and also makes it to be acoustically strange.

Generally, sinusoids are wave form graphs. Therefore, any phenomenon that has a periodic behavior or characteristics of a wave is capable of being modeled by sinusoids. This is including numerous simple actions like blood pressure in the heart, a pendulum, motion of an engine's piston-crankshaft, a child's swing, a Ferris wheel, tides, hours of daylight through out a year, (visible) shape of the moon, seasons, and sounds.

Examples of sinusoids are biorhythms of a person. The proponents of biorhythms are claiming that our every day lives are greatly influenced by the rhythmic cycles. These cycles are capable of interacting to make an indication of vigorous and inactive phases not only in the physical but also in emotional and mental aspects of humans. If one conducts a web search to establish biorhythm software, it will give out a series of waves. The people who use biorhythms do not state that biorhythms predict nor explain events. They however state that biorhythms recommend how we can cope up with them.

Another familiar instance of an action that is capable of being modeled by a sinusoid is the pendulum's motion. When we plot time against the angle that the arm of the pendulum makes with a vertical line indicating the location of the pendulum at rest will generate a sinusoid.

Cyclical behavior is also widespread in the business world. As there are periodic changes in the temperature of places, there are also seasonal changes in the demand for surfing equipment, snow shovels among several other things. The graph below gives a cyclical feature in employment at securities firms in the .U.S.

Law of sines

The law of sines which is also known as the sine formula or sine rule is applied in trigonometry. The equation tries to show the relationship between the lengths of the triangle's sides and the sines of the angles of the triangle. Given an example like this;

The law states that:

a, b, and c represents the lengths of the triangle's sides while a, B, and C. represents the opposite angles .on a number of situations, the reciprocal of this equation is used to state this equation. Thus;

The sine law is sometimes used in the computation of the remaining sides of the triangle given two angles and a side of the triangle. This technique is known as triangulation. Besides, it can be applied if two sides of the triangle and also one of angles that are not enclosed are given. In that given case, the formula can give two probable values for the angle that is enclosed. This always leads to an ambiguous case.

Law of cosines

The law of cosines is also referred to as the cosine formula or sometimes as cosine rule. It is a statement about a universal triangle that tries to relate the lengths of the sides of the triangle to the cosine of one of its angles. Using notation as in Using figure, the law of cosines states that represents the angle that is contained between sides of lengths a and b and opposite the side of length c.

The cosine rule generalizes Pythagorean Theorem. This theorem applies only to right triangles: when the angle ? is a right angle i.e. It measures 900 or ?/2 radians, cos (?) = 0, and therefore the law of cosines will reduce to;

The cosine rule is useful in the computation of the third side of a triangle if two sides and also their enclosed angle are given. Besides, it is used in the computation of the angles of a triangle when all the three sides of the triangle are given.

The following formulas also state the cosine rule.

Fibonacci numbers

Introduction

In mathematics, the Fibonacci numbers refers to the numbers that are in these given integer sequence;

Through definition, the beginning two Fibonacci numbers are 0 and 1. Besides, each succeeding number is the addition or the sum of the preceding two numbers. Other sources do not include the original 0, as an alternative, they begin the progression with two 1s.

Mathematically, the sequence Fn of Fibonacci numbers is always defined by the recurrence relation

Using seed values

The Fibonacci sequence has been named after Leonardo of Pisa. He was referred to as Fibonacci. Fibonacci 1202 volume which was called Liber Abaci made the introduction of the sequence to Western European mathematics (Laurence, 2002) despite that fact that the sequence was autonomously described in Indian mathematics (Goonatilake, 1998; Parmanand, 1985; Rachel, 2008 and Knuth, 2006).

Fibonacci numbers are widely applied in the examination of financial markets, in strategies like Fibonacci retracement. They are also widely applied in computer algorithms like the Fibonacci search method and also the Fibonacci pile data structure. The easy recursion of Fibonacci numbers has also stimulated a group of recursive graphs which are referred to as Fibonacci cubes and are used for interconnecting corresponding and also distributed systems. Besides, they come out in biological settings (Douady and Couder,1996) like the branching of trees, leaves arrangement on a stem, the spouts of fruits in a pineapple (Judy and Wilson,2006) flowering of artichoke, and the uncurling fern and also the arrangement of the pine cone (Brousseau,1969).

Fibonacci number patterns are always encountered. They occur so regularly in nature that we always get that the phenomenon is always called the "law of nature."

The petals of a pine cone always spiral in two directions. The number of petals going around once is usually a Fibonacci number. Seeds on a sunflower seeds are also showing the Fibonacci spiral. The patter is also found in pineapples.

Fibonacci sequence is shown by petals on many flowers

Number of Petals

Flower

3 petals lily and iris

5 petals buttercup, columbine vinca, larkspur and wild rose

8 petals

Delphinium and coreopsis

13 petals ragwort, cineraria and marigold

21 petals aster, chicory and black-eyed Susan

34 petals plantain, pyrethrum and daisy

55 petals

Daisy and the family of asteraceae

89 petals

Daisy and the family of asteraceae

Why these arrangements occur

Plants are always not aware of this sequence. They only grow in the most efficient ways. In the scenario of the leaf arrangement and phyllotaxis, a number of the scenarios might be related to maximizing each leaf's space and also the standard amount of light that falls on each of them. A tiny advantage would also come to take control over numerous generations (Grist, n.d)

Fibonacci sequence and population growth in animals

Fibonacci sequence copies the population growth pattern of animals. Beginning from one distinct offspring or 1 animal. After a period of one year, it matures up and is capable of reproducing .In a single year; it is capable of reproducing one offspring. Now they become 2 animals. In one single year, the mother will reproduce one fresh offspring and the offspring that is given birth to in the preceding year matures up. They now become 3 animals. In another year, the mother and the now mature offspring will each reproduce one offspring and the offspring that comes from the last year will become mature. There will be 5 animals now. The trend will go on. (Fuzzy, 2010)

Rational Root (zero) Theorem

The Rational Zero Theorem provides a list of probable rational zeros of the polynomial function. The theorem provides all potential rational roots of the polynomial equation. Not all numbers in the list shall be a zero belonging to the function, however all rational zeros belonging to the polynomial function shall come out somewhere in the list.

The rational root theorem is also a test that is capable of being used to get the probable number of rational solutions or sometimes roots of the polynomial equation having coefficients which are integers.

Degree

The degree of a monomial equals to the sum of the exponents of individual variables that appears in the monomial. For example, the degree of x2yz3 is 2 + 1 + 3(Beckmann, 1976)

A polynomial is a monomial. It can also be said to be the algebraic sum or sometimes the difference of monomials. A polynomial degree is the greatest of the degrees of its terms after the combination of like terms. The leading coefficient is described as the coefficient of the term with the greatest. The polynomials which are having one, two or even three terms are referred to as monomials, binomials and trinomials in that order (Buchanan, 2010)

The degree of a monomial can simply be defined as the exponent or power that the monomial is raised to. If there exists three or more monomials that are being added or subtracted so as to make a polynomial, and each of them has a degree and the monomial having the highest degree are representing the whole degree of the polynomial.

The fundamental theory of algebra

It is one of the most essential results in mathematics.

The Fundamental Theorem of Algebra is practically basic spontaneously. It states that provided with any polynomial that is not constant in the field C, we can always get a root of the polynomial that is provided. In spite of the fact that it is scarcely applied directly in mathematical proofs, it is always an essential feature to the other theorems. For example, in showing evidence that all angles cannot be trisected by use of a straight-edge and compass only, the Fundamental Theorem of Algebra is applied if the roots of any polynomials are got. The theorem is what makes the other big theorems in math to work, like the Hilbert Nullstelensatz Theorem. Additionally, also the Axiom of Choice applies the Fundamental Theorem of Algebra if proven by Zorn's Lemma, which was introduced in the year1935, and which is similar to the Axiom of Choice (Fraleigh, 2003)

The Theorem was proved first by Carl Frieddrich Gauss (1777-1855).The Fundamental Theorem of Algebra tells us that when we have completely factored a polynomial: On one side, the polynomial has completely been factored if only all its factors are linear or are irreducible quadratic. On the contrary, every time a polynomial has been factored into linear or irreducible quadratics only, then it has completely been factored because both linear factors and irreducible quadratics are not capable being factored any more over real numbers. The theorem is however not constructive since it does not tell us the way to factor completely a polynomial.